Linear Sobolev Type Equations and Degenerate Semigroups of Operators [Reprint 2012 ed.] 9783110915501

Table of contents :
Chapter 1. Auxiliary material
1.1. Banach spaces and linear operators
1.2. Theorems on infinitesimal generators
1.3. Functional spaces and differential operators
Chapter 2. Relatively p-radial operators and degenerate strongly continuous semigroups of operators
2.0. Introduction
2.1. Relative resolvents
2.2. Relatively p-radial operators
2.3. Degenerate strongly continuous semigroups of operators
2.4. Approximations of Hille–Widder–Post type
2.5. Splitting of spaces
2.6. Infinitesimal generators and phase spaces
2.7. Generators of degenerate strongly continuous semigroups of operators
2.8. Degenerate strongly continuous groups of operators
Chapter 3. Relatively p-sectorial operators and degenerate analytic semigroups of operators
3.0. Introduction
3.1. Relatively p-sectorial operators
3.2. Degenerate analytic semigroups of operators
3.3. Phase spaces for the case of degenerate analytic semigroups
3.4. Space splitting
3.5. Generators of degenerate analytic semigroups of operators
3.6. Degenerate infinitely differentiable semigroups of operators
3.7. Phase spaces for the case of degenerate infinitely continuously differentiable semigroups
3.8. Kernels and images of degenerate infinitely differentiable semigroups of operators
Chapter 4. Relatively σ-bounded operators and degenerate analytic groups of operators
4.0. Introduction
4.1. Relatively σ-bounded operators
4.2. Relative σ-boundedness and relative p-sectoriality
4.3. Relative σ-boundedness and relatively adjoint vectors
4.4. Degenerate analytical groups of operators
4.5. Sufficient conditions of the relative σ-bounded ness
4.6. The case of a Fredholm operator
4.7. Analytical semigroups of operators degenerating on the chains of relatively adjoint vectors of an arbitrary length
Chapter 5. Cauchy problem for inhomogeneous Sobolev-type equations
5.0. Introduction
5.1. Case of a relatively s-bounded operator
5.2. The case of a relatively p-sectorial operator
5.3. Case of a relatively p-radial operator
5.4. Strong solution of Cauchy problem
5.5. Cauchy problem for an equation with Banach-adjoint operators
5.6. Propagators
5.7. Inhomogeneous Cauchy problem for high-order Sobolev-type equations
Chapter 6. Bounded solutions of Sobolev-type equations
6.0. Introduction
6.1. Relatively spectral theorem
6.2. Bounded relaxed solutions of a homogeneous equation
6.3. Bounded solutions of the inhomogeneous equation
6.4. Examples
Chapter 7. Optimal control
7.0. Introduction
7.1. Strong solution of Cauchy problem for an equation with Hilbert-adjoint operators
7.2. Problem of optimal control for an equation with relatively s-bounded operator
7.3. Problem of optimal control for equation with a relatively p-sectorial operator
7.4. Barenblatt–Zheltov–Kochina equation
7.5. System of ordinary differential equations
7.6. Equation of the evolution of the free filtered-fluid surface
Bibliography
Index

Citation preview

INVERSE A N D ILL-POSED PROBLEMS SERIES

Linear Sobolev Type Equations and Degenerate Semigroups of Operators

Also available in the Inverse and Ill-Posed Problems Series: Ill-Posed and Non-Classical Problems of Mathematical Physics and Analysis Editors: M.M. Lavrent'ev and S.I. Kabanikhin Forward and Inverse Problems for Hyperbolic, Elliptic and Mixed Type Equations AG. Megrabov Nonclassical Linear Volterra Equations of the First Kind AS. Apartsyn Poorly Visible Media in X-ray Tomography D.S.Anikonov.V.G. Nazarov, and I.V. Prokhorov Dynamical Inverse Problems of Distributed Systems V.l. Maksimov Theory of Linear Ill-Posed Problems and its Applications V.K. Ivanov.V.V.Vasin andV.P.Tanana Ill-Posed Internal Boundary Value Problems for the Biharmonic Equation MAAtakhodzhaev Investigation Methods for Inverse Problems V.G. Romanov Operator Theory. Nonclassical Problems S.G. Pyatkov Inverse Problems for Partial Differential Equations Yu.Ya. Belov Method of Spectral Mappings in the Inverse Problem Theory V.Yurko Theory of Linear Optimization I.I. Eremin Integral Geometry and Inverse Problems for Kinetic Equations A.Kh.Amirov Computer Modelling in Tomography and Ill-Posed Problems MM. Lavrent'ev, SM.Zerkal and O.ETrofimov An Introduction to Identification Problems via Functional Analysis A. Lorenzi Coefficient Inverse Problems for Parabolic Type Equations and Their Application P.G. Danilaev Inverse Problems for Kinetic and Other Evolution Equations Yu.E Anikonov Inverse Problems ofWave Processes A.S. Blagoveshchenskii Uniqueness Problems for Degenerating Equations and Nonclassical Problems S.P. Shishatskii, A. Asanov and ER. Atamanov

Uniqueness Questions in Reconstruction of Multidimensional Tomography-Type Projection Data V.P. Golubyatnikov Monte Carlo Method for Solving Inverse Problems of Radiation Transfer V.S.Antyufeev Introduction to the Theory of Inverse Problems A.L Bukhgeim Identification Problems ofWave Phenomena Theory and Numerics S.I. Kabanikhin and A. Lorenzi Inverse Problems of Electromagnetic Geophysical Fields P.S. Martyshko Composite Type Equations and Inverse Problems A.I. Kozhanov Inverse Problems ofVibrational Spectroscopy A.G.Yagolo, I.V. Kochikov, GM. Kuramshina andYuA. Pentín Elements of the Theory of Inverse Problems AM. Denisov Volterra Equations and Inverse Problems A.L Bughgeim Small Parameter Method in Multidimensional Inverse Problems A.S. Barashkov Regularization, Uniqueness and Existence of Volterra Equations of the First Kind A. Asanov Methods for Solution of Nonlinear Operator Equations V.P.Tanana Inverse and Ill-Posed Sources Problems Yu.EAnikonov, B.A. Bubnov and G.N. Erokhin Methods for Solving Operator Equations V.P.Tanana Nonclassical and Inverse Problems for Pseudoparabolic Equations A.Asanov and ERAtamanov Formulas in Inverse and Ill-Posed Problems Yu.EAnikonov Inverse Logarithmic Potential Problem V.G. Cherednichenko Multidimensional Inverse and Ill-Posed Problems for Differential Equations Yu.EAnikonov Ill-Posed Problems with A Priori Information V.V.Vasin andALAgeev Integral Geometry ofTensor Fields VA. Sharafutdinov Inverse Problems for Maxwell's Equations V.G. Romanov and S.I. Kabanikhin

INVERSE AND ILL-POSED PROBLEMS SERIES

Linear SobolevType Equations and Degenerate Semigroups of Operators

G.A. Sviridyuk and V.E. Fedorov

///!/SP/// UTRECHT · BOSTON

2003

VSP

Tel: + 3 1 3 0 6 9 2 5 7 9 0

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Fax: +31 3 0 693 2 0 8 1

3 7 0 0 A H Zeist

[email protected]

The Netherlands

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2003

First p u b l i s h e d in 2 0 0 3

ISBN 90-6764-383-1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

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Contents

Chapter 1. Auxiliary material 1.1. Banach spaces and linear operators 1.2. Theorems on infinitesimal generators 1.3. Functional spaces and differential operators Chapter 2. Relatively p-radial operators and degenerate strongly continuous semigroups of operators 2.0. Introduction 2.1. Relative resolvents 2.2. Relatively p-radial operators 2.3. Degenerate strongly continuous semigroups of operators 2.4. Approximations of Hille—Widder—Post type 2.5. Splitting of spaces 2.6. Infinitesimal generators and phase spaces 2.7. Generators of degenerate strongly continuous semigroups of operators 2.8. Degenerate strongly continuous groups of operators Chapter 3. Relatively p-sectorial operators and degenerate analytic semigroups of operators 3.0. Introduction 3.1. Relatively p-sectorial operators 3.2. Degenerate analytic semigroups of operators 3.3. Phase spaces for the case of degenerate analytic semigroups . . . 3.4. Space splitting

1 1 6 9

13 13 15 20 25 31 36 42 46 49

55 55 57 59 65 68

vi

G. A. Sviridyuk and V. E. Fedorov

3.5. Generators of degenerate analytic semigroups of operators . . . . 3.6. Degenerate infinitely differentiable semigroups of operators . . . . 3.7. Phase spaces for the case of degenerate infinitely continuously differentiable semigroups 3.8. Kernels and images of degenerate infinitely differentiable semigroups of operators Chapter 4. Relatively σ-bounded operators and degenerate analytic groups of operators 4.0. Introduction 4.1. Relatively σ-bounded operators 4.2. Relative σ-boundedness and relative p-sectoriality 4.3. Relative σ-boundedness and relatively adjoint vectors 4.4. Degenerate analytical groups of operators 4.5. Sufficient conditions of the relative σ-bounded ness 4.6. The case of a Fredholm operator 4.7. Analytical semigroups of operators degenerating on the chains of relatively adjoint vectors of an arbitrary length Chapter 5. Cauchy problem for inhomogeneous Sobolev-type equations 5.0. Introduction 5.1. Case of a relatively σ-bounded operator 5.2. The case of a relatively p-sectorial operator 5.3. Case of a relatively p-radial operator 5.4. Strong solution of Cauchy problem 5.5. Cauchy problem for an equation with Banach-adjoint operators 5.6. Propagators 5.7. Inhomogeneous Cauchy problem for high-order Sobolev-type equations Chapter 6. Bounded solutions of Sobolev-type equations 6.0. Introduction 6.1. Relatively spectral theorem 6.2. Bounded relaxed solutions of a homogeneous equation 6.3. Bounded solutions of the inhomogeneous equation 6.4. Examples

74 77 81 84

87 87 89 92 95 97 101 108 Ill

119 119 121 131 135 139 . 143 150 154 159 159 160 163 170 177

Contents

vii

Chapter 7. Optimal control

183

7.0. Introduction 7.1. Strong solution of Cauchy problem for an equation with Hilbert-adjoint operators 7.2. Problem of optimal control for an equation with relatively σ-bounded operator 7.3. Problem of optimal control for equation with a relatively p-sectorial operator 7.4. Barenblatt—Zheltov—Kochina equation 7.5. System of ordinary differential equations

183

192 194 195

7.6. Equation of the evolution of the free filtered-fluid surface

197

Bibliography

201

Index

215

183 188

Chapter 1. Auxiliary material

1.1.

BANACH SPACES AND LINEAR OPERATORS

A set X is called a linear or vector space over the field of real (complex) numbers if 1) the addition operation is defined: to any elements x,y G X there corresponds a definite element χ + y G X called their sum; 2) x + y = y + x;

3) χ + (y + z) = (x+ y) + z; 4) there exists a zero element OG A1 such that χ + 0 — x; 5) for any χ E X there exists — χ G X such that χ + (—χ) — 0; 6) the operation of multiplication by a number is defined: to any χ G X and any number λ G R (C) there corresponds a definite element Xx Ε X; 7) Χ(μχ) = (Χμ)χ; 8) 1 · χ = χ\ 9) Χ(χ + y) = Xx + Xy, 10) (λ + μ)χ = Χχ + μχ. The elements of a linear space will be called vectors or points.

G. A. Sviridyuk and V. E. Fedorov

2

Remark 1.1.1. It is easy to show that the zero and inverse elements are uniquely defined. A linear space X is called normed if to any χ G X a nonnegative number IMI* (the subscript will sometimes be omitted) called the norm of χ is assigned so that the following axioms are satisfied: !) IMI > 0;

2) ||:r|| = 0 if and only if χ = 0; 3) ||λχ|| = |λ| · IMI for any λ G R (C); 4) ||a; + y|| < IMI + IMI for any x,y E X. To any real linear space X there corresponds a complex linear space X consisting of all possible formal sums ζ = χ + iy, where x, y G X and i is the imaginary unit. Clearly, X C X. Such an inclusion of X in the space X is called the complexification of the Banach space X. A sequence of vectors C X is called convergent to a vector χ Ε X, which is denoted as χ = l i m ^ o o II —> 0 as η —> oo. The set ST(xo) = {x G X | ||a; — a;o|| < r} is called open ball with radius r > 0 centered at the point xq G X. A set A C X is called bounded if 3K G R+

Va; G A IMI
X, which bijectively maps dom A~l = im^4 onto dom A. The operator A~l is linear. An operator A is called continuously invertible if there exists an operator A~l 6 X). Theorem 1.1.5. The operator A~l exists and is bounded on im A if and only if there exists m G R+ such that for all χ G dom A ||Aa;||y > m||:r||;r· Theorem 1.1.6. Let X and y be Banach spaces, the operator A G C{X] y), and im^l = y and A is invertible. Then the operator A is continuously invertible. Theorem 1.1.7. Let X be a Banach space, A G C{X), and ||A|| < 1. Then the operator I — A is continuously invertible and

» ' ' - ^ ' ^ i H i W A linear operator A : dom A —>• is called closed if it follows from {xn} C dom A, limn^oo Xfi — and linin—Kx, Axn = y that χ G dom A and Ax = y. The set of closed operators A : dom A —• y with domains of definition dense in the space X will be denoted by Cl(X;y). The set of operators Cl{X\ X) will be denoted by Cl(X). Theorem 1.1.8. An operator A belongs to the space £(X;y) only if it is closed and defìned on the whole space.

if and

Chapter 1. Auxiliary material

5

Theorem 1.1.9. If an operator A is closed and invertible, then the operator A~1 is closed. Let us introduce a graph norm || · || s = || • ||Λ· + definition dom A of a linear closed operator A.

•

on the domain of

Theorem 1.1.10. If an operator A : dom A X, dom A C X, is linear and closed, then the normed space dom A is a Banach space with respect to the graph norm, and the operator A G £(dom A). Let X be a complex Banach space and let an operator A : dom A — X , dom A C X, be linear. A complex number Λ is called a regular point of the operator A if the operator XI — A is continuously invertible (there exists the operator (XI — G C(X)). The set of all regular points of the operator A is called the resolvent set of the operator and is denoted by p(A). If λ G p(A), then the operator R\(A) = (XI — A ) - 1 is called the resolvent of the operator A. The spectrum of the operator A is the set σ(Α) = C \ p(A). Theorem 1.1.11. trum σ(Α) is closed.

The resolvent set p(A) is open, and the spec-

Theorem 1.1.12. The spectrum of a continuous operator A lies in the cirde{AGC||À|-as »&> called the spectral radius of the operator A and rc(A) < ||>l||£(;r)· Theorem 1.1.14. Let X be a Banach space, an operator A G C(X), and |λ| > τσ(Α). Then X G ρ(Α). An operator function -Α(λ) : C —> C(X) is called analytic at a point Ao if it is expanded in some neighbourhood of the point λο into a power series

G. A. Sviridyuk and V. E. Fedorov

6

Α(λ) = Y ^ = 0 A k ( \ — \o) k convergent in this neighbourhood. Note that the notions of analyticity in the sense of the uniform and strong convergence of the series are equivalent. Theorem 1.1.15. λ G p(A).

R\(A)

is an analytic function of λ at any point

Remark 1.1.2. Let an operator A G C>{X) and |Λ| > νσ(Α). Then, based on Theorem 1.1.7, the following expansion can be readily obtained:

k-0 Theorem 1.1.16. Let X be a Banach space and an operator A G C(X). Then σ(Α) φ 0. An operator A is called idempotent if A 2 = A. Projector is an idempotent operator A G C.(X). On a space X there exists a projector A if and only if X — Xo ® X1, where A\xo = Ο, A\xi = I. The space £(/t;M(C)) is called adjoint to X and is denoted by X'. Its elements are called functionals. If to every element χ G A" an element χ G Χ" is assigned by the rule x ( f ) = f(x) V/ G X', then it is clear that X C X". A space X such that X — X" is called reflexive. A sequence {x n } C X is called weakly convergent to a: E X and this fact is denoted as χ = ω-ΐχπίη-^οο xn if limn-^oo f{xn) — f(x) for any functional

f ex'. 1.2.

THEOREMS ON INFINITESIMAL GENERATORS

Let A" be a Banach space and let an operator A : dom A —> X, dom A C X, be linear and closed. Consider the Cauchy problem z(0)

=

XQ,

XQ

G dom A

for an operator-differential equation χ — Ax. The question of solvability of this problem on the semiaxis R+ = {0}UR+ (on the axis R) is equivalent to the problem of finding a semigroup (group) of operators generated by the operator A.

Chapter 1. Auxiliary material

7

Definition 1.2.1. A semigroup of linear continuous operators is a mapping X- : R+ ->· C(X) such that XsX1 = Xs+t for all s, t G R+. A semigroup will be identified with the set { X 1 | t € R+}· A semigroup { X t \ t G R+} is called nondegenerate if X o = I and strongly continuous if for any t G R+ lim4_».tXs = Xt, lims_>o+-^s = A nondegenerate strongly continuous semigroup is called strongly continuous (Co)-semigroup (or a (Co)-continuous semigroup). Definition 1.2.2. An infinitesimal generator of a nondegenerate semigroup of operators {X1 \ t G R+} is the operator t->o+

t

defined only on these vectors χ for which the above limit exists. In this case the operator A is said to generate the semigroup {Xl | t G R+}. Let us introduce the denotation Ra^ = {c G R | c > a}. Definition 1.2.3. An operator A G Cl(V) satisfying the conditions 3a G R V/i G Ra,+ 3K G R+

VMGla,+

Vn G Ν

μ G p{A),

| | ( ^ ( ^ ) ) η | | £ ( ν ) < Κ/{μ - α)η

will be called radial. Theorem 1.2.1 [Hille—Yosida]. An operator A is radial if and only if it generates a strongly continuous (Co)-semigroup. Definition 1.2.4. A group of linear continuous operators is a mapping X- : R C(X) such that XsX1 = Xs+t for all s,t G R A group will be identified with the set { X 4 | t G M}. A group {X1 I t G R} is called nondegenerate if Xo — I and strongly continuous if for any t e l lim^-^f Xs = Xt. A nondegenerate strongly continuous group is called a strongly continuous (Co)-group (or (Co)-continuous group). Definition 1.2.5. The infinitesimal generator of a nondegenerate group of operators { X 1 \ t G R} is the operator t-fO

t

G. A. Sviridyuk and V. E. Fedorov

8

defined on those vectors χ for which the above limit exists. In this case it is said that the operator A generates the semigroup { X 1 \ t G R}. Let Ka = {c £ R I \c\ > a}. Definition 1.2.6. If an operator A G Cl(V) satisfies the conditions 3a G R 3Κ G R+

Ν/μ G Ra

Vn G Ν

G Ra

μ G ρ(Α),

||(Λ μ (^)) η || £ ( ν ) < Κ/(|μ| - a) n ,

it will be called biradial. Theorem 1.2.2. An operator A generates a strongly continuous group if and only if it is biradial. A group { X 4 I t G R} of operators is called analytic if it can be analytically continued to the whole complex plane in the variable t with retaining its group property. Theorem 1.2.3. The semigroup generated by an operator A G £•{%) can be continued to an analytic group. Conversely, the generator of an analytic group is a bounded operator A G C{X). A semigroup { X 1 \ t G R} of operators is called analytic if it can be analytically continued to a certain sector containing R+ in the variable t with retaining its semigroup property. Definition 1.2.7. An operator A G Cl(V) is called sectorial if it satisfies the conditions 3a G R

30 G (π/2, π)

SafiiA) = {μ G C I I arg (μ - α)| < θ, μ φ a} C ρ(Α), 3Κ G R+

V ^ ^ t A )

\\ΙΙμ(Α)\\ε{ν)μ

_

M)

_lL{XL _

Hence by virtue of Lemma 2.1.1 and the continuity of the operator L there follows the assertion of the theorem. • Remark 2.1.4. Prom (2.1.4) and (2.1.5) it follows that right (left) L-resolvents of the operator M commute. Now write identities (2.1.2) in the following form: M^L-Mr^/xL^M)-/, (//L — M)-1M

(2.1.7) = μΒ%{Μ) - I .

Hence there obviously follows an identity M^L-M^L^LfaL-M^M.

(2.1.8)

Remark 2.1.5. Prom the identity (2.1.7) it immediately follows that for every μ G pL(M) the operators M(pL — M)~l and (μL — M)~XM are linear and can be extended to continuous ones on spaces Τ and U respectively.

G. A. Sviridyuk and V.E. Fedorov

18

L e m m a 2.1.2. Let points Χ, μ E pL{M). (i) i m R

L

X{M) =

i m R L ( M ) , imL

L

X {M)

Then

= imL L ß {M);

(ii) ker R{{M) = ker Ζ, k e r l ^ ( M ) = {Mu \ u E dom Μ Π ker Ζ,}. Proof. On the strength of the linearity of the operator (μΣ — M ) - 1 it is clear that kerL C kerR^(M). The inverse inclusion can also be readily obtained. If R]¡{M)u = 0, then Lu = (/iL - M)0 = 0. If L^(M)f = 0, then Lv = 0, where ν = A f ) - 1 / € dom M . Hence / = (μL — M)v = M(—v) = Mu. Let us prove the invertible inclusion. f = Mu = M{-v) -- (μL - M)v, Lj¡{M)f = L{-u) = 0, since u e kerL. The assertions of the lemma on the images of L-resolvents immediately follow from identities (2.1.4), (2.1.5). • Ordered set {φ\, ψ2, •..} C U will be called a chain of vectors of the eigenvector ψο E ker L adjoint with respect to operator M (or simply M-adjoint) vectors if Lipq+1 = M(pq,

ipq £ ker L,

q = 0,1,... .

A chain is finite if there exists such an M-adjoint vector φ ρ , that either φν 0 dom M, or Μφρ 0 imL. In particular, the eigenvector φα has no M-adjoint vectors if either φο 0 dom M, or Μ ψ o 0 imL. The power of the finite chain is called its length . When a chain is infinite we say that it has an infinite length. A linear span of all the eigen and M-adjoint vectors of the operator L is called M-root lineal of the operator L. If the M-root lineal is closed, it is called an M-root space of the operator L. Remark 2.1.6. There is an obvious analogy with the conventional terminology which is evident for the case of the existence of the operator M " 1 e C{T\U). L e m m a 2.1.3. (i) Let φρ be an M-adjoint vector of the eigenvector φο of an operator L, and let the point be μ E pL(M). Then ~Κμ{Μ)φρ where {φι,...,φρ} an operator L.

= ιρρ-ι + μψρ-2 + ... + μρ~1φο,

is the corresponding chain of the M-adjoint vectors of

Chapter 2. Relatively p-radial operators

19

(ii) Let φρ be an adjoint vector of the eigenvector ψο of an operator ϋμ{Μ), and let the point μ € pL{M). Then -Lipp = Μ(ψρ-1 + μφρ-2 + ... + μρ~1ψο), where {φι,..., ator R¿(M).

φρ} is the corresponding chain of adjoint vectors of an oper-

More often it would be more convenient to use the following, rather than the lemma itself: Corollary 2.1.1. In conditions of Lemma 2.1.3 (i) there holds a representation (~Κμ{Μ))ρφρ

— φο·

Lemma 2.1.3 and Corollary 2.1.1 are proved by mathematical induction using the obvious recursion (recurrence) relation —Lipq+i + ßLipq = (pL — M)· + ο ο .

(2.3.7)

Therefore, (ui -

= jT ¿

MUjr'u)

ds = j*

-

GM)uds.

28

G. A. Sviridyuk and V. E. Fedorov

Hence II U{u - UluWu < tma

||t/^U£(W)||G(A)n

-

0(μ)η\\u (2.3.8)

+oo μ μ-»+οο

Vu0 G U°

U°u° = 0

by the construction of the semigroup.

Chapter

2. R e l a t i v e l y p-radial

operators

29

Let us prove that the semigroup allows equation (2.3.1). {R%{M))P+2V, then

Let u l =

ft

(Ufi — I ) u = f υ*μΟ{μ)ν> d s . JoΌ l

Tending μ

+oo to zero by virtue of (2.3.7) we obtain that

{Ul - I ) u x = Γ U s ( R j } ( M ) ) p + l ( ß L - M ) ~ l M v d s , Jo if a convergence U ^ G ( ß ) u l under μ —>• +oo is used uniform in with respect to s G [0, t ] , which can be given as follows:

IIυ>μα{μ)*} - U s { R L ß { M ) Y + l { ß L - M ) ~ l M v \ \ u < IIυ°μΟ(μ)ηι - U l { R L ß { M ) Y + l { ß L - M ) ~ l M v \ \ u + IIΐΐμ(Rß(M))

p+1

ι

s

(ßL - Μ)~ Μυ - U {Rjj{M))

(2.3.9) p+1

{ßL - M)~lMv\\. (2.3.10)

It follows from (2.3.4) that the term (2.3.9) does not exceed

Κ\\0{μ)ηι - {.RLß{M)Y+\ßL- M)-lMv\\u. The last expression tends to zero when μ —> +oo by virtue of (2.3.7). The term (2.3.10) tends to zero by the construction of the semigroup construction. Thus, by using the strong continuity of the semigroup and the fact that (Rß(M))p+1(ßL - Μ ) ~ ι Μ υ E U 1 , we have from the theorem of the mean that U1 = U ° ( R ß ( M ) ) p + l ( ß L -

tlim

M)~LMV

= (Rjj{M))p+1(ßL - M)~lMv.

(2.3.11)

By acting on (2.3.11) by the operator U s let us obtain differentiability of the semigroup on the right of this element u l at the point s G K+. In order to prove leftwise differentiability at this point we must consider the following expression:

-t

u =

t

-it 1 ,

s > t > 0,

30

G. A. Sviridyuk and V. E. Fedorov

and turn to the limit at t —> 0+, using the uniform boundedness of the semigroup. Thus, according to (2.3.11) ^-Ut{RLß{M))p+2v

= Ut(Rß(M))p+1(ßL

-

Μ)~ιΜυ.

iXJj

Let us act on the last identity by the operator R^(M). Note that if u G dom M 0 + im (fí¿(M))P + 1 , MU^u — F^Mu ->· FlMu

under

μ ->• +oo,

because Mu G U° 4- im (L¿(M)) P + 1 according to identity (2.1.8). (Proof of the pointwise convergence of the operators

^

k=0

to operators of the solving semigroup of equation (2.3.2) on the set of Τ at μ —> +00 is analogous to the proof of convergence of operators U^.). Since the operator M is closed we have that Vu 6 dom M 0 + im {RLß (M)) p + 1

Ulu G dom M,

MUlu =

F^u. (2.3.12)

By the construction and by virtue of (2.3.12) U1 commutes with operators i?¿(M) and (aL — M ) _ 1 M at the corresponding u, therefore, we obtain from (2.3.11) using identities (2.1.8), (2.1.4), i?¿(M) j t [ / V = {aL - M ) _ 1 M f / t u 1 .

(2.3.13)

Clearly, for G U° Us(u° + ul) = Usul. Therefore, the vector in the identity (2.3.13) can be replaced by the vector u — u° + u1 U° + i m { R ^ ( M ) ) P + 2 . Thus, the function u{t) = U*u is a solution equation (2.3.1) at any u from lineal U° + im {R%(M))P+2 dense in Ü Lemma 2.2.8. •

u1 G of by

Remark 2.3.1. Obviously, if a semigroup {U1 \ t G R+} is solving for equation Lib = Mu, where an operator M = M — aL is (L,p)-radial with a constant a = 0 (Definition 2.2.1), then the semigroup of the original equation Lii — Mu will be

C h a p t e r

2.

R e l a t i v e l y p-radial

31

operators

Accordingly, for this semigroup an exponential estimate \\Ψ'\\ £ ( μ ) 0+fc—+00K

In addition, using the above-mentioned uniform convergence and identity (2.4.5) it can be shown that \/u G Ul

Ü°u = lim ϋ*η = lim lim UÍu = lim lim UÍu = u. t->0+ t-»0+ fc—>oo fc->oot-vO+

We obtain that Ü° = P. Obviously,

It follows from (2.4.2) that l i m ^ + 0 0 {U¡/k)k = (Üt/k)k. \\(u¡/k)ku-

(Üt/k)hu\\u

Indeed, at u G Ü

= Ι ]Γ (ullk)k-m-l{&ik)m(ullk m=0

-&'k)u\ U

oo in (2.4.10), (2.4.11) and use the strong continuity of the family of operators {U* \ t G R+}with respect to t, we obtain as desired. Further, let u = (Rjj(M))p+2v for some β E and υ G U. Then \\U¡u - &u\\u = \im \\U\u-UluWu «—>00 - Ä

i1 =

+

Ï) K3f2

rtww·

-

2{p+l)ßP-^{M){{ßL

-

W-Wvh,

by virtue of (2.4.7). vi« = ( H i i ü U ^ j f l p · p+1

N

- E^-fiK+D/tW) k=0 = u + (p + p+l P+1

+Σ

k=2

- {' +

- m)-'M)p+1„

k (RLß(M)r2-kwk

l)R(p+lyt(M)(Rß(M))p+1wi

/ \k Μ ï+1 Κ + 1 ) Α ( ) )

(RLß(M)r2~kwk.

C

Here, vectors wk are the same as before (p +

l)Rlp+l)lt{M){RLß{M)f+lWl =

t{RLß{M))^wl+tR\p+l)lt{M){RLß{M)Yw2.

Prom the definition of (L,p)-radiality t'l{tRlp+l),t{M){R^M)YW2)\u
P~kìk(RÌ(M))p+1~k{R{(M))k(\L

- M)-1

Mu

k=0

using identity (2.1.4). Hence I I ^ R L ß ( M ) r + 1 u - (XRLx(M)r+1u\\u


oo

= w,

as it follows from the proof of Theorem 2.5.3, from which the notation L~l = (R%l(Mi))p+l{kLi - Mi)~l is taken. Hence ||(fci?fc(Si)) p+2 u - u\\u + \\Si((kRk(Si)r+2u - MkRk(Si))*+2u

- «)||Μ

- u\\u + \\(kRk(Si))p+2Siu

- 5i«|| m

0

at k —> oo, as desired. Thus, the generator A coincides with the continuous operator Si on the lineal im (Rμ(5ι))ρ+2 dense in the Banach space dom S\. It may be uniquely

44

G . A.

S v i r i d y u k and

V. E.

Fedorov

extended to the whole space in continuity. Therefore, Au = S\u Vu G dom Si. By virtue of the (L,p)-radiality for every μ G M+ there exists a resolvent ( μ Ι - S i ) _ 1 = Rj^ (Mi) G C { U ) . The resolvent of an infinitesimal generator of a uniformly bounded Co-semigroup {C/f j ¿ G R+} for μ G M+ exists and equal to its Laplace transform 1

l

roo ( μ Ι - Α ) ~ \ =

/

e ' ^ U i u d t

Vu

G U

1

.

Jo

Thus, the operator μΐ — A maps both dom Si and dom A into the space Ul bijectively. Hence it follows that dom Si = dom A. (For the semigroup {F/ | t G R+} the infinitesimal generator is obtained similarly.) • On applying the results of the classical theory of semigroups, we immediately obtain the following corollaries. Corollary 2.6.1. and

Τι

axe

In

the

conditions

of

Theorem

2.6.1

the

operators

Si

radial.

This fact immediately follows from the Hille—Yosida theorem. Corollary 2.6.2.

In

the

conditions

of

Theorem

2.6.1

{μ

G C | Re μ G

P (M). l

Rf} C

Proof. Let us take μ from the right semiplane, then by virtue of Lemma 2.2.4 and Theorem 2.5.3 (μΣ

-

1

Μ ) "

/

=

( μ Μ ^ Σ

=

(μΗ

-

=

(μ10 -

0

-

Μ

0

) ~

η - ' Μ ^ Η ΐ

I ) ~

1

k

M

M ¿ \ I -

ι

{ 1 -

-

Q)f

Q)f +

Ρ =

Y , t

Q)f

+

+

(μΐ

(μΐ

-

{μΏχ -

-

L î

l

M

x

) Q f

M i ) ~

l

L ^ Q f

S i ^ L ^ Q f

reo J

'

H

k

ö

1

(

I

- Q ) f +

e - ^ L ^ Q f d t

Vf

G

Τ .

(2.6.4)

Jo

fc=o

In doing so we used the nilpotency of the operator H and Theorem 2.6.1.

• Making use of the equality Ulu = U[Pu for all u G U, we obtain Corollary 2.6.3. In {U1

I t G 1 + }

{ { F

t

the

I t G 1 + } )

conditions can

be

of represented

Theorem as

2.6.1

the

semigroup

Chapter 2. Relatively p-radial operators 1 lim ω^+cc 2m / 1 (**/ = Hm — \ w-H-oo 2m Utu=

45

ρη+ίω / e^RJSAPudu Vu e U y 7 _ iw ρ+ιω . e ^ R ^ Q f d ß Vf e F) J~_ ^ J iuj >η—ιω

Vi G 1+ V7 G ! + . Remark 2.6.1. The notion of a strongly (7,p)-radial operator M in the case of U = Τ coincides with the notion of a radial operator. Indeed, in the case of ( i » - r a d i a l i t y U° = {0}, U1 = U, operator M = Ll 1M\ is radial by virtue of Corollary 2.6.1. The opposite statement is obvious. Definition 2.6.1. A closed set V C V is called a phase space of equation (2.3.3), if (i) any solution ν = v(t) of equation (2.3.3) lies in V, that is v(t) E V Vi 6 I + ; (ii) for every vq from some set dense in V there exists a unique solution of the Cauchy problem u(0) = vq for equation (2.3.3). Theorem 2.6.2. Let an operator M be strongly (L,p)-radial. Then Ul (Tl) is a phase space of equation (2.3.1) ((2.3.2)). Proof. Let us take the solution u(t) = u°(t) + u 1 ( i ) of equation (2.3.1), which, according to Theorem 2.5.2, Lemma 2.2.1 and Corollary 2.5.2 is split into a system of two equations L0u° = Mqu°, 1

1

Liù = Miu ,

(2.6.5) (2.6.6)

defined on the subspaces U° and U l respectively. By virtue of Lemma 2.2.4 equation (2.6.5) is equivalent to the equation Hù° = u°.

(2.6.7)

According to Lemma 2.2.5, 0 = Hp+1ù° = Hpu°. After differentiating the last identity using the continuity of the operator H and identity (2.6.7), we get 0 = Hpii° -

Hp-luü.

46

G. A. Sviridyuk and V. E. Fedorov

By repeating this procedure ρ — 1 more times we obtain 0 = Hu° = u°. Hence it follows that any solution of equation (2.3.1) lies in U1, and with allowance for Corollary 2.5.2 it follows that any solution of equation (2.3.2) f(t) is in Tl As a consequence of Theorem 2.5.3, equation (2.6.6) is equivalent to the equation ώ1 = SlU\ and from Theorem 2.6.1 it follows that for every UQ G dom M\ vectorfunction U1UQ is a unique solution of the Cauchy problem u(0) = UQ for this equation and consequently also for equation (2.3.1), since U1UQ = U0+tuo = U°Utu0 = PUtu0eU1. Similarly, making use of Theorem 2.6.1 it can be shown that for every /o 6 dom T\ = L[dom M{\ vector-function Ftfo is a unique solution of the Cauchy problem /(0) = /o for equation (2.3.2). • Remark 2.6.2. Clearly, the phase space of equation (2.0.2) will also be Ul. Remark 2.6.3. Prom the uniqueness of the solution of the Cauchy problem there follows the uniqueness of the solving semigroup. Indeed, two solving semigroups, for example, equations (2.3.1) must coincide on the set of dom Μχ dense in the space Ul. And since the operators that enter the semigroup are continuous and, in addition, turn to zero in the semigroups coincide on the whole space U. In particular, we observe the coincidence of semigroups obtained by the approximation of the Yosida-type or by the approximations of Hille-Widder-Post type.

2.7.

GENERATORS OF DEGENERATE STRONGLY CONTINUOUS SEMIGROUPS OF OPERATORS

Here we shall obtain the results opposite to those obtained in the previous paragraphs of this chapter. To this end, in terms of semigroups we formulate five conditions that follow from the strong (L,p)-radiality of the operator M. Then we will show that these conditions are sufficient for the strong p-radiality of the operator M with respect to the operator L.

Chapter 2. Relatively p-radial operators

47

(CSI) There exist two strongly continuous and uniformly bounded semigroups {U^te 1 + } and {F1 | t G I + } of operators Ut G C{U), Fl e C(T). Let us assume that Ρ = U°, Q = F°. According to semigroup property, Ρ and Q are projectors. Let us introduce the following notation: U° = ker P, UL = i m P , JF° =_kerQ, = imQ;_we have U = U° ®U\ Τ = 1 Θ J" . By {U{ I t G 1 + } and {F¡ | t G 1 + } let us denote restrictions of the respective semigroups on the subspaces U l and T l . The restrictions are nondegenerate semigroups and by the Hille—Yosida theorem they have infinitesimal generator, which will be denoted as S\ and T\ respectively. According to the classical theory they are radial operators. (CS2) There exists linear homeomorphism L\ : Ul —> Tx, L\S\ = T\L\.

such that

The last equation implies that Li[dom 5i] = dom T\. (CS3) There exists a bijective operator Mo G Cl(U°] Prom the bijectivity there follows the existence of the operator M 0 _1 G (CS4) There exists an operator Lq G C{UO\T°) such that the operator H = M^Lo is of nilpotency degree not exceeding ρ G No. (CS5) L = L0(I - P) + LXP, dom Mo + dom S ι.

M

= M0{I

- P) + L&P,

It is easily seen that the operator L G

dom M

=

and the operator M G

CL{U]F).

Theorem 2.7.1. An operator M is strongly (L,p)-radial if and only if all the conditions (CS1)-(CS5) are satisfíed. Proof. It was shown in the previous paragraphs of this chapter that the conditions (CS1)-(CS5) follow from the strong (L,p)-radiality of the operator M. Let us show the reverse. Let μ G K+, then

{μΐ

- M ) " 1 = (//Lo - MO)-1

(I - Q) + ( μ ΐ χ -

= - ¿ßkH k=0

k

L X S ^ Q

M ^ ( I -Q)

+ (μΐ -

S.r'L^Q.

48

G. A. Sviridyuk and V. E. Fedorov

Similarly, we obtain the following identities itf(M)

— Ρ) + (μΐ - S,)-1 Ρ,

k=0

( Ä f M ( M ) ) " = 0 ( I - Ρ) + n ( Ä M ( S i ) ) n P , k=0 LfrM)

= 0(μ0 - I)-\I

-Q) + (μΐ -

T\)~lQ

= -G ¿ /,*G*(I - Q) + (μΐ - n r ' Q , k—Q (Lfß>p)(M)r

= o(I -Q) + Π ( R ^ T Q , k=0

where μ/C G K+, k = 0 , . . . ,p, and the operator G = LQMQ1 = MQHMQ1 is of nilpotency degree not higher than p, which follows from condition (CS4). In addition to the nilpotency of the operators H and G here we also used the fact that the resolvents commute. Further, Rlm{M){\L

- M)-1 =

fl(ßkI-Sl)-1(XI-Siy1L^Qi k=0

M (XL - M)~lLLM{M)i

= (λ/ - Τ ι ) - 1 Π (/i*/ - T . r ^ Q f , k=z 0

since the operator T\ commutes with its resolvent. The vector f is taken from the lineal T = dom Ti, which is dense in Τ because dom Τί = T y . Prom the above relations and the radiality of the operators S ι and T\ there follows the theorem statement. • Remark 2.7.1. If the operator L is continuously invertible, the conditions (CS2)-(CS5) become trivial and allowing for Remark 2.6.1. Theorem 2.7.1 turns into the Hille—Yosida theorem and therefore is its generalisation. Remark 2.7.2. It follows from Theorem 2.7.1 that for every ρ G Ν a strongly (L,p)-radial operator M is strongly (L,p + g)-radial. Indeed, the number ρ in the condition (CS4) can always be increased.

Chapter 2. Relatively p-radial 2.8.

operators

49

D E G E N E R A T E STRONGLY CONTINUOUS GROUPS OF OPERATORS

Let U and Τ be Banach spaces, operator L G C(U\T), and operator M G Definition 2.8.1. An operator M is called p-biradial with respect to the operator L (or, briefly, (L,p)-biradial) if (i) 3a G 1 + Υμ e Ι α μ e

Pl(M));

(ii) 3K G 1 + Vn € Ν Υμ* 6 Κα max{||(iîf M (M)) n || £ ( w ) ,||(Lf M (M)r|| z ; ( ^ ) } < K[f[(\ßk\ k-0 Here μ =

-

α)η}~\

(μ0,μι,...,μρ).

Remark 2.8.1. Let there exist an operator L~l G C{T\U). If in this case the operator L~lM G CÍ(U) (or, which is equivalent, the operator MLG Ci{T)) is a generator of a nondegenerate strongly continuous group (Hille and Phillips, 1957; Yosida, 1965), the operator M is (L,p)-biradial. When ρ = 0, the opposite is also true. Remark 2.8.2. (L,p)-biradiality of the operator M is equivalent to the (L,p)-radiality of the operators M and —M. Hence immediately follows the validity of all the results of the second paragraph under the condition of (L,p)-biradiality of the operator M. The solution of equation (2.3.3) will be called a vector-function υ G C^M; V) corresponding to it. Definition 2.8.2. Mapping V~ : Κ -» £(V) is called a group of solving operators (solving group) of equation (2.3.3) if (i) Vw G V V«,t G E VsVlv = Vs+tv, (ii) v(t) = Vtv is the solution of equation (2.3.3) for every ν from the lineal dense in V . A group is called exponentially bounded if 3WGR

3C G K+

Vi G Κ

IIV'II^V) < (7βω|ί|.

50

G. A. Sviridyuk and V. E. Fedorov

Theorem 2.8.1. Let an operator M be (L, ρ)-biradial. Then there exists an exponentially bounded strongly continuous solving group of equation (2.3.1) ((2.3.2)) considered on the subspace U (T). Proof. By virtue of (Z/,p)-biradiality of the operator M correctly defined are semigroups of operators {U\. G C{Ul), t > 0} and {F[ G C{Ul), t> 0}, where U\+ = s-lim Ulμ = s-lim e ^ ™ μ-++οο μ^+οο

± * ' η! V η—0

V μ(

ϋ+1

"

)

f ,

(2.8.1)

η—0 and semigroups of operators {Ut G C{Ul), l

l

{F _ G C{F ),

t > 0},

(2.8.3)

t > 0}.

(2.8.4)

The first and the third semigroups (the second and the fourth) coincide with accuracy to the substitution M for — M. Let us extend semigroups (2.8.1), (2.8.3) by zero to the subspace U° and semigroups (2.8.2), (2.8.4) by zero to J=°. Then for every vector u0 G U° 4- im (Rj¡(M))P+2, f0 G + im (Lj¡{M))P+2 functions of t U\u0, F { / 0 , UÍuo, Fíf0 are solutions of respective equations (2.3.1) and (2.3.2), R^i-M)

^ = —(aL + M ) _ 1 M , at

(2.8.5)

+M)-lf.

L^(-M)ft=-M(aL

The identities of the semigroups will be: the projector U+ =

= l

Ρ to i m Ä f M ( M ) - i m Ä f M ( - M ) = i m ( - l ) P ^ R ^ p ) ( M ) = U along k e r R ^ p j ( M ) —U° — k e r R ^ p ^ ( - M ) for the first and the third semigroups and the projector

= F° = Q to i m L ^ p ) ( M ) =

along ker L^p)(M)

=

!F° for the second and the fourth semigroups. Let us show that the family of operators {Ul G C{Ü) I υ1 = U\, t > 0;

U* = Ul\

t < 0}

(2.8.6)

Chapter

2. Relatively p-radial

operators

51

is a strongly continuous group. Let us consider an operator Gl — Let us take Uo E i¿o

the functions R%(M)

^

at

im(Rß(M))P+2. U+UQ,

G'uo

ULUQ

U+UÍ.

As has already been mentioned, for such

resolve equations (2.3.1), (2.8.5). Therefore,

= (OLL - M)~lMUVU^UQ

+ RLa{M)U\

= {aL - M^MUlUiuo

- υ^ία(-Μ)

^

= (aL - M)~lMU+U^UQ

- (aL - M)'1

¿

at

UIUQ

UÍuQ

MU^UÍuo

= 0.

Here we used the commutation of the operators i ? ¿ ( M ) and (aL — M ) _ 1 M with U+ and Ut,

which follows from the construction of the semigroup.

Thus, G*uo = const = G°u0 = u0 for every u0 E im ( i ^ ( M ) ) p + 2 C U1.

The

operators Gt are continuous, therefore, by virtue of Lemma 2.2.8 the identity Gtuo — UQ can be extended to all the subspaces Ul.

By the construction

of semigroup Glu = 0 for u E

€ U°, u1 e W 1 , then

Glu = ul = Pu =

If u = u° + u 1 ,

U°u.

Similarly, it can be shown that UtU+u — U°u for u E U. The group character of the family (2.8.6) is verified directly, and the exponential boundedness < Ke a W follows from the construction. It remains to verify that the group solves equation (2.3.1) at t < 0. Indeed, for u E im (Rjj(M))p+2

RLa(M)

I

U'u = -Rta{-M) = {-aL

jt

UZtu

+ M)~l(-M)Ultu

= (aL -

The group { F 1 \ t E R } is constructed similarly.

M)~lMUtu.

•

R e m a r k 2 . 8 . 3 . In addition, it is easily seen that the first of the groups resolves equation (2.0.2). The constructed groups are defined not over the whole space.

From

Remark 2.8.2 and Theorem 2.5.1 it follows T h e o r e m 2 . 8 . 2 . Let the space U (T) (L,p)-biradial.

Then U = U° θ U

l

(T =

be reflexive, and operator M ©

be

T ). x

In order to split spaces that are not reflexive, let us impose additional conditions on the operators L, M using a procedure similar to that used in Section 2.5.

52

G. A. Sviridyuk and V. E. Fedorov

Definition 2.8.3. An operator M is called strongly (L,p)-biradial the right (on the left) if it is (L,p)-biradial and for all u G dom M I I R L M { M ) { \ L - M)-lMu\\u < const (u) [(|A| - α) Π(|μ*| - a) k-0

on

-1

(there exists a lineal Τ dense in J 7 , such that for all f G Τ

IIM(XL - M)-lLLM{M)f\\r

< const(/) [(|λ| - a)

k-0

- a)

-ι

for every λ , μ ο , μ ι , . . . , μ ρ G 1β·

Remark 2.8.4. The strong (L,p)-biradiality of the operator M on the right (on the left) is equivalent to the strong (L,p)-radiality of the operators M and — M on the right (on the left). According to Theorem 2.5.2 the following results are obvious: Theorem 2.8.3. Let the operator M be strongly (L,p)-biradial right (on the left). Then U = U° Θ Ul [Τ = θ

on the

Provided the operator M is strongly (L,p)-radial on the right (on the left) the solving group of equation ( 2 . 3 . 1 ) ( ( 2 . 3 . 2 ) ) is specified over the whole space U (F), and its identity is the projector Ρ (Q). Corollary 2.8.1. Let an operator M be strongly (L,p)-radial right and on the left. Then (i) Vu G U LPu = QLu; (ii) Vu G dom M (Pu G dom M) Λ (MPu = QMu).

on the

Corollary 2.8.2. In the conditions of Corollary 2.5.1 the operator Li G ύ(ΙΑι\Τι), and the operators Mk G Cl(Uk\Tk), k = 0.1. Remark 2.8.5. The statements of Theorem 2 . 8 . 3 , Corollaries 2 . 8 . 1 , 2.8.2 is true under more weak assumptions: (L,p)-biradiality and strongly (L,p)-radiality on the right (on the left). Definition 2.8.4. An operator M is called strongly (L,p)-biradial, is strongly (L,p)-biradial on the left and

if it

Chapter 2. Relatively p-radial operators

IIRfßtp)(M)(XL

- M)-l\\c{J:,u)

53

< Κ [(|λ| - α) f [ ( H - α)" ' k=0

for all λ , μ 0 , . . . ,/¿p G R α · Remark 2.8.6. The strong (L,p)-biradiality of the operator M is equivalent to the strong (L,p)-radiality of the operators M and —M . Remark 2.8.7. The strongly (L,p)-biradial operator M is strongly (L,p)-biradial on the right. Remark 2.8.8. Let an operator L be continuously invertible, and an operator S = L~lM E Cl{U) be a generator of a strongly continuous (Co)-group. Then the operator M is strongly (L,p)-biradial. Theorem 2.8.4. Let the operator M be strongly (L,p)-biradial. there exists an operator L^1 G C(Tl]Ul).

Then

The restriction {U{ \ t G R} ({F¡ \ t G Κ}) of the group {Ul \ t G R} ({F* I t G R}) onto subspace Ul {Tl) is a nondegenerate strongly continuous group. Let us introduce the following notation Si - L^1 Mi : dom Si Τι = M i L f

1

: dom 7\ ->·

U1,

dom Si - dom Mx, ,

dom Ti = Li[dom Mi]

provided the operator M is strongly (L,p)-biradial. Theorem 2.8.5. Let an operator M be strongly (L,p)-biradial. Then the infìnitesimal generator of the group {U{ \ t G R} ({F* | t G R}) is the operator Si (Ti). Proof. The nondegenerate group {C/f | t G R} consists of two nondegenerate semigroups in accordance with (2.8.6). According to Remark 2.8.6 and Theorem 2.6.1 the infinitesimal generators of these semigroups are respectively operators L ^ l M i , By the construction of the group {U[ \ t G R} we find that its generator is the operator S\. Similarly we find the generator of the group {F* | t G R}. •

54

G. A. Sviridyuk and V. E. Fedorov

Remark 2.8.9. The statements of Theorems 2.8.4, 2.8.5 is also true under more weak assumptions of (L,p)-biradiality and strongly (L,p)-radiality. Finally, determine the necessary and sufficient conditions of the strong (L,p)-biradiality of the operator M in group terms. (CG) There exist two strongly continuous and exponentially bounded groups {UÉ 6 £(U) \ t G K} and {Ft G C(T) \ t G K} of operators kernels. Let us assume that Ρ = U°, Q = F°. Clearly, Ρ and Q are projectors. Let us introduce the following denotations: U° = kerP, Ul — imP, = 1 1 ker Q, = imQ; we have U = θ U , Τ = JF° Θ F . By {U\ \ t G R} and {Ρ* I t G IR} let us denote the restrictions of the respective groups on the subspaces U 1 and . The restrictions are nondegenerate groups and have infinitesimal generators S\ and T\ respectively. Theorem 2.8.6. An operator M is strongly (L,p)-biradial if and only if all the conditions (CG), (CS2)-(CS5) are satisfied. Proof. From the results of the previous paragraphs there follows the necessity of the conditions (CG), (CS2)-(CS5). Their sufficiency is shown using the procedure similar to that employed in the proof of Theorem 2.7.1 making use of the estimates on the norms of the resolvents of nondegenerate groups generators: \\{μΙ - S ^ W ^ for every μ G Κ, |μ| > α.

R.

•

The following lemma and theorem are proved using the procedure similar to that of obtaining the analogous results in Chapter 1.

Chapter 3. Relatively p-sectorial

operators

59

Lemma 3.1.2. Let an operator M be (L,p)-sectorial. Then the lengths of all chains of relatively adjoint vectors are bounded by a number p. Theorem 3.1.1. Let operator M be (L,p)-sectorial.

Then

0) k e r i ? f M ( M ) n i m R ^ p ) ( M ) = {0}, (u)

3.2.

kerL

L ) ( M ) n i m L f M ( M ) = {0}.

D E G E N E R A T E ANALYTIC SEMIGROUPS OF OPERATORS

Similar Section 2.3, let us introduce into consideration a pair of equations equivalent to equation (3.0.2) Ra(M)ù = (aL - M)~lMu,

(3.2.1)

L^(M)f

(3.2.2)

= M(aL-M)~1f.

Both of these equations will be considered as specific interpretations of the equation (3.2.3)

Av = Bv,

where operators A, Β E £(V); V is some Banach space. Further vector function ν E C 1 ( R + ; V ) satisfying this equation will be called the relaxed solution of equation (3.2.3).

Definition 3.2.1. A mapping V• E C 1 (K + ; £(V)) is called a semigroup of solving operators (or briefly, a solving semigroup)

(i) v s v t

=

of equation (3.2.3), if

ys+t V s . t e R f ;

(ii) for every ι>ο £ V the vector function v(t) = Vtvo is the relaxed solution of equation (3.2.3). As in Section 1.3 let us identify the semigroup with the set { V 1 \ t E M+}. A semigroup {V1 \ t 6ffi_)_}will be called analytic, if it admits some analytic extension to a certain sector containing a ray K+, while retaining its properties (i), (ii), and will be called uniformly bounded, if 3CGR

VÍGK+

||ν*||£(ν) < C.

60

G. A. Sviridyuk and V. E. Fedorov

Remark 3.2.1. Note that the fact that a solving semigroup of equation (3.2.3) is with an identity is not postulated. Theorem 3.2.1. Let an operator M be (L,p)-sectorial. Then there exists an analytic and uniformly bounded solving semigroup of equation (3.2.1) (equation (3.2.2)). Proof. Let contour Γ C Sg(M) be such that | argμ| —>· θ for μ —> oo, μ ΕΓ. Consider an improper integral of a Dunford—Taylor type (3.2.4) Note the absolute convergence of integral (3.2.4) for every ί G 1+ and the fact that this integral can be analytically extended to the sector{T G C | I arg τ I < θ — π/2, τ φ 0}, which can be easily derived from the inequality cos (arg μ + arg τ) < 0. Let a contour Γ' C Sg(M) be such that | arg^| —> θ for μ —> oo, μ G Γ', and in this connection the domain containing the contour Γ remains on the left of the contour Γ' when it is traversed in the positive direction. Then by Theorem 2.1.1 and Cauchy theorem

Hence

by virtue of Fubini theorem, theorem of residues and identities (2.1.4). Let us establish a uniform boundedness of the semigroup {U* \ t G R+}· We fix t G M+, then U* = J7Û*1)' = (

2

f

f JTq

JT\

...!

RLm{M)

JTp

xexp(£ g=0

C h a p t e r

3 .

R e l a t i v e l y

p - s e c t o r i a l

o p e r a t o r s

61

Here, s = t/(p + 1), and the region containing the contour Γ 9 _ ι , remains on the left when the contour Γ 9 , q = 1,2,... ,p is traversed in the positive direction; and | a x g p \ —> θ when μ9 —>• οο, μ9 G Γ ς , ç = 0,1,... ,ρ. q

Hence τ \\U

\ c ( u )


t, then, evidently, ker V1 C ker Vs. We now fix t > 0 such that V*

t Vsφ = 0. By the theorem on the uniqueness of an analytic function Vsψ = 0 for all s from the sector of the semigroup analyticity. Remark 3.2.3. Note the obvious relationships i m P C imF* C im V' Vi < s. Let us denote by Ul {Tl) closure in the norm of the space U {T) of the lineal i m Ä f M ( M ) ( i m ¿ f M ( M ) ) . Theorem 3.2.2. Let operator M be (L,ρ)-sectorial. and im F~ = Tl.

Then im U' = Ul

Proof. Let \ q G Slg(M), q = 0 , 1 , . . . , p . Then, successively applying identity (2.1.4) to (3.2.4), we obtain U* =

J é* άμ - RLxp{M)^-

= (-ir

1

ilf

M

J (μ - \P)RLß (M)e^ άμ

( M ) ¿ ί Π (μ - Xq)RÌ(M)e^ J r 5=0

άμ.

This implies that for every vector u G U the limit limi_>o+ υ ι η must lie in U l by virtue of the closedness. Let us show that for every u G U1 there exists a limit lim Utu = u. Let us first take u G i m ( M ) ,

(3.2.6)

i.e., u = R^Qp^(M)v for some aq G

Sg(M), q — 0 , 1 , . . . , p , lying on the right of the contour Γ, and υ E U (Theorem 2.1.2). Then U*u =

UtR^a^(M)v

Chapter 3. Relatively p-sectorial

operators

63

by virtue of identity (2.1.4) and the Cauchy theorem. From Definition 3.1.1 there follows the existence of a limit * τ lim U u = --Rf t->0+

Ru(M)v , χ 1 f ^ v( M ) — / μΚ άμ. ^ 2m JT μ - αρ

Evaluating the integral on the right by the theorem of residues we obtain ( lim

Thus, limit (3.2.6) exists on of the uniform boundedness the Banach—Steinhaus that The relation imF- = T l

υ*ν = Β$ α φ ) {Μ)υ = η. the lineal i m d e n s e in U 1 . By virtue of the semigroup {U1 \ t G M+} it follows from limit (3.2.6) exists on the whole subspace U l . is proved similarly. •

We next turn to studying kernels of semigroups {¡7' 11 g R+} {Ft 11 e R+}. Let us assume U° = ker U , J70 = ker F~ and by LQ (M 0 ) let us denote the restriction of the operator L (M) on U° (dom Μ Π U°). Lemma 3.2.1. Let an operator M be (L,p)-sectorial. G and M0 : dom

LQ

Proof. Let φ G

Then

operators

then by virtue of (3.2.4) and (3.2.5) we have FtL^

= FtLip = LU1 ψ = 0.

Let ψ G dom M DU 0 , then by virtue of (2.1.8), (3.2.4) and (3.2.5) we have F'Mοψ =

= MU1 ψ = 0.

• Introduce the notation σ$(Μ) as Lo-spectrum of an operator MQ. Then σ$(Μ) = 0.

Lemma 3.2.2. Let operator M he (L,p)-sectorial. Proof. integral

Let ψ G U°, then by virtue of (2.1.2) we have a convergent

1 r ^L -M)~l 2m J ρ λ—μ

_

etlt

Μΰ)ψ

=

M y l

W

+

^

=

Similarly, for every ψ (Ξ F 0 we have (ALo -Mo)±[

2m Jr

{μ1Γ

λ—μ

άμ = -e·"φ .

64

G. A. Sviridyuk and V. E. Fedorov 1

This implies that for every λ G C there exists an operator (XLQ — Mo) equal to the restriction of the operator 1

2m Jr

on •

μ - λ

p

v

(3.2.7)

'

(Here, the contour Γ C Sjf(M) is the same as in (3.2.4) and (3.2.5)).

Corollary 3.2.1. In the conditions of Lemma 3.2.2 there exists an operator M0-1 G £(Jr0;W°). Proof. The desired operator is equal to the restriction of operator (3.2.7) on for Λ = 0. • Theorem 3.2.3. Let an operator M be (L,p)-sectorial. and J* = ker L^p)(M). Proof. Let φ G ker

Then U° —

then

U*

1. By differentiating (3.3.8) with respect to t and acting by the operator R%(M) we obtain ρ (.Ra(M))p Σ aru{r+1) = aR%{M)u - u τ-0

(3.3.10)

by virtue of (3.3.9). By substituting equality (3.3.8) into the first term on the right-hand side of (3.3.10) we get u{t) =

P+1 {RLa{M))^Y,bru(T\t). r—0

Thus, by theorem 3.2.2 u(t) € Ul Vi G K+. (The statement of the theorem on coincidence of the image of semigroup (3.2.5) with a phase space of equation (3.3.2) is proved similarly.) • Remark 3.3.5. It follows from Theorem 3.3.2 that any solution of equation (3.0.2) with (L,p)-sectorial operator M is analytic in a sector containing a positive semiaxis.

3.4.

SPACE SPLITTING

Definition 3.4.1. Let {V* \ t G IR+} be a semigroup defined on a Banach space V. An operator E G £(V) is called an identity of this semigroup if E = s-limt_+o+ VK

Chapter 3. Relatively p-sectorial operators

69

Remark 3.4.1. As can easily be seen, the identity E of the semigroup {V I t G IR+} is a projector to V and at the same time ker E = ker Va and im E = im y·. 1

Remark 3.4.2. The problem of the existence of spaces' splitting into kernels and images of degenerate analytic semigroups is closely connected with the existence of identities of these semigroups. Indeed, by virtue of Theorems 3.2.2 and 3.2.3 splitting of a space U = U° ®Ul (T = JF0 © T1) is equivalent to the existence of identity of the semigroup {U1 \ t € R_|_} {{F* I t G 1+}). Theorem 3.4.1. Let a space U (T) be refìexive and let an operator M be (L,p)-sectorial. Then there exists an identity of semigroup (3.2.4) (semigroup (3.2.5)). Proof. The proof of splittings U = U° ® W1, Τ = ® T1 is similar to the proof of Theorem 2.5.1. Allowing for Remark 3.4.2 we obtain the desired. • Definition 3.4.2. An operator M is called strongly (L,p)-sectorial on the right (on the left), if it is (L,p)-sectorial and

I I R \ ß p ) ( M ) ( \ L - M)~lMu\\u

ρ < const/|λ| J J \μη\

Vu G dom M ,

9=0 O

where const = const (u) (there exists a lineal Τ dense in Τ such that

IIM(\L - M ) - 1 L ¿ i P ) ( M ) / | | ^ < const / | λ | f [ |μ,| 9=0

V/ e > ,

where const = const(/)); λ,μ ς € Sg(M), q = 0 , 1 , . . . ,p. Theorem 3.4.2. Let an operator M be (L,p)-sectorial on the right (on the left). Then there exists an identity of semigroup (3.2.4) (semigroup (3.2.5)).

70

G. A. Sviridyuk and V. E. Fedorov Proof. Let u G dom M and s > t > 0. Then

Usu - Ulu = υ\υ$-1

M)~lMue^( äs - t ) dX

- I)u = U ^ Í {XL 2m JT

^χ

Ρ X Mu exp ^

MqT + H s

^ Ρ · ' ' ^Ml

j

where τ = t/(p + 1), and contours Γ 9 , q = 0,1,... ,p, and Γ are the same as in the proof of Theorem 3.3.1. From the strong (L,p)-sectoriality of the operator M on the right we have il / p \ ( Y[(vqL - tM)~lL\ (uL - (s - t)M)~lMu

p

< const / \ v \ JJ \uq\ U

9=0

9=0

for every s > t > 0. Hence Ρ IIU u - U uII < {s - t) const ί — \dv\ T T [ |e"'| \dvq\, J Γ' U 9=0^ s

l

where ν — A(s — t), vq = μητ, q = 0,1,... ,p. On choosing contours Γ'9, q = 0 , 1 , . . . ,p as independent of r and contour Γ' independent of s — t, by virtue of Cauchy criterion we obtain the existence of the limit Pu = lim¿_>o+ Ulu for every u G dom M. In view of the uniform boundedness of the semigroup {U1 I t G K+} this implies the existence of the projector P = s-]imUt. t->o+

(3.4.1)

Q = s-limF 4 i->0+

(3.4.2)

(Existence of the projector

is established similarly).

•

Corollary 3.4.1. Let an operator M be strongly (L,p)-sectorial on the right and on the left. Then (i) Vu G U

LPu = QLu;

(ii) Vu G dom M

(Pu G dom M) A (MPu = QMu).

Chapter 3. Relatively p-sectorial operators

71

Proof. The assertion (i) is obvious by virtue of the continuity of the operator L and identity LU1 = FtL. Let us prove the assertion (ii). Let u G domM, then it follows from (3.4.1) that limt_>o+ Ulu = Pu, and from (3.4) and (3.2.5) it follows that lim MUlu = lim F1 Mu = QMu. i—f0+ -f0+ t-+o+ Hence the closedness of the M suggests (ii).

•

Let us denote by L\ (Mi) the restriction of an operator L (M) on Ul (dom Mi = MdW 1 ). By dom Mo we will respectively denote dom Μ ΠΚ°. By virtue of Corollary 3.4.1 is true Corollary 3.4.2. In the conditions of Corollary 3.4.1 the operator L\ G C{U1·, T l ) and the operator Mk G Cl{Uk; Tk), k = 0,1. Proof. It follows from corollary 3.4.1 (ii) that impair" elements u e dom M, i.e., Pu G dom M, it follows that the set Uk Π dom M is dense in Uk, Corollary 3.4.2 the operator Mk : dom Μ Π Uk closed and densely defined. •

projector Ρ "does not if u G dom M. Hence k — 0,1. Therefore, by Tk, k = 0,1, is linear,

Remark 3.4.3. It follows from Corollary 3.2.1 that the operator MQ is bijective. Let us obtain the existence of the operator LJ" 1 G C{Tl-,Ul). To this end, under the assumption of (L,p)-sectoriality of the operator M let us construct an operator (3.4.3) where the contour Γ C Sg(M) is the same as in (3.2.4) and (3.2.5). The integral converges by virtue of Lemma 3.1.1. Lemma 3.4.1. Let an operator M be (L,p)-sectorial. Then the family {R I t G 1+} can be analytically extended into the sector {r G C | | arg τ| < θ - π/2} Rs+t = UsRt = RsFt Vs,i G M+. l

The proof is similar to the proof of Theorem 3.2.1, only instead of identity (2.1.4), the identity (2.1.3) should be used.

72

G. A. Sviridyuk and V. E. Fedorov

Lemma 3.4.2. Let an operator M be strongly (L,ρ)-sectorial right and on the left. Then im R* C U1 and ker R* = Vi € M+.

on the

The proof is obvious, in view of inclusion ker i?4 C ker F 4 = relation Rl = PRÏQ, which follows from Lemma 3.4.1 and Theorem 3.4.2. Definition 3.4.3. An operator M is called strongly (L,p)-sectorial, is strongly (L, ^-sectorial on the left and I\(XL - M)-lLfßtp)(M)\\c{r.u)

if it

< const / (|λ| f [ |μ,|)

for every λ,μ 9 G S%{M), q = 0 , 1 , . . . ,p. Remark 3.4.4. The strongly (L,p)-sectorial operator M is strongly (L,p)-sectorial on the right. Remark 3.4.5. Let there exist an operator L~l e C{T\U). Then from the sectoriality of the operator L~lM (or ML~l) there follows strong (L,p)-sectoriality of the operator M. For ρ = 0, the reverse is true. Lemma 3.4.3. Let an operator M be strongly (L, p)-sectorial. W^WcirU) < const

Then

Vi G M+.

Proof. Let t > s > 0. By Lemma 3.4.1,

where contours Γ 9 , Γ C Sg(M), q = 0 , 1 , . . . , ρ , are the same as in the proof of Theorem 3.2.1, and τ = (t — s)/(p + 1). Pursuing further reasoning similar to that used in the proof of Theorem 3.2.1, we obtain the desired. •

Chapter

3. Relatively

p-sectorial

operators

T h e o r e m 3 . 4 . 3 . Let operator M be strongly there exists an operator Lj" 1 G C(Tl]Ul).

(L,p)-sectorial.

73 Then

P r o o f . Introduce the notation R\ for the restriction of the operator Rl on Tl. By virtue of Lemmas 3.4.1 and 3.4.3 for every s > t > 0 and / G F1 we have R{f

- R[f

= RUF3-*

-

Q)f.

Hence by virtue of Lemma 3.4.3 we obtain \\Rlf - R\f\\ < const ||(F s_i - Q)f\\r

0

at s —> 0 + . It implies the existence of a limit LT1 = s-limR\ G C(Tl-,Ul) 1 t->o+

(3.4.4)

•

It follows from relations RtL = U* and LRt = Fl Vi G R+. that L f 1 is the desired operator. • Let us assume Si = L~[lM\, dom S\ — d o m M f l W 1 ; T\ = M i L j - 1 , dom Ti = L\[dom S\]. It follows from Theorem 3.4.3 that operators Si and Ti are linear, closed and densely defined. The restriction {U[ | t E K + } ({Fl I t G 1 + } ) of the semigroup {U* \ t G 1 + } {{F1 \ t G 1 + } ) to the semispace U l is a nondegenerate analytic semigroup. C o r o l l a r y 3 . 4 . 3 . Let an operator M be strongly (L,p)-sectorial. Then the operator S ι G Cl (Li1) (Τι G Cl(!F1)) will be the infinitesimal generator of the semigroup of operators {U{ \ t G K+} ( { F / | t G 11+}). In the proof integral representations of semigroup (3.2.4), (3.2.5), Theorem 3.4.3 and identities = Rß(S ι),

Lj?(Mi)

=

Rß(Tι)

are used. Making use of theorem on generators of analytic semigroups we immediately obtain C o r o l l a r y 3 . 4 . 4 . Let an operator M be strongly (L,p)-sectorial. operators Si and Τι are sectorial and σ ( 5 ι ) = σ(Τι).

Then

R e m a r k 3 . 4 . 6 . Let there exist an operator L~l G C(JA\T). From Corollary 3.4.4 it follows that the operator M is strongly (L,p)-sectorial when the operator L~lM is sectorial (or, which is equivalent, operator ML"1).

74

3.5.

G. A. Sviridyuk and V. E. Fedorov

GENERATORS OF DEGENERATE ANALYTIC SEMIGROUPS OF OPERATORS

In the previous paragraphs of this chapter we determined the mapping of a set of operator pairs (L,M), where the operator M is strongly (L,^-sectorial, into a set of pairs of degenerate analytic semigroups ({C/f | t G R+}, { F 4 1 1 e R+}). Here, we will establish the existence of the reverse mapping. To this end, let us prove the sufficiency of some necessary conditions of the strong (L,p)-sectoriality of the operator M. Thus let U and Τ be Banach spaces. Let us formulate the conditions. (ASI) There exists a pair ({U* \ t G 1 + } , {F1 \ t G 1 + } ) of analytical in some sector {r G C | |argr| < θ — π / 2 } , 0 G (π/2, π), strongly continuous and uniformly bounded on R+ semigroups of operators U1 G C(U), F1 G C{T). From (ASI) there follows the existence of identities Ρ — U° and Q = F° semigroups of operators { U t \ t G and | t G M+} respectively. Obviously, keri/=kerP,

ker i*1" = ker Q,

im/7' = i m P ,

imF' =

imQ.

Let us assume ker P = U°,

im P = Ul,

kerQ = JF°,

imQ^^1.

Introduce the notation {U\ \ t G R+} ({F¡ \ t G R+}) for the restriction of the semigroup {U1 \ t G 1 + } ({F 4 | t G 1 + } ) on Ul (V 1 ). Since ker = {0} and ker F[ = {0} by the construction, then by virtue of classical results there exist sectorial operators S\ : dom S\ —» U1 T\ : dom Ti Tl (in this case it is possible to take sectors S$(S i) = Sg {T\ ) ) such that (3.5.1) (3.5.2) where t G R f , and s-lim¿_>o+ U{ — I, s-lim¿^o+ F{ = I. (AS2) There exists a toplinear isomorphism L\ : Ul — T l , such that L\S\ = T\L\. Prom (ASI), (AS2) there follows the existence of a closed densely defined operator M\ = L\S\ = T\L\, dom M\ = dom S\.

75

Chapter 3. Relatively p-sectorial operators

(AS3) There exists a linear, closed, densely defined bijective operator M 0 : dom M 0 dom M 0 C W°. (AS4) There exists an operator Lo G £(W°;.7r0), such that the operator H — MQ 1 LQ G C(U°) is nilpotent and the degree of its nilpotency does not exceed some number ρ e NoFrom (AS4) there follows the existence of a nilpotent operator G = LQMQ1 G £(J r 0 ) with nilpotency degree not more than p. (AS5) L = L0(I-P) M = M0(I

- P) + Μι Ρ,

+

L1P,

dom M = dom M 0 + dom M i .

Lemma 3.5.1. Let all the conditions of (ASI)-(AS5) be satisfìed. Then the operator M is (L,p)-sectorial, and for the semigroup {Ut \ t G K + } and {Ft I t G I + } there hold formulas (3.2.4) and (3.2.5) respectively. Proof. Since the operators H and G are nilpotent, then ρ ρ {μΗ-Ι)-χ

= -Y^μ,lHq 9=0

and

(pG - I ) ' 1 = - Σ ßqGq 9=0

-Q)

+ (μΐ -

for every μ G C. Hence ρ ^L - M)"1

= -Σμ'Η'ΜϊΗΐ 9=0 -

Mo"1

Ρ Σ μ*α*(Ι 9=0

Si)~lL^Q

-Q)

+

-

nr'Q

for every μ G S'a (Si) = Se(Ti). Therefore, (L,p)-sectoriality of the operator M follows from the sectoriality of the operators Si and T\. Further, it follows from (3.5.1) that U* =

j ^ L i - Mi)~lLxe^

= ¿

/ ( -

άμΡ

-

P)

+ = ¿

Similarly, from (3.5.2) there follows (3.2.5).

- Mi)-lLiP)&ut j

RÌ(M)e^ •

άμ

άμ Vi G Κ+ ·

76

G. A. Sviridyuk and V. E. Fedorov

Theorem 3.5.1. Conditions (AS1)-(AS5) the operator M is strongly (L,p)-sectorial.

are satisfied precisely

when

Proof. Necessity of these conditions for the (Z,p)-sectoriality of the operator M was established earlier in this chapter. It remains to prove their sufficiency. Let us first show strong (L,p)-sectoriality of the operator M on the left. o · o Let us assume T= + dom T\. By the construction, the lineal Τ is dense o in T. Let f GJF, then M(XL - M)-xLfM(M)/

= T,(XI - Τι)-1

f[(ßqI

-

9=0

= Here Qf G dom T\ by the construction, therefore,

T^Qf

Ρ (XI-T1)-1l[faqI-T1)-1T1Qf. 9=0

\\M{\L-MYlLLM{M)f\\T = H(A7 — Τ ι ) - 1 Π Ο ν - Τ ! ) -

1

^ / ^ < const \\T.Qfy / (|A| f [ K | )

9=0

9=0

by virtue of sectoriality of the operator T\. Further, ||(ÀL-M)- 1 Lj i i P ) (M)|| £ ( ^ ) < const II(λ/ -

Τι)-1

ρ ρ 1 Π ( μ 9 / - Τι)" 1| < const / (|λ| Π |μ,|J. 9=0

9=0

Remark 3.5.1. When the operator L is continuously invertible, conditions (AS2)-(AS5) are trivial and allowing for Remark 3.4.6, we can assert that Theorem 3.5.1 generalises the Solomyak—Yosida theorem. Remark 3.5.2. It follows from Theorem 3.5.1 that for every q e Ν a strongly (L,p)-sectorial operator M is strongly (L,p + g)-sectorial, since the number ρ in condition (AS4) can always be assumed large.

Chapter 3. Relatively p-sectorial operators

77

Remark 3.5.3. Since from conditions (AS1)-(AS5) there follow conditions (CS1)-(CS5), a strongly (L,p)-sectorial operator M is strongly (L,p)-radial.

3.6.

DEGENERATE INFINITELY DIFFERENTIABLE SEMIGROUPS OF OPERATORS

Let U and Τ be Banach spaces, operator L G C(U-,T), and operator M G Cl{U\T). Definition 3.6.1. An operator M is called (ρ, α, ß)-sectorial with respect to operator L with the number ρ G No (or, briefly, (L,p,a, β)-sectorial), if (i) there exist constants α G M and a G (0,1] such that the set Σ α = {μ G C | Re (μ - α) > - | Im (μ - α)| Ω } C pL{M)· (ii) there exist constants Κ G M+ and β G (0, α] such that /

p

max{||i^ > p ) (M)|| £ ( w ) , | | L f M ( M ) | | £ ( ^ } < Κ / J ] Iwt - R}. L e m m a 3.6.2. Let an operator M be (L,p, α, ß)-sectorial. Then the lengths of all chains of relatively adjoint vector are restricted by the number p. L e m m a 3.6.3. Let an operator

M be (L,p, α, β)-sectorial.

(i) kerR^p)(M)

nimR^jM)

(») ker Llp)(Μ)

ΓΊ i m L f ß p ) ( M ) = {0}.

Then

= {0};

Theorem 3.6.1. Let an operator M be (L,p, α, β)-sectorial. Then there exists an infìnitely differentiable solving semigroup {Ut \ t G R + } ({Ft I t G R + } ) of equation (3.3.1) ((3.3.2)), and its operators are specifìed by integrals of the Dunford—Taylor type ' = òlTL¿M)el"""

(F'

u

- h

f

r

d

"

)

·

''

(3 6 3)

where res,

Γ = {μ = 1 — (1 + |y|)Q + iy \ y G M}.

(3.6.4)

Proof. It is easily verified that Γ C Σ. Let us consider an integral ^ ¿ ^ y ^ M ^ d / i ,

t e R f , fc = 0 , l , . . . .

(3.6.5)

Using the Cauchy theorem let us show that in operators Glk contour Γ can be substituted by Γ' obtained from the contour Γ (3.6.4) by the right shift parallel to the axis x. To this end, it must be shown that integrals over the horizontal lines Γ+ = {μ = χ + iR \ χ € [1 - (1 + R)a, b - (1 + R)a]}, Γ_ = {μ = χ - iR I χ G [1 - (1 + R)a,b - (1 + Ä) a ]} connecting the

Chapter 3. Relatively p-sectorial operators

79

contours Γ and Γ' tend to zero when R —> oo. Indeed, under sufficiently large R according to Lemma 3.6.1 [

Jr+

μ^1μ(Μ)β^άμ

= Cl /

0, |μ| = v V + (1 - (1 + y)a)2 < c(l + y).

(3.6.8)

It follows from (3.6.6) that the integral G\ converges uniformly on t on every interval of the kind (to,T), 0 < ίο < T. Hence it follows that mapping U' : M+ —» C{U) is infinitely differentiate: dk ^

= 01

k = 0,1,2....

Inclusion im Ul C dom M

Vi G K+

(3.6.9)

holds by virtue of the integral representation of the semigroup, closedness of the operator M and the identity J ΜΕ^Μ)εμίάμ

= J μΏ^{Μ)βμί

άμ - J Lé11 άμ = LG\.

We showed that MU* - LG\. This implies that the semigroup {U1 | t G R+ } is solving for equation (2.3.1). For {F1 I t E R+ } the theorem is proved similarly.

•

Remark 3.6.4. It follows from (3.6.6) that ll^ll < Ce t i ( p + 1 ) ( ^- 1 ) / Q .

(3.6.10)

Clearly, similar inequalities are satisfied also for the operators Fl. Remark 3.6.5. Let a semigroup {U 1 \ t G K+} be solving for the equation Lii — Mu where the operator M = M — aL is (Σ,ρ,α,β) sectorial with a curvilinear sector Σ centred at the origin of co-ordinates (see Definition 3.6.1 and Remark 3.6.1). Then the semigroup of equation Lù — Mu will be a family of operators {Ut \ t G K+ } Ut = eat{jt

=

_L f 2πι 7r

¿μ

=

_L [ R{(M)eXtdX, 2πι ,/jv

Chapter 3. Relatively p-sectorial

operators

81

where contour Γ' was derived from the contour Γ with the shift on the constant α parallel to the real axis. At the same time H C 4 ^ ^ ) < (a+l)tt(p+l)(ß-l)/a_

Ce

Remark 3.6.6. In the conditions of Theorem 3.6.1 the following relations are obvious LU1 = FtL,

Μυ*υ, — FtMu

Vu E dom M

Vi G M+.

They follow from identity (2.1.8) and inclusion (3.6.9). The constructed semigroups, generally speaking, have no analytic extension to a domain. In addition, estimate (3.6.10) suggests the possibility of an unbounded increase in the norms of the semigroup operators under t

0+.

Example 3.6.1. Let us take U = Τ = h, L = I, M ( { a „ } ) = { - ( \ / ñ + 1 + ( - 1 ) n i n ) a n } , where { α η } G h (Hille and Phillips, 1957). Then the operator Μ (Ζ, 0,1/2,1/2) is sectorial. Semigroups in this case have the form J7(i)({a n }) = F ( i ) ( { a n } ) = { e - ^ + i + C - i ) " » " ) ^ } . It can readily be shown that they have no analytic extension from the positive semiaxis.

3.7.

P H A S E SPACES F O R T H E C A S E OF D E G E N E R A T E I N F I N I T E L Y CONTINUOUSLY D I F F E R E N T I A B L E SEMIGROUPS

Let us give a definition of the phase space. Definition 3.7.1. A closed set tion (3.2.3), if

V

C V is called α phase

space

of equa-

(i) any solution v(t) of equation (3.2.3) lies in V, i.e., v(t) 6 V Vi 6 M+; o (ii) for every VQ of the lineal dense in V, there exists a unique relaxed solution of problem (3.2.3), (3.3.7). Theorem 3.7.1. Let the operator M be (Σ,ρ,α,β) sectorial. Then the phase space of equation (3.3.1) ((3.3.2)) is imRf JM) (im L^ AM)).

G. A. Sviridyuk and V. E.

82

Fedorov

Proof. Let us show the existence of Cauchy problem relaxed solution for equation (3.3.1) for all UQ € By Theorem 3.6.1, the vector function UtUQ | K+ — U is a relaxed solution of equation (3.3.1). Therefore, according to the definition of solution it remains only to show that for every UQ G i m l i m t _ > o + Υ*Υ,Ο = UQ. Let us take vector UQ = By Theorem 2.1.2 arbitrary points μο,...,μρ can be taken from the Lresolvent set pL(M). We will choose the points outside the contour Γ satisfying (3.6.4). Then due to the continuity of the right L-resolvents and identity (2.1.4) utuo

^édi

R L x { M ) e X t d X R

= TT tf (M) (-L / 11 >\2niJr

(^

μρ-Χ

{ M ) v

ext dX + Riμ

»κ

2πι]τΧ-μρ

!

dX). ) (3.7.1)

The last integral is zero by the Cauchy theorem. Tend t —»· 0 + , then I - U%JrÌRÌk(M)±

[

In this case we can turn to the limit under the integration sign, because it converges uniformly with respect to the parameter t G [0,1] according to Wierstrass criterion. Let us prove that. Let us bound from above the module of the exponential function in (3.7.1) by a constant. Let us construct a circle SR(0) of radius R > 2\μρ\ centred at point 0. As Γ^, let us take a part of contour Γ that lies inside the circle 5^(0) and the right part of the circle SR(0) intercepted by the contour Γ. We obtain a closed contour that can be traversed in the negative direction. By we will denote the part of the contour Γ outside the circle SR(0), and the left part of the circle SR(0). In view of (L,p, a, /0)-sectoriality of the operator M, the integral over the contour Γ ^ tends to zero under R oo. Indeed, f


r, 2a/(C + 2) < r. If we take, say, α = 2r, C > 2, the conditions of (L,p)-sectoriality will be satisfied for a certain θ G (π/2, π) with a constant Κ = C p + 1 max {||P||i/, IIQII^r}, (for the right (L,p)-resolvent it is proved similarly using the identity (4.2.1) and with nilpotency degree not greater than for the ρ operator H). O Let us take a lineal J70 Θ Li[dom M\] as T- By construction, it is dense in the space J7, since the lineal Li[dom Mi] is dense in Ul because L\ is a o homeomorphism. Let the vector / ET, then it follows from identity (4.2.2) and the nilpotency of the operator G that IIM(XL - M)-lLLM{M)f\\r

= WT.R^)

R ^ Q f y k=0

k=Q

C{T1)·

Here, we used the fact that the operator commutes with its resolvent, and the vector Qf 6 dom T\ by the choice of / . It should be noted using the reasoning similar to the above that in the sector S%e(M) constructed by us there is a strong (L,p)-sectoriality of the operator M on the left. Its strong (L,p)-sectoriality is proved analogously. •

Chapter 4. Relatively σ-bounded operators 4.3.

95

RELATIVE σ - B O U N D E D N E S S A N D RELATIVELY A D J O I N T VECTORS

Let us consider a M-root lineal of the operator L for the case when the operator M is (L, a)-bounded. Theorem 4.3.1. point oo be

Let an operator M be (L,a)-bounded,

and the

(i) an essential singularity of the L-resolvent of the operator M. the M-root lineal of the operator L is contained in U°;

Then

(ii) a pole of order ρ ζ Ν of the operator M L-resolvent. Then the M-root space of the operator L coincides with U° and consists of M-adjoint vectors of the operator L whose height does not exceed p; (iii) a removable singularity of the operator M L-resolvent. Then ker L = U°, im L = and any eigenvector of the operator L has no M-adjoint vectors. Proof, (i) Let ψρ be a M-adjoint vector of the eigenvector ψο of the operator L. Then by Lemma 2.1.3 (i) we have ρ

1 fv~x Ψ ν = - Τ ~ · / X > V P - 9 - i d / i = 0. 2 m Jr^o

(4.3.1)

(ii) It follows from (4.3.1) that the M-root lineal of the operator L is always contained in U°. Let φ E U°. Let us assume Ψρ = ψ, 1

Ψρ—Ι = Ηφ,

...,

ψ0 = Ηρψ.

(4.3.2)

Since Η = M Q L Q the set { 1=1 g=0

Σ Σ ω Χ 1=1 ς=0

= °·

Then ¿=1 9=0

1=1

Hence alp = 0, I — 1 , 2 , . . . , k. It implies that

1=1 9=0

(=1

i.e., a l p = 0, I = 1 , 2 , . . . , k . If this procedure is applied furtheron, this gives us a contradiction. Hence, in particular, follows that the eigenvector φο and all its M-adjoint vectors entering into some single chain are linearly independent. It follows from Theorem 4.3.1 (iii) that the image im L and kernel ker L of the operator L are complemented in the spaces U and Τ respectively. Such operators are called double-splitting (Borisovich, Zvyagin, and Sapronov, 1977). Theorem 4.3.2. Let an operator M 6 ύ{ΙΑ\Τ) be σ-bounded with respect to a double-splitting operator L. Then every eigenvector of the operator L has a chain of M-adjoint vectors of a ßnite length. Proof. Let there exist a vector φο 6 ker L \ {0} which has chains of M-adjoint vectors only of an infinite length. Let us denote by coimL = U θ ker L some algebraic and topological complement to the kernel of the operator L, and by L the restriction of the operator L to coimL. By Banach

Chapter 4. Relatively σ-bounded operators

97

theorem, the operator L : coim L —l· imL is a toplinear isomorphism. The set {ψι,ψ2·, · · •}, where φ9+ι — L~1QMipq, q = 0,1,... is an infinite chain of M-adjoint vectors of the vector φο· This implies that a series 00

Φ= 9—0 converges absolutely and uniformly outside the circle {μ £ C | |μ| < WL^QMWc^} , where Q : Τ —> imL is some projector, whereas the vector ψ G ker(μΣ — Μ). The latter contradicts the (L, a)-boundedness of the operator M. •

4.4.

DEGENERATE ANALYTICAL GROUPS OF OPERATORS

As in the previous chapters, we will substitute the equation Lù = Mu for a pair of equations equivalent to it R^(M)ù = (aL — M)~lMu, 1

L%{M)f = M(aL - Μ ) " / ,

(4.4.1) (4.4.2)

which will be regarded as specific interpretations of the equation Av = Bv,

(4.4.3)

where operators A, Β G £(V), and V is some Banach space. The solution of the equation (4.4.3) is then a vector function ν E C^K; V) satisfying this equation. Definition 4.4.1. The mapping V· E C ^ K ^ V ) ) is called a group of solving operators of equation (4.4.3), if

(i) VSVt =

Vs, t E R;

(ii) for every VQ E V vector function v(t) = Vtvo is the solutions of equation (4.4.3). Let us identify the group with its set of values { V 1 \ t Ε M}. The group {V1 I t Ε Κ} will be called analytical, if it can be analytically extended to the whole complex plane retaining its properties (i) and (ii) from Definition 4.4.1.

98

G. A. Sviridyuk and V. E. Fedorov

Theorem 4.4.1. Let an operator M be (Σ,σ)-bounded. Then there exists an analytic solving group of equation (4.4.1) (equation (4.4.2)). Proof. Let the contour Γ = {μ G C | |μ| = r > α}. Let us consider the integral of Danford—Taylor type ßt U* = - ί - f Ru(M)e 2m JT μ

άμ,

t G R.

(4.4.4)

As can readily be seen, the mapping U~ G C°°(R;£(V)) in an obvious way can be extended to the whole complex plane C. Let the contour Γ' = {μ G C I |μ| = r' > r}. Then by repeating the proof of Theorem 3.2.1 we obtain UlUs = Ut+S Vi, s G C. Further, let i¿o G U. Then

at

- M)"1

U'uo = ^-.{aL ¿m

f μΣϋ^Μ^οε^ Jρ ^

άμ = (aL -

M^MU^o

by Cauchy theorem. (Let the contour Γ C C be the same as in (4.4.4). Then it is similarly easy to verify that the mapping Fl =

f L{{Μ)βμί άμ, 2m Jr

t G R,

(4.4.5)

will be an analytical solving group of equation (4.4.2)).

•

Remark 4.4.1. Projectors Ρ and Q from (4.1.1) are obviously the identities of solving groups {Ul \ t G R} and {Fl ( e l ) respectively. Therefore, Ul = UlP =

f {μΣχ - M\)~lL\Peßt 2m JT

= etsP,

(4.4.6)

where the operator S = L~[lM\ G C(Ul) by Theorem 4.1.1. Similarly, Fl = F*Q = etTQ, where Τ =

G C{Fl).

(4.4.7)

Chapter 4. Relatively σ-bounded operators

99

Definition 4.4.2. The set kerV- = { t ; e V | V * = 0 Vi G R} is called a kernel, and the set imV· = {v G V I ν = V°v} is called an image of the analytical group {V1 | t G R}. Obviously, ker V = k e r V \

imV' = imVt

Vi € R,

therefore Definition 4.4.2 is correct. In view of Remark 4.4.1, the following corollary holds true Corollary 4.4.1. In the conditions of Theorem 4.4.1 kerU-=U°, Definition 4.4.3. tion (4.4.3), if

imU'=Ul\

kerF =F°,

imF~ =

.

A set V C V is called a phase space of equa-

(i) any solution ν = v(t) of equation (4.4.3) lies in V, i. e., v(t) G V Vi G R; (ii) for every vo G V there exists a unique solution ν G C^R; V) of Cauchy problem υ(0) = VQ for equation (4.4.3). Theorem 4.4.2. Let an operator M be (L,a)-bounded, whereas σο is a removable singularity or a pole of order ρ G Ν of the L-resolvent of the operator M. Then the phase space of equation (4.4.1) (equation (4.4.2)) coincides with the image of the solving group (4.4.4) (solving group (4.4.5)). Proof. As in the previous chapters, belonging to the subspace U l will be proved only for solutions of the class C P+1 (R;W), where ρ is the order of the pole or zero for the case of a removable singularity. For an arbitrary solution from C^RjW) in the case of a pole, the statement has been proved by Fedorov (2001b). Let u = u(t) be the solution of equation (4.4.1). Let us assume u — (I - P)u + Pu = u° + ul. Then by Theorem 4.4.1 we have (off - I)~lHu0

H = MQ1LQ ,

= (oH - /)" V , 1

1

[αϊ - 5 ) " ¿ = (αϊ - S)'

1

1

Su , S = L^Mi.

(4.4.8) (4.4.9)

G. A. Sviridyuk and V. E. Fedorov

100

Let oo be a pole of order ρ 6 Ν of the operator M L-resolvent. Then we obtain from (4.4.8) by successive differentiation o = HP+l

dtp+l

u0 = Hr^u° dtP

= ... = Hù° = uQ.

Therefore, if u = u(t) is the solution of equation (4.4.1), then u = ul (t) E U1 Vie R Further by Theorem 4.1.1 operator S G C(Ul). Therefore, for every ií¿ G Ul there exists a unique solution of Cauchy problem u(0) — uj, of the form u(t) = etsul. Thus, U1 is the phase space of equation (4.4.1) and at the same time (see Corollary 4.4.1) it is also an image of group (4.4.4). (The theorem statement with respect to equation (4.4.2) and solving group (4.4.5) is established similarly). • E x a m p l e 4.4.1. Let U = Τ — loo- We will write the vector u G U as u — (ui, i¿2, · · ·). Operators L, M G C{U·,Τ) will be determined by the formulas Lu = (u2,i¿3/2,u4/3, ...), M = I. Since jk

_ ( u k+1 v-k+2 2u fc+3 2 · 3ufc+4 V k\ ' (k + 1)!' (k + 2)!' (k + 3)!

\ '

then I I Lk\\c(u) = Therefore, the series oo (/zL-M)-1 = - ^ ß k L k=0

k

converges uniformly and absolutely on any compact in C. This implies that the operator M is (L, a)-bounded and oo is the essential singularity of the operator M L-resolvent. It is readily seen that Cauchy problem Lii = Mu,

u(0) = 0

has one more solution in addition to the obvious stationary solution: u(t) = (í,í 2 /2,í 3 /3, . · ·),

« e C 1 ((—1,1 )·Μ).

Chapter 4. Relatively σ-bounded operators

101

Remark 4.4.2. Further for brevity the cases "oo is removable singularity" and "oo is pole of the order ρ E W will be combined in the case "oo is nonessential singularity". The order of such a singularity will be taken to be the order of the pole or zero for the case of a removable singularity.

4.5.

SUFFICIENT CONDITIONS OF T H E RELATIVE σ - B O U N D E D N E S S

Let U and Τ be Banach spaces, and operators L,M E In applications, it is rather difficult to verify the (L, a)-boundedness of the operator M only on the basis of Definition 4.1.1. Therefore, we will define some necessary conditions of the (L, cr)-boundedness of the operator M, connecting only operators L and M, and will prove their sufficiency. ( A l ) The length of any chain of M-adjoint vectors of any eigenvector of the operator L is bounded by a number ρ Ε Ν. Let us denote by U° an M-root lineal of the operator L. (A2) U° is a subspace complemented in U. Let us denote by U 2 = Kohi 0 some algebraic and topological complement and let us assume JF° = M[U°], T2 = M[U2]. (A3) Τ = Τ* Θ Τ 2 . Let us denote by MQ the restriction of the operator M in U°. (A4) There exists an operator MQ1 Ε

£.{Τι\ΙΑι).

Let us denote by Lo the restriction of the operator L in U°. By construction, the operator Lo E £(U°]!F0). Let us assume Η =

MQ1LQ

E £(U°).

Lemma 4.5.1. Let all the conditions (A1)-(A4) be satisfìed. Then the operator Η is nilpotent, its nilpotency degree not exceeding the number p. Let us denote by ¿2 (M2) the restriction of the operator L ( M ) in the subspace U2, and by P2 (Q2) let us designate a projector along U° (J70) in U 2 {T 2 ). Note that by Banach theorem there exists an operator L^ 1 E C{T2\U2). Let us introduce into consideration an operator S9¿ equal to the restriction of the operator MQ1{I — Q)M in U2 and an operator T2 equal to the restriction of the operator L^Q2M in U2. Obviously, 5° G £(U2]U°), T2 E C{U2). Let us construct a set Ul = {uEU

\ u = (I - R)u2, u2 E U2} ,

102

G. A.

Sviridyuk

and

V. E.

Fedorov

where the operator ρ

R = Y^H q SlT¡eC{U 1 - ) U Q ). 9=0 Lemma 4.5.2. is a subspace

in U,

Let

all

the

conditions

supplementary

to

(4.5.1)

(A1)-(A4)

be satisfied.

Then

U

l

U°.

Proof. Let us construct an operator P\ = (I — R)P2 and show that Pi is a projector along U° in Ul. The continuity of the operator Pi is evident, and the idempotency follows from the fact that P2R = O by construction. Thus, Pi is a projector, and im Pi = U1 and kerPi D U°. Let u G kerPi, then u = (I - P2)u + P2u = u° + U1. It implies that Pi it = P\u2 = u2 - Ru2 = 0. Since P 2 Pi = P 2 , P2P\u

= P2u2

= 0 , i . e . , u2

Let us assume

T

Lemma 4.5.3. is a subspace

l

= 0.

=

Let

•

1

L\U }. all

the

conditions

in T , supplementary

to

(A1)-(A4) T

0

be satisfied.

Then

T

l

.

Proof. Let us construct an operator Qχ = L P i L ^ Q i and let us show that Qi is a projector along to F1. The continuity of the operator Q\ is evident. Further, QÌ

= LPl{L^Q2L)PlL^1Q2

= L(P1P2)P1L^1Q2

=

Q1,

since Q2L = LP2 = L2P2 by construction, hence P2 = L^QiL. The equality P\P2 = Pi is evident. Thus, Q\ is a projector, and at the same time kerQi D imQiC^"1. Let / € kerQi, then f = (I - Q2)f + Q2f = f° + f2. Hence, Qif = Q i f 2 = 0. Let us assume u2 = L ^ f 2 , then Q i f 2 = LPiu2 = L(u2-Ru2) = f 2 + g, where g = -LRu2 e J70. It implies that Q2Qif2 = / 2 = 0. Thus, kerQi = F Q . Let / G

then / = Qif 1

Lu1

= LPxu2

= QxLPxL^f2

=

L P i L ^ f

= L P

x

L ^ f

2

.

2

=

Hence / ,

i. e., imQi = F · • Let us denote by L\ (Mi) a restriction of the operator L (M) in U1. By construction and by Banach theorem there exists an operator Lj"1 G

Chapter 4. Relatively σ-bounded operators Lemma 4.5.4. Let all the conditions (A1)-(A4) operator Mx G C{Ul]Tl).

103

be satisfìed.. Then the

P r o o f . Since the operator M :U —l· F is linear and continuous, it will be sufficient only to show the existence of the action M : U l —>· Let v} G Ul,

then

ρ = M(/-^(M0-1Lo)9M0-1(/-Q2)M2(L2

Mu1 = M(I-R)u2

1Q2M2)9)U2

9 =o

ρ

= (J - ^(LoM^ni

- Q2){M2L^lQ2y)Q2Mu2

.

9=0

Q\Mul

= LP1LÎ1Q2MU1

= L(I

-

H)L^LQ2Mu2

ρ = (Q2 - ^(LoM^ni

-

Q2){M2L^Q2)q)Q2Mu2.

9=0

Thus, Mu1 = QiMu1.

•

Let us denote by S\ the restriction of the operator L^LM\P\ operator S\ G C(Ul),

The

consequently,

3a G M+ where p(Sx)

in U1.

Υμ G C

(|μ| > α) =>· (μ G p(Si)),

is the resolvent set of the operator S\. Note in addition that

by virtue of the nilpotency of the operator Η

9=0

Theorem 4.5.1. Let all the conditions (A1)-(A4) be satisfìed. Then the operator M is (L, a)-bounded, and oo is a nonessential singularity of the operator M L-resolvent. Proof. Let μ G C, |μ| > α. Then (/iL - M ) ' 1 = (μ£ 0 - M o ) ' 1 { I - Q i ) + = (μΗ - I)~1Mq1(I

p = - Σ 9=0

•

-

M ^ Q i

- Qx) + (μΐ -

S^L^Q, oo

μ Ί Ρ Μ ^ (J - QO + Σ 9—1

^qSrlL-xlQi.

104

G. A. Sviridyuk and V. E. Fedorov

Remark 4.5.1. Projectors Pi and Qι coincide respectively with projectors Ρ and Q from (4.1.1). Theorem 4.5.1 establishes sufficient conditions of the operator M (L, σ)boundedness. Nevertheless, it is still highly laborious to verify the conditions (A1)-(A4) in applications. Therefore, let us show the conditions sufficient for the operator M (L, a)-boundedness in the case of double-splitting operator L, which will prove simpler that the conditions (A1)-(A4). Thus, let L G C(U] T) be a double-splitting operator and the operator M e £{U;F). ( B l ) Every chain of M-adjoint vectors of any eigenvector of the operator L has the length equal to ρ G Ν. Let us denote by coim L = U θ ker L some algebraic and topological complement to the kernel ker L. Let L be the restriction of the operator L in coim L. By Banach theorem, there exists an operator L~l G £(im L; coim L). Let condition (Bl) be satisfied, then there exist lineáis U0q = L-lM[U0q-1},

U00 = ker L,

q=

l,2,...,p.

Obviously, M[U0p} Π im L = {0}. (B2) M[U°p] ® im L = T. Thus, let conditions (Bl), (B2) be satisfied. Let us denote by Po (Qp) a projector along coimL (imL) in U00 (M[U0p}). Let us assume Qp = I — Qp and construct an operator A = L~lQpM. Note that A[U0q] =U0q+\

q = l,2,...,p,

A[U0p] = { 0}.

Hence it follows that (4.5.2) Let us construct an operator D — QPMAP : U M[U0p]. By definition D[U00] = M[U0p] and D G C(U; M[U0p]). In addition, it can be seen that ker D Π U00 = {0}. Let us denote by D the restriction of the operator D in U 00 . By Banach theorem, it follows from the above that there exists an operator D~l : M[U0p] U00. Let us construct operators Pq = AqD~1QpMAp~q,

g = 1,2,.

· ·

,Ρ·

(4.5.3)

Chapter 4. Relatively σ-bounded

operators

105

L e m m a 4.5.5. Let the conditions (Bl) and (B2) be satisfìed. operators Pq :U —> U0g are projectors, and at the same time Pq[U0r} = { 0},

9,r

= 0,l,...,p;

Then

q + r.

Proof. Obviously, Pq G £(U), q = 0 , 1 , . . . ,p. In addition, P92 = A"(D-l{QpMAp))D-lQpMAp-'i

= Pq .

Further, Pq[U0r] = AqD^1 QpMAp~g[U0r]

= {0},

q φ r,

and finally, the relation imP g = U0q, q — 0 , 1 , . . . ,p, is obvious. Let us assume

U°

=

Θ U 0p

•

.

9=0

L e m m a 4.5.6. Let the conditions (Bl) and (B2) be satisfìed. Then U° is an M-root lineal of the operator L and the condition (A2) is satisfìed. Proof. Let us construct an operator P = ¿P

9

.

(4.5.4)

g=0

By Lemma 4.5.5, Ρ is a projector to U°. Thus, the condition (A2) is satisfied. Let us show that U° is an M-root lineal of the operator L. Let a vector φ G ker L \ {0} (C U°) have a chain of M-adjoint vectors such that there exists a vector (pq 0 U°, 1 < q < p, but ψτ+ι, · · ·} such that r centred at zero, intercepted by the initial contour, and the branches of the initial contour remaining outside the circle. Let the new contour retain the previous notation. Then for every φ e ker U~ ΙΊ dom M, f 6 ker

1 f e ^ p 2 mjr μ-Χ

1 Γ 2m J γ

t E M+

L_M)-iLe(»-X)t

d Ύ

'

*

_At

= φ- e C/V = Ψ,

{XL M)

- 2VÍÍY

Ϊ ά μ

—Χ 1

f e^-^4

e~xt

Γ

, =

.

f-e-»F'f

by Cauchy theorem, continuity of operator L and closedness of M.

=f

116

G. A. Sviridyuk and V. E. Fedorov Further, (μΣ -2πϊf JT

9¥*) + Σ β Ω λ λ ί / ( λ - λ λ ) ( η 0 ι ^ ) ^ λ=λ*. k=\ oo ' + ^ {eaxkt^x~x^ fc=l

- l)

(/.\—VkVk) „ aXk

Here, {(fib} and {λ^} axe sets of orthonormalised eigenfunctions and their respective eigenvalues of the homogeneous Dirichlet problem for the Laplace operator in the domain Ω, numbered with respect to nonascending of the eigenvalues allowing for their multiplicity. The primed summation symbol denotes the absence of terms with numbers k such that λ =

G. A. Sviridyuk and V. E. Fedorov

126

Proof. The proof consists in applying Theorem 5.1.1. Note only that since

and the series in the right-hand sides are uniformly and absolutely convergent in the norm of the space li, termwise integration is admissible. • Remark 5.1.2. From here on, the absolute term / = f{x) is considered independent of t. This is done to simplify calculations and to obtain a more interesting result. Namely, set Aj in this case appears to be a phase space. Example 5.1.3 [Linearized system of Oskolkov equations]. Let Ω C Mn be a bounded domain with the boundary Un- the orthoprojector associated with this splitting. The o restriction of the projector Π to the space Η2 Π H 1 C L2 is a continuous _ o O O operator Π : Β2 Π Η 1 ->• Η2 Π Η 1 . Let us thus represent the space tfili1 as a direct sum Θ Ηξ, where IHÇ = kerll, = imIL There occur dense embedding £ C 1HÇ and continuous dense embeddings i f C I i and IHÇ C Ηπ-. The space IHÇ consists of vector functions equal to zero on L 2 with a discrete negative spectrum σ(Α) of finite multiplicity condensing only at minus infinity. (ii) Formula Β : u -4 vV2u Β : Ήξ © E f t

specifies the linear continuous

operator

->L2.

(iii) Formula C = u —> — V ( V - u ) specifies the linear continuous C : Eg θ IHÇ

operator

L 2 , while i m C = H » , ker C = Kg.

Let us assume Σ = I — Π and denote by A ( Β ) the restriction of the operator Σ A (ΣΒ)

on Eg.

L e m m a 5.1.4. The operator A : Kg —H^ is linear and continuous; its spectrum σ(Α) is discrete, negative, of finite multiplicity, condensing only at minus infinity. Let us assume Ax — I — χΑ. Let us choose the parameter χ such that χ - 1 ^ σ ( Α ) Π σ(Α). Let us denote by Αχσ (Αχτ) restriction of the operator Σ Λ χ ( I M - ^ o n H Ç (H,). L e m m a 5.1.5. Let χ - 1 0 σ(Α) Π σ(Α). ΙΗΙσ (Αχπ

: ΙΗ^ —)· ΗΙ^) is a toplinear

Then the operator Αχσ

: fflÇ

isomorphism.

Proof. The proof of the lemma for the operator Αχσ follows from Lemma 5.1.4. Therefore, let us prove the lemma only for the operator Α χ τ . The operator Αχτ : H^ -> Kg is continuous by construction. Let us show that it is bijective. First, let us establish injectivity. Let / π G ker Αχπ. Then A~lU = ησ Ε Ήζ, i.e., / π = ιισ. Hence Αχσησ = 0, consequently, η σ — 0, and therefore / π = 0. Now let us establish the surjectivity. Let u n G 1HÇ. Let us assume ua = -Α~0ΣΑχιιπ G Η ξ . ΤΙιβηΣΛχίίσ+Σ-Αχίί,Γ = 0, ΙΙΑχιισ+Ι1Αχΐίπ = /π. Hence Αχ{υ,σ + ηΈ) = / π , i.e., ησ + ιιπ = A^fπ, uπ = Λ χ π / π . • By using the natural isomorphism of the direct sum and Cartesian prodO uct of Banach spaces, let us represent the spaces tf ΠΗ 1 and L 2 as ïï£ x IHÇ and Ho- χ IH^ respectively. Let us assume U = Eg χ Ης; χ Hp,

J~ — H(j χ H^ χ Hp,

The element u G U is of the form u = (ησ,ηπ,ηρ),

Hp = H«·. ησ = Ση, ηπ = Πΐί,

Up = Vp; and the element / G Τ has the form / = (/σ,ίπ,ίρ), U = Π/.

fa — Σ / ,

128

G. A. Sviridyuk and V. E. Fedorov Lemma 5.1.6. (i) Formula

L =

(

Aχσ

Π ΑΧΣ

ΣΑΧΠ

0\

UAXU

o

O

V

o /

specifìes the linear continuous operator L : U ker L = { 0 } x { 0 } x Bp, im L = Β , x ΙΗ^ χ { 0 } ; (ii) formula Μ —

/ΣΒ

ΣΒ

IIB

ΠΒ

-I

C

Ο

V Ο

T.

If χ'1

i σ(Α),

then

θ\

specifies the linear continuous operator M : U -

T.

The proof is obvious in view of Lemmas 5.1.3-5.1.5. The reduction of problem (5.1.13), (5.1.14) to the Cauchy problem (5.1.1) is completed. Lemma 5.1.7. Let χ - 1 0 cr{A) Πσ(Α), then the operator M is (L,o)bounded, while oo is the pole of order 1 of the L-reslovent of the operator M. Proof. By Lemma 5.1.6 the operator L is double-splitting. Therefore, to prove the lemma in view of Theorem 4.5.2 it is sufficient to show that each vector φ G ker L \ { 0 } has exactly one M-adjoint vector, and that JF01 © i m L = T. Thus, let ψ € ker L \ { 0 } , then by Lemma 5.1.6 (i) vector φ — (0,0, φρ), φρ Φ 0. Hence Μφ = (0, — φρ,0) G imL. Let us find the vector ψ 0 ker L \ { 0 } such that Lip = Μφ. For it we have the following system of equations Αχσψσ

+ ΣΑχψπ

= 0,

ΤΙΑχφσ + ΠΑχψπ =

-ψρ.

(5.1.15)

Hence by Lemma 5.1.4 ψσ = —Α~^ΣΑχ·φπ. Therefore, if φρ = 0, then φ σ = 0, and, consequently, ψρ = 0. However, ψρ φ 0 by the condition, and, consequently ψρ φ 0. Hence ( Μφ =

ΣΒ(φσ

+ φπ)

ΠΒ(φσ + φπ)-φρ

\ since by Lemma 5.1.3 (iii) Οφπ φ 0, if φ φ 0.

I ^ imL,

Chapter 5. Sobolev-type

equations

129

Now we have only to prove the existence of the vector φ G ker L \ {0} satisfying the system (5.1.15). To this end, consider the operator (ΣΑ~ιΣ L~l =

Π-Αχ 1 Σ V

ΣΑ"1 Π Α-χπ

Ο

Ο

G C(U),

l

Since /Σ l

L~ L =

O

O

π

lo

o

LL~

(ν = 0

0

{o

o

π

the components φσ and ψπ of the vector φ can be obtained from the equalities φσ = —ΣΑ~1ψρ, φπ - —Αχπψρ, and the component φρ can be chosen arbitrarily. Let us now verify the statement J®1 Θ im L = T. Let us assume = ker L,

coimL = i g χ Εξ χ {0} .

By using the operator L~l we obtain J^00 = M[U00} = {0} χ H* χ {0} C im L, U 01 =

= Σ Α " 1 ^ ] χ Αχ,pH*] χ {0}.

Since Αχ7Γ[Η^] = EÇ by Lemma 5.1.4, U 0 1 = Σ Α ^ Α ' ^ Έ ξ ] χ Εξ χ {0} C coimL. Hence T01 = M[U01} = ΣΒ(ΣΑ-1Α~1+Ι)[Έξ]

x ηΒ(ΣΑ~1Α~1

+ /)[E£] x C[Ήξ].

Since 1 ΣΑχ^Αχχ + 1 — ΣΑχΎ 1 Αχτ Α χ π Α γ Ι — (ΣΑ^,1 + Π Α Ύ 1~+ ^χπ^χπ χ χ/ ) Α Χ7χΓ^ — χ "ΧΤΓ)

then JF01 = Σ.5ΣΑ" 1 Α~,}(7 -1 [Hip] χ Π ^ Σ Α ^ Α ^ ό " 1 ^ ] xBpÇC im L,

G. A. Sviridyuk

130

and V. E.

Fedorov

where the operator C~l is inverse to the restriction C of the operator C on The existence of a continuous inverse operator C~~l : IHI^ —>· IH^. follows from Lemma 5.1.3 (iii). Further, let us assume /o

o

O

O

O

\o

O

Π

Po = where

3 P¡2 \

o\ ,

3 Π O

Pi —

\o

O

O/

P¡2 = ΈΑ~ιΑ~1\ O

o

O

Π

\o

o

Qf o

Qo =

(o

\ Qi =

o \o

/

O Ql3\ o Qf o π

)

where Qf = ΣΒA~lA~lC~l, Qf = Π B A - l A ~ l C ~ \ Qf - - Q f . It can be readily verified that the operators Pjt : U Uok, Qk : Τ -> k = 0,1 are projectors, while Po Pi = P\Pq — O, QoQi = QiQo = Û. Therefore, the operator Q = I — Q\ is also a projector, and at the same time im Q = im L, ker Q = T01. It implies that Θ im L = T. • As in the previous case, we will confine ourselves to the case / = f ( x ) . Therefore, according to Remark 5.1.2 we find the phase space of problem (5.1.13), (5.1.14) Af = {ueU\(I-

Q)(Mu

+ f ) = 0},

where the vector / = ( / σ , / π , 0), / σ = Σ / , / π = Π / ; and the projector I — Q by virtue of Remark 4.5.2 is equal to Qo + Qi· Since QoQi = QiQo = O, ((Qo + Qi){Mu

+ f ) = 0) O ((Qo(Mu + / ) = 0) Λ (Qi(Mn + / ) = 0)).

Further Q1(Mu + f ) =

QfC

C

QfC

C

C

(ησ\ = 0

o j \UP)

precisely when η π = 0, which implies that Qo(M(col (Μ σ ,0,up)) + f )

( =

O

O

o

Π BY,

Π Ρ Π + Qf Π

-Π

O

o

\ (ησ\ 0

o / \upJ

/ 0 \

+

U

voy

= o

Chapter 5. Sobolev-type

equations

131

precisely when U B u a + / π = u p . Ultimately we obtain Af = {ti 6 U I HBua +

= up, ν,π = 0} .

Thus, we have proved Theorem 5.1.4. For every f G L 2 and every UQ G Af there exists a unique solution u € C^M; Af) of problem (5.1.13), (5.1.14); and the set Af is the full complete affine manifold, homeomorphic to the subspace U1 = {u e U I ι% = 0, Π Β η σ = Up}.

5.2.

T H E CASE OF A RELATIVELY p-SECTORIAL OPERATOR

Let operators L G C{U] T) and M G while the operator M is a strongly (L,p)-sectorial. Let I = [0, a) be some half-interval. Let us take a vector function / G C°°{Ia]T) and let us consider the Cauchy problem Lu = Mu + f ,

u(0)=uQ.

(5.2.1)

By Theorems 3.3.1, 3.4.3 and Corollary 3.2.1, problem (5.2.1) is split into two problems HÙ0 - u° + Mo" 1 / 0 ,

u°(0) = ug,

Ú1 = Su1 + Lf1/1,

u1(0)=«o·

(5.2.2) (5·2·3)

By Corollary 3.2.3 the operator H is nilpotent with its nilpotency degree not exceeding p. Therefore, problem (5.2.2) is solved exactly in the same way as problem (5.1.2). That is, condition «δ = - έ ^ Μ

0

-

1

^ ( 0 )

(5.2.4)

>

9=0

is necessary and sufficient for the existence of a unique solution u° G Cl{Ia-,UQ) of this problem, representable in the following form: u(t) = -j2H'>MQ-1^-(t), 9=0

te Ia.

132

G. A. Sviridyuk and V. E. Fedorov

Let us consider problem (5.2.3). According to Corollary 3.4.4, the operator S : dom S —>· Ul is sectorial. Hence by virtue of the classical results there follows the existence of a unique solution ul G Cl(Ia\Ul) of problem (5.2.3) with locally Holderian function f1, of the form rt

ul(t) = etsu¡ + Γe^-^L-1/1^) Jo Ό

ds,

t G Ia.

(Here, {etS | t G R + } is a semigroup generated by a sectorial operator S). Similar to Section 5.1, let us introduce into consideration the set Af = {ue

dom M I (/ - Q) (Mu +

^ " ( 0 ) ) = θ}. 9=o

Since (I — Q)MPu = 0 for every u € dom M any vector u G Af is of the form 5=0 l

where u is an arbitrary vector from dom M\. It is obvious that condition (5.2.4) is satisfied if the vector UQ G Af. Thus, we obtained the proof of Theorem 5.2.1. Let an operator M be strongly (L,p)-sectorial. Then for every f : Ia Τ such that f° G C p + 1 ( / a ; T) and f1 : Ia Τ is locally Holderian and for any UQ G A¡ there exists a unique solution u G Cl (Ia;U) of problem (5.2.1) of the form u(t) = - ¿ HqMQl ^ - ( t ) + υ% dt9 g^O

+ Γ β^ϋΓ1/ J 0

1

W ds.

Remark 5.2.1. If the operator M is (L, a)-bounded, while oo is a nonessential singularity of the L-resolvent of the operator M, then, obviously, formulas in Theorems 5.1.1 and 5.2.1 coincide. Example 5.2.1 [Linear system of Navier—Stokes equations]. Let η = 2,3, be a bounded domain with the boundary d f l of the class C°°. In the cylinder Ω χ M+ let us consider the Cauchy—Dirichlet problem

ficK",

ν(χ,0) =ϋο(χ),

χ G Ω,

«(χ,ί)=0,

(χ, ί) 6 0Ω χ R f ,

(5.2.5)

Chapter 5. Sobolev-type

equations

133

for the system of equations vt = uV 2 v - Vp + / ,

0 = V · v.

(5.2.6)

Before we proceed to studying problem (5.2.5), (5.2.6), let us replace the incompressibility equation 0 = V · υ by equations 0 = V(V • υ). Note that by Green's and Gauss theorems we will obtain the system of equations equivalent to the initial one. Thus, let us consider the problem for a linear system of Navier—Stokes equations represented as: (5.2.7)

vt — uV ν — Vp + / .

Let the spaces H 2 , H 1 , L 2 , IH^, H^·, and operators Π, A, B, C have the same sense as in Section 5.1. Let us assume the spaces U = Τ — IHI^ χ IHI^ χ Hp, Bp = H^. The element u Ç.U is of the form u = (ησ,ηπ,ηρ), 7 and the element f £ J is of the form / = ( / σ , / π , / ρ ) · Let us assume Σ = I — Π. As is readily seen, the formula L = diag{E, Π, O} specifies a linear continuous operator L : U —> J7, while imL = Η^ χ H^ χ {0}, kerL = {0} x {0} x Bp. Lemma 5.2.1. The formula

M =

\

( ΣΒ

ΣΒ

ΠΒ

UB

-I

c

o J

Vo

specifies a linear closed densely defined operator M : dom M C U dom M = Hg χ χ Op.

T,

Let us proceed to studying (L,p)-sectoriality of the operator M. For this purpose, let us construct a L-resolvent of the operator M. Since (Σ(μΙ-Β) μΣ-Μ

=

—ELB \

Ο

-ΣΒ

O

Π ( μ ΐ -Β)

I

-C

(5.2.8)

Ο/

we must more closely study operators comprising the matrix (5.2.8). Introduce the notation Βσ restriction of the operator ΣΒ on . The operator Βσ : —• Ho- is linear and continuous while its spectrum σ(Βσ) is discrete, of finite multiplicity and condenses only to —oo. Let {φ^} be a family of orthonormalised in the sense of Ho- eigenfunctions of the operator Βσ,

G. A. Sviridyuk and V. E. Fedorov

134

numbered with respect to nonascending of the eigenvalues {Ajt} allowing for their multiplicity. Then Σ(μ7 - £ ) Σ = μΣ - Βσ — Σ^-^μ - Ajt)(-, where ( ·, · ) is a scalar product in Hb. Let μ φ X&, then there exists a continuous operator {μΣ-Βν)-1

oo =

k=1

Y/(ß-\k)-1(-,

k

- ß X l

λX ~- λ X* k

{•,'(9)(0)} 9=0

there exists a unique strong solution of problem (5.5.1), (5.5.2), of the form ρ f'(t) = - YiGyM^il t^o

rt

- P')u'^{t)

f

+ F' /Ó + /

tft-'u'is)

ds.

Jo

Theorem 5.5.2 implies the obvious Corollary 5.5.3. If Banach spaces U and Τ are reñexive, then the operator M is strongly (L,p) -sectorial precisely when the operator M' is strongly (L',p)-sectorial.

148

G. A. Sviridyuk and V. E.

Fedorov

Prom Theorem 3.2.1 and Corollary 5.5.3 there follows the existence of the semigroups Un = ¿

'

F t

=

J

Òi

(M') exp (μί) άμ,

t G I+,

( M , ) eXP

1 €

acting in the spaces li' and T' respectively. The identities of these semigroups, projectors Ρ' and Q' respectively, split the spaces W and T' into a direct sum U'=U'°(BUn,

=

where U'° = kerP', U'1 = imP', JF'° = ker Q',

= im Q'.

Corollary 5.5.4. Let Banach spaces U and Τ be reflexive, and the operator M be strongly (L,p)-sectorial. Then (i) U° and Ul are annihilators U'1 and U'° (ii) J70 and Tl axe annihilators Tn and

respectively; respectively;

If an operator M be strongly (L,p)-sectorial, then the statements of the Remark 5.5.1 are also true. By Theorems 3.4.2, 3.4.3, 5.5.1, Corollaries 3.4.2, 5.5.3, 5.5.4 and Remark 5.5.1 problem (5.5.1), (5.5.2) will split into two problems G'f'° = f'° + u'°,

f'°{0) = fo,

(5.5.6)

f'^T'f'

fn(0)

(5-5.7)

+ u'1,

= fó\

where the operators G' = (L 0 M 0 "*)' = M¿ _1 Z/ 0 G C{F°), Τ = ( M j L " 1 ) ' = operators LQ, MQ, M[ are restrictions of operators L' and M' on the subspaces J7'0 and T'X respectively, and the vectors F'° = (I— Q')f, f'1 = Q'f, = M'0-L{I-P')u', un = L'{LP'u'. Obviously, problem (5.5.7) is solved using standard technique, because from Corollaries 3.4.4, 5.5.3 and the definition of a linear operator T" there follows its sectoriality. Thus, there holds L e m m a 5.5.5. Let Banach spacesU and J7 be reflexive, the operator M be strongly (L, p)-sectorial, and the function u G W¡+\U'1). Then for 1 l every /Q G T there exists a unique strong solution of problem (5.5.7), of the form

Chapter 5. Sobolev-type f'1(t) = F[tf^+

Jo

equations

149

ftF[t-snn(s)dsi

where {F^ \ t G R + } is the restriction of the semigroup {Fn onT'1.

| t 6 M+}

Lemma 5.5.6. Let Banach spaces U and Τ be reflexive, operator M be strongly (L,p)-sectorial, and vector function u E < q < oo. Then if

/οΛ = -Σ(^) β « Λ(ϊ) (0), Q will be later more exactly defined. We shall retain the notation l\ = (a, b) 3 0. Let there exist an operator A~l G £ ( £ ; V ) , then equation (5.6.2) is trivially reduced to the equivalent equation «(») = Cn-iv^-V

+ ... + Civ' + C0v + h,

(5.6.3)

where Cm = A~1Bm G £(V), m = 0 , 1 , . . . , η — 1, h — A~~lg. If an operator pencil C = ( C n _ i , . . . , C\, Co) is polynomially bounded, i. e., 3a G M_)_ Υμ G C (|μ| > a) =» (μ η Ι - . . . - μ Ο χ - Co)" 1 € £(V), then, as is well-known, (see, for instance, Gokhberg and Krein, 1965), for every vm G V, m — 0, l , . . . , n — 1, and h G C(/¿; V) there exists a unique solution ν G C™(/¿; V) of problem (5.6.1), (5.6.3), representable as n-l

ri V

v

«( 0

V/i G C

(|μ| > α) =Φ· Δ~ G C(G,V).

Remark 5.6.1. If there exists an operator A - 1 G £(Q,V), then the pencil Β is polynomially A-bounded precisely when the pencil (A~1Bn-1,... ,A~lBi, A~1BQ) is polynomially bounded Theorem 5.6.1. The following statements are equivalent (i) the operator M is (L, a)-bounded; (ii) the pencil Β is polynomially

A-bounded.

152

G. A. Sviridyuk and V. E. Fedorov

The proof immediately follows from Lemma 5.6.1 while the constants α and α may be taken equal. The term solution of a homogeneous equation (5.6.2) means here a vector function υ G C n (R; V), satisfying this equation. Definition 5.6.2. Mapping V· G C n (R;£(V)) is called a propagator of a homogeneous equation (5.6.2), if for every vector υ E V the vector function v(t) — Vtv is the solution of this equation. Let a pencil Β be polynomially A-bounded. Let us consider equation (4.0.1), where operators L and M are represented by matrices (5.6.5). By using the representation of the L-resolvent of the operator M from the proof of Lemma 5.6.1 and Theorem 4.4.1, it is easy to construct a solving semigroup of equation (4.0.1)

U'=

uh =

d

^k--

where

V

™

=

é¡L

- • • · - J W ) ^ Ίμ, m = 0 , 1 , . . . , η — 1,

(5.6.6)

are propagators of homogeneous equation (5.6.2) and at the same time by Theorem 5.6.1 the contour Γ C C may be chosen the same as in Theorem 4.4.1. This proves Theorem 5.6.2. Let a pencil Β be polynomially Α-bounded. Then along the entire axis R there exist propagators of the homogeneous equation (5.6.2) of the form (5.6.6) that can be analytically extended to the whole complex plane. Remark 5.6.2. If there exists an operator A~l € V), then the propagators of homogeneous equations (5.6.2) and (5.6.3) coincide. Otherwise, as it will be shown later, their properties are quite different. Similar to the construction of the group {Ul \ t € M} projectors Ρ and Q from Lemma 4.1.1 can be readily constructed, while Ρ = U°. The projector Ρ splits the space U into a direct sum U = U°@Ul while if UQ G Ul,

Chapter 5. Sobolev-type

equations

153

then

It follows from Theorem 4.1.1 that L(I - P) = L0{I - Ρ) and MP M\P. In addition, lim(/iZ - M)~l(I

-Q)

= -Mq1{I

- Q),

lini(L - \M)~lQ

=

=

L^Q.

Therefore, it is in principle possible to construct an operator Κ = Mq1(I - Q)L0(I - Ρ) = Η (I - P), which is nilpotent exactly when the operator Η is nilpotent. It follows from the above that Theorem 5.6.3. Let a pencil Β be polynomially same time let the operator Κ be nilpotent. Then (i) any solution υ = v(t) of the homogeneous inclusion

bounded,

and at the

equation (5.6.2) satisfies the

V i G Κ; (ii) for any initial data vm, m — 0 , 1 , . . . , η — 1, satisfying the equalities (5.6.7) there exists a unique solution υ = v(t) of problem (5.6.1) for homogeneous equation (5.6.2), and it is of the form n-l v(t)

= Σ

V™v™

v< e

R·

771=0

Let Vo G kerA \ {0}. By formulas Αφk+i = k = 0,1,..., jt = 0, I < k, let us introduce into consideration B-adjoint vectors of the vector i/o· If it is required that ψ^ φ ker A, k — 1 , 2 , . . . , then, as it can easily be seen, the vector i/'Jfc will be B-adjoint precisely when the vector = col {ip k -n-i,^k-n-2, • • · where Vfc-z = 0 under I > k, is the M-adjoint vector of the vector ψο = col ( 0 , . . . , 0, V>o) € ker L \ {0}. Hence follows the theorem Theorem 5.6.4. The length of the chain {ψι,ψ2,. • •} of B-adjoint vectors of the vector ψο E ker A \ {0} is fìnite precisely when the length of the chain {ψ\, ψ2, · · · } of the M-adjoint vectors of the vector ψο = col ( 0 , . . . , 0, ψο) is fìnite.

154

G. A. Sviridyuk and V. E. Fedorov

Recall the earlier employed condition connecting the operators L and M. ( A l ) The length of any chain of M-adjoint vectors of any operator L eigenvector is bounded by the number ρ G No. Let us also formulate a condition (Al) The length of any chain of B-adjoint vectors of any vector ψο G ker A \ {0} does not exceed a certain number ρ G NoCorollary 5.6.1. Conditions (Al) and (Al) are equivalent. Corollary 5.6.2. Condition (Al) is equivalent to the nilpotency of the operator Κ. Remark 5.6.3. Without loss of generality it can be assumed that ρ from the condition (Al) and the nilpotency degree of the operator Κ coincide. Corollary 5.6.3. Let an operator A be a Fredholm operator and let the condition (Al) be satisfied. Then the statements of Theorems 5.6.2 and 5.6.3 also hold true. Since the operator A is a Fredholm operator precisely when the operator L is a Fredholm operator, the proof immediately follows from Theorem 4.5.2. 5.7.

I N H O M O G E N E O U S C A U C H Y PROBLEM FOR HIGH-ORDER SOBOLEV-TYPE EQUATIONS

Let us consider an inhomogeneous problem (5.6.1), (5.6.2). By the solution of this problem we understand the function ν G C n (7¿, V), satisfying equation (5.6.2) and condition (5.6.1). Let a pencil Β be polynomially Α-bounded and let the condition (Al) be satisfied. Let g G Cp+1(Iba,Q). Taking / = col (0,...,0,g), we arrive at problem (5.1.1), where the operators L, M are given by the matrices (5.6.5). By Theorem 5.6.4 and Corollaries 5.6.1, 5.6.2 for the operators L, M the conditions of Theorem 5.1.1 are satisfied. According to this theorem, the solution u — col (v,v',... , u n _ 1 ) of problem (5.6.1), (5.6.2) can be represented as a sum of singular u° = col (υ°, v° , . . . , υ0" ) and regular u 1 = col ( υ 1 , . . . , υ χη ) parts, while the both parts are found respectively by the formulas (5.1.5), (5.1.4). Therefore, let us represent the solution υ = v(t) of problem (5.6.1), (5.6.2) as ν = + υ 1 , and let us call v° a singular part 1 and υ a regular part of ν and let us calculate each of the parts separately.

Chapter 5. Sobolev-type

equations

155

Let us begin with the singular part. Let us construct matrices Wq = K1MQ1{I -Q), q = 0 , 1 , . . . ,p, and by Wtg, I = 0 , 1 , . . . ,n - 1, denote the elements of these matrices in the extreme right column. Let vectors VQ, i>i, . . . , u n _i G V satisfy the conditions ¿I

n_1

y

m=0

t=o

= ^ +

^

9=0

jq ' dt*

/ = 0 , 1 , . . . ,n — 1,

i=0

(5.7.1)

then the vector function p

dfl

=

te[0,T),

(5.7.2)

9=0

is the singular part of the solution ν = v(t) of problem (5.6.1), (5.6.2). Further, applying Theorems 5.6.2 and 5.1.1, let us construct the regular part ν 1 = υ 1 (ί) of the solution υ = v(t) of problem (5.6.1), (5.6.2) vl

(t)

= y/Vivm n^o

= vl+

[*ν*-τ9(τ)άτ,

tell

(5.7.3)

Jo

where Vt = (2m)~ l fr Δ~ιβμί άμ, ί G i , in (5.6.6). Thus, by Theorem 5.1.1 there holds

contour Γ C C is the same as

Theorem 5.7.1. Let a pencil Β be polynomially Α-bounded and the condition (Al) be satisfìed. Then for every g G Cp+l(ll G) and any vq,vi, . . . , v n - i G V, satisfying the condition (5.7.1), there exists a unique solution of problem (5.6.1), (5.6.2) representable as υ = v° + vl, where vk — vk(t), k = 0,1 is given by the formulas (5.7.2) and (5.7.3) respectively. As has already been told, the operator A is Fredholm precisely when the operator L, given by matrix (5.6.5) is Fredholm. Therefore, by Corollaries 5.6.1, 5.6.3 and Theorem 4.6.1 there holds Corollary 5.7.1. Let an operator A be a Fredholm operator and let the condition (Al) be satisfìed. Then the statement of Theorem 5.7.1 holds true. Remark 5.7.1. On the face of it, it is rather hard to find operators W¡9 to verify condition (5.7.1). However, the search will be significantly facili-

156

G. A. Sviridyuk and V. E. Fedorov

tated if formula (5.7.2) is used. Indeed,

5 At) -- Σ WJ

»(0 = - Σ Ό·

9=0

9=0

(5'7·4)

In view of the arbitrariness in choosing g = g(t), let us take g(t) = go = const, then from (5.7.4) we obtain W^ = O. Let us take g(t) = got, then Wl = WQ, g(t) = got2, then W2 — Applying this procedure further, let us determine all W±, q = 0 , 1 , . . . ,p, by the relation

9=0

9=0

etc. Specifically, if η = 2 and ρ = 0, conditions (5.7.1) are transformed to the form

V0°v° + V i V = υο + w§g( 0),

at

+ vy)

t=0

=

v\

(5.7.5)

where W§ is an operator in the upper right corner of the matrix M 0 _ 1 (I—Q). If, on the other hand, η = 2 and ρ = 1, it follows from (5.7.1) that F 0 V + Vi0«1 =

+ < 5 ( 0 ) + WoV (0),

jt(V¿v° + Vy)\t=o = v' + W 0 V(0),

(5.7.6)

where WQ is the operator in the upper right corner of the matrix M 0 _ 1 (/ —

Q)LQ{I — P)MQ1(I — Q).

E x a m p l e 5.7.1. Let Ω C R m be a bounded domain with the boundary 9Ω of the class C°°. In the cylinder Ω χ (0,T) let us consider the following equation (λ" - Δ Η = α ( Δ - λ > , + β(Α - Χ)υ + g.

(5.7.7)

For equation (5.7.7) let us state the initial boundary value problem v(x,0) = vo(x),

ü(M)=0,

vi(x), χ G Ω; (χ,t) e dû χ (0,Τ).

vt(z,0) =

(5.7.8)

On reducing problem (5.7.7), (5.7.8) to problem (5.6.1), (5.6.2) let us assume V = Η2(Ω) Π H1 (Cl), G = L2{iï). Operators A, Bx and B0 are specified by the formulas A = λ " — Δ , B\ = Δ — λ', 5ο = Δ — Λ. As is easily seen, the operator A is a Fredholm operator due to its being self-adjoint. Let us denote by σ ( Δ ) the spectrum of the Dirichlet problem for the Laplace operator Δ in the domain Ω.

Chapter 5. Sobolev-type

equations

157

Lemma 5.7.1. Let one of the following conditions be satisfìed:

Proof. The condition (i) will not be considered in view of ker A = {0}. Let condition (ii) be satisfied. Let us consider an orthonormalised (in the sense Ζ^(Ω)) basis {ψ^, . . . , ψι0} kernels ker A. Since the vector Β\ipQ = λ(Δ - \")t}>Q + α (Λ" — is orthogonal to the lineal im A, k = 1,2 any vector from kernels ker A has no B-adjoint vectors, where the pencil Β = ( B I , B Q ) . This implies that in the condition (Al) it may be assumed that ρ = 0. Let condition (iii) be satisfied. Here, = 0 and the vector Βοψ§ + B\0 = β(Α - A")¿o + ß(x" - χ)Ψο i s orthogonal to the lineal im A. This implies that in the condition (Â1) it may be assumed that ρ = 1. • From Corollary 5.6.3 there follows the existence of propagators 00

1

v

¿ =Σ

Γ i

- λ'>+«(λ*

- λ'))^4

άμ,

where Δ μ * - ((Afc - Χ")μ2 + α(Xk - λ')μ + ß{Xk - λ ) ) " ^ · , and \ k and tpk are eigenvalues and corresponding eigenfunctions of the Dirichlet problem for the Laplace operator in the domain Ω, numbered with respect to nonascending of allowing for their multiplicity. Theorem 5.7.2. Let condition (i) of Lemma 5.7.1 be satisfìed. Then for every g G C([0,T); G) and for any υο,υι G V there exists a unique solution ν e C 2 ([0,T); V) of problem (5.7.7), (5.7.8), which can be represented in the following form

where V1 = ΣΓ=ι(2^)"1

fr

άμ.

G. A. Sviridyuk and V. E. Fedorov

158

On considering conditions (ii) and (iii) it is necessary to find projectors Ρ and Q. In this case they coincide, although they act in different spaces. Since 00

! i (Xk — λ")μ + a(Afc — Λ') 1 ß(h

V

-

μ

λ)

the both projectors can be found by the theorem of residues. Making use of this fact, we can find L0(I - Ρ) and M 0 _1 (7 - Q). Therefore, by Corollary 5.7.1 there holds true Theorem 5.7.3. Let condition (ii) of Lemma 5.7.1 be satisfìed. Then for every g G C 1 ([0,T); G) and any i>o, i>i G V, satisfying conditions (5.7.5), there exists a unique solution ν G C 2 ([0,T); V) of problem (5.7.7), (5.7.8), representable as

Theorem 5.7.4. Let condition (iii) of Lemma 5.7.1 be satisfìed. Then for every g G C 2 ([0,T); G) and any VQ, V\ G V satisfying conditions (5.7.6), there exists a unique solution υ G C 2 ([0,T); V) of problem (5.7.7), (5.7.8), representable as 11

M

11

V(t) = - Σ o ¿Ζ 9(t) + Σ ^ 9=0 k=0 W

/·« V t

+ / J

°

~

T

^ ) dr.

Chapter 6. Bounded solutions of Sobolev-type equations

6.0.

INTRODUCTION

In this chapter conditions are considered sufficient for the existence of bounded solutions of homogeneous Lu = Mu

(6.0.1)

Lù = Mu + f

(6.0.2)

and inhomogeneous

linear Sobolev-type equations for the cases of (L, a)-boundedness and strong (L,p)-sectoriality of the operator M. In Section 6.1, a relatively spectral theorem is proved generalising the well-known (see, for example, Riesz and Nagy, 1972) spectral theorem. In Section 6.2, bounded solutions of equation (6.0.1) are studied, and in Section 6.3, those of equation (6.0.2). In Section 6.4, examples are presented. Most of results in the chapter were obtained by Sviridyuk and Keller (1997). Section 6.1 was written by Keller (1996). Similar result was obtained by Rutkas (1975) before.

G. A. Sviridyuk and V. E. Fedorov

160

6.1.

RELATIVELY SPECTRAL THEOREM

Let operators L E C{U\T) and M € CL{U-,T), and let σ Ι { Μ ) be an ¿-spectrum of the operator M, for which the following condition is satisfied: σι{Μ) = σ[{Μ) U σ^(M), while there exists a bounded contour Γ C C, bounding the domain containing a f (M), and (M) lies outside this domain.

(6.1.1)

Then by Theorem 2.1.1, there exist integrals of F. Riesz type

Lemma 6.1.1. Let condition (6.1.1) be satisfied. Then the operator Ρ \ U -+U (Q : Τ -Λ Τ) is a projector. Proof. From condition (6.1.1) and the closedness of the relative spectrum GL(M) there follows the existence of a closed contour Γ' C C, Γ' Π σι(Μ) = 0 bounding the domain containing the contour Γ. From the analyticity of the right L-resolvent of the operator M there follows that

Further, using the same reasoning as in the proof of Lemma 4.1.1 we obtain the desired result. (The proof with respect to the operator Q is performed in a similar way). • Relatively spectral projectors Ρ and Q split the spaces U and F in the following way U = i m P ® ker Ρ,

Τ = imQ Θ ker Q.

Let us assume im P = U\

ker P = U2,

imQ = f \

ker Q = T2,

and let us denote by L¡. the restriction of the operator L to the subspace Uk] and by Mk let us denote the restriction of the operator M to dom Μ Π Uk, k = 1,2. Theorem 6.1.1. Let condition (6.1.1) be satisfied. Then (i) Lk -Uk

Tk, k = 1,2;

(ii) Mk : dom M η Uk

Tk, k = 1,2.

Chapter 6. Bounded solutions of Sobolev-type equations

161

Proof. The statement (i) follows from the construction of operators Ρ and Q, since LP — QL. Hence L(I — Ρ) — (I — Q)L, i. e., the action of the operator L is split Lk : Uk -> Fk, k = 1,2. The proof of the statement (ii) is analogous if identity (2.1.8) is employed. • Introduce the notation a L k (Mk) for the L^-spectrum of the operator Mk, k = 1,2. Theorem 6.1.2. Let condition (6.1.1) be satisfìed. Then the relative spectrum of the operator M decomposes in the following way: a^(M)=aLk(Mk),

k = 1,2.

Proof. Let us denote by Ωχ a domain bounded by the contour Γ, and let us assume Ω2 = 0\Ωχ. By the definition of the operators Lk -U k —»• Ä = 1,2, it follows that σ%(Μ) C aLk{Mk), k = 1,2. Now let us show that aLk(Mk) C σ£(Μ). Obviously, if Λ ^ σ£(Μ), then λ φ aLk(Mk), k — 1,2. Thus, let Λ 0 σ%(Μ), and let us demonstrate the existence of continuous operators (XLk — Mk)-1 : Tk —> Uk, equal to the restriction of the operator

to the subspaces Tk and k = 1,2 respectively. Indeed, by virtue of identity (2.1.2) we have

Λ G Ωχ; λ G Ω2; and

Λ G Ωχ; λ G Ω2. Therefore,

(Μ) = aLk (Mk).

•

G. A. Svmdyuk and V. E. Fedorov

162

Corollary 6.1.1. In conditions of Theorem 6.1.2 (i) there exists an operator L\ 1 G L{Tl\U1)] (ii) the operator Mi G C{Ul\Tl). Proof, (i) The operator LJ"1 is equal to the restriction of the operator

to T1. Indeed, from the construction of the operators Ρ and Q we obtain for every / G Τ λ

L^JjjiL-

Μ)"1/ άμ = ¿ j L^L - M)"1/ dß = Qf = f

and for every u EU1

— f (/iL - Μ) -1 άμΣη = -J— f (/zL - Μ)~ιΣηάμ = Pu = u. 2m yr

2m yr

It is obvious from the obtained that the restriction of the operator ¿ ¿ ( μ Ι - Μ Γ

1

* .

to U1 is the inverse operator of the operator L\. (ii) Note that projector Ρ maps vectors from dom M to dom M, i.e., Pu G dom M, if u G dom M. On considering for every u G dom M the Riemann sum ¿=1 and taking into account that for every μ^ G pL(M) Rj¡k(M)u G dom M, we obtain Pnu G dom M. In addition, by the definition of the Riemann integral under η —> oo Pnu —» Pu; and by virtue of the operator M closedness, we obtain that Pu G dom M. From the already mentioned action of the projector P, closedness of the operator M and identity (2.1.2) we have

Mxu = MPu = M — [ Β%{Μ)νάμ μ 2m Jr

=

éî i

ΜΚί(Μ>ά»=éï

= ¿T ^LRi(M)udß,

(6.1.3)

Chapter 6. Bounded solutions of Sobolev-type

equations

163

if u G dom M. Since the lineal P[dom M] is dense in U1, and the integral on the right in ( 6 . 1 . 3 ) specifies the continuous operator, the operator Μ χ G C{Ul]Tl). •

6.2.

B O U N D E D R E L A X E D SOLUTIONS OF A H O M O G E N E O U S EQUATION

Let the operators L G C(U\ T), M G Cl{U]T). Let us consider the Cauchy problem Lu = Mu, u(0)=u0. (6.2.1) If the operator M is strongly (L,p)-sectorial, then by Theorems 3.2.2, 3.3.2, and 3.4.2 the phase space V of equation (6.2.1) is the image of the semigroup {U* I t e 1 + } (3.2.4). Definition 6.2.1. The subspace V' C V is called an invariant space of equation (6.2.1), if for every uo G V the relaxed solution u — u(t) of problem (6.2.1) lies in V', i.e., u{t) GV'MtG I+. Theorem 6.2.1. Let an operator M be strongly (L,p)-sectorial and let condition ( 6 . 1 . 1 ) be satisfìed, then there exist invariant spaces of equation ( 6 . 2 . 1 ) . Proof. Let Ρ G £(U) be a relatively spectral projector defined in (6.1.2). Let us assume that Un = im Ρ Π V, U12 = k e r P Π V, Ulk are the closed subspaces in U, k — 1,2, and V is the phase space of equation (6.2.1). Then there holds the following expansion:

UT =

L· i RiW>eßt + éï ST Ri{M)eßt

Ά

Μ

=

+

Ή'

( 6

·

2

·

2 )

where Γχ C pL{M) is a closed contour bounding the domain Ωχ, which contains a part of the spectrum tff(M), and the contour Γ2 C pL{M) is of the form Γ2 = Γ U ΐ γ . Here, ΐ γ is the contour Γι oriented in the opposite direction, and the contour Γ is the same as in (3.2.4). As it can easily be seen, each term in (6.2.2) is a solving semigroup of equation (6.2.1), while by Theorem 3.2.2 imí/¿ = Ulk, k — 1,2, and by Theorem 3.3.2 Ul = Un @U12. • Remark 6.2.1. The semigroup {U{ \ t G K+} can be extended to the group {U[ I t G M}. Indeed, by virtue of the closedness of the contour Γχ,

164

G. A. Sviridyuk and V. E. Fedorov

the operator Μ η is (Lu, a)-bounded, where L u (Mu) is the restriction of the operators L and (M) to the subspace Un (dom Μ Π Un). Definition 6.2.2. Let Ul be the phase space, and let Ulk, k — 1,2, be the invariant spaces of equation (6.2.1), while U l = U n θ U 12 . The relaxed solutions u = u(t) of equation (6.2.1) have exponential dichotomy (e-dichotomy), if there exist constants Nk, v¡¡ G k = 1,2, such that (i) (ii) | | « 2 ( ί ) | | Μ < ^ 2 β - ^ - ' ) | | « 2 ( β ) | | Μ > ί > β , where uk G Ulk, Λ = 1,2. The exponentially dichotomic behaviour of the relaxed solutions consists in the following: the phase space of equation (6.2.1) decomposes into a direct sum of invariant spaces while the relaxed solutions from U 12 , exponentially decrease (remaining in U 12 ), and relaxed solutions from U n , exponentially increase (remaining in Theorem 6.2.2. Let an operator M be strongly (L,p)-sectorial and let there exist a number ω G M+ such that aL(M) Π {μ G C | —ω < Re μ < ω} = 0. Then the relaxed solutions u = u{t) of equation ( 6 . 2 . 1 ) have e-dichotomy. Proof. Let the contour Γχ be contained in the right half-plane and bound the part of the Li-spectrum of the operator Mi, which lies in this half-plane, and let the contour Γ2 lie in the left half-plane and bound the part of the Li-spectrum of the operator Mi, which is in this half-plane, and at the same time arg μ ±θ under |μ| —> +oo, μ G Γ2. Then there holds expansion (6.2.2), while by virtue of remark 6.2.1 the semigroup {U{ | t G K+} can be extended to the group {U[ \ t G Κ}. Therefore, if u x (s) G Un and s > t, then u^t) = Ul~su1(s) = e-^'-**

f

M

Rj (M)eT^t~s^u1 (s) dr,

where τ — μ —νι, while R e r > 0, r G Γ'ι· Consequently, 0 < v\ < min Re μ, and we can assume v\ — ω — e, e > 0. Hence we readily obtain the estimate ||ux(i)b
const ||u|L.

roo


T 1 is locally Holderian, then the vector function rb i{t) = [' Gt~sf(s) ds ur Ja for any a, b G R, a < b, and every t G (a,b) satisfies the inhomogeneous equation (6.3.1). Indeed, let us differentiate the equality u{t) = j* G^'f^ds

+ j* G t _ 7 ( s ) ds

Chapter 6. Bounded solutions of Sobolev-type equations

173

and act on the result by the operator L. This yields b

dCt~s L ——— f(s) ds

/

t

JQt — S L ——— f(s) ds = Mu(t) + f(t).

/

(G4). Since the L-spectrum of the operator M does not intersect the imaginary axis, the reasoning similar to that in the proof of Theorem 6.2.2 leads to the existence of constants Ν, ν G R+. such that 11(3*11
e>lt

d

^

R

*

=

é ï I

T

^

L

~

Μ)

~1βμί

άμ

'

where the contour Γ2 can be selected lying in the left half-plane and tifsf(s) = i ^ ~ 7 1 2 ( s ) , the solution (6.3.11) will be bounded on R+. If in addition UQ G U 1 2 , the vector function (6.3.11) will be a unique bounded relaxed solution of the Cauchy problem u 12 (0) = UQ for the third equation (6.3.8). On comparing (6.3.9), (6.3.10), and (6.3.11) and recalling that (I — U$)u0 = UQ + UQ1, by virtue of (6.3.7) we obtain the desired. • If we impose more stringent requirements on the operators L and M, we will be able to get information about bounded solutions of equation (6.3.1) along the entire axis R. Theorem 6.3.2. Let an operator M be (Σ,σ)-bounded, while 00 is a nonessential singularity of the order ρ of the L-resolvent of the operator M. Let the vector function f : R Τ be such that f° G C P + 1 (R; and f 1 e 1 CÍR;^" ). Equation (6.3.1) has a unique solution u G C^RjW) bounded on R precisely when (i) the L-spectrum of the operator M does not intersect the imaginary axis; (ii) the vector function f° and all its derivatives up to the order ρ inclusive are bounded along the entire axis R; (Hi) the vector function f 1 is bounded on R. At the same time this solution u = u(t) is of the form u{t)=

roo Ρ / Gt~sf{s)ds-1}2HqMÖ1f0{q)(t)· q=0

(6-3.12)

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G. A. Sviridyuk and V. E. Fedorov

I f , moreover, Ρ

roo

u0= / G-sf(s)ds-Y/H'}M^f0^(0), J— 00 η

(6.3.13)

then the vector function (6.3.12) is a unique solution of problem (6.3.1) bounded on M. P r o o f . The proof will be started from the sufficiency. Equation (6.3.1) will be reduced to the equivalent system of equations Hù° = u° + Μ0_1/0(ί),

Lxùl

= Mm1 + f1.

Since Gt~sf(s) ξ Gt~sfl(s) for every s,t G R, then as it follows from the properties (G1)-(G4), the vector function (6.3.12) is a solution of equation (6.3.1). We have from the property (G4) and the conditions of the theorem: /•oo

ρ e~^s~^\\f(s)\\uds +

\Ht)\\ux

xk*(?)(o)}

there exists a unique strong solution of problem (7.1.1), (7.1.2), of the form

y*(t) = -j2(RH)9(MorHl-P^*(qHt)+exp(T*Ht)y*0+ 9=0

f R*^x*{s)

Jo

ds.

It naturally follows from Theorem 7.1.1 that L e m m a 7.1.5. Let X and y be Hilbert spaces. Then the operator M is strongly (L,p)-sectorial when and only when the operator M* is strongly (L*,p)-sectorial.

Chapter 7. Optimal control

187

From Theorem 3.2.1 and Lemma 7.1.5 there follows the existence of semigroups X

"

=

^

/

{M

*)eßt

dßi

t e

1 +

Y

*t

=

é ï i

{M

*)eßt

άμ

*G

1 + 1

"

'

acting on spaces X* and respectively. The identities of these semigroups (projectors P* and Q* respectively) split the spaces X* and y* into a direct sum x* = x*°®x*\ y* = y*°®y*1, where X*° = ker Ρ*, X*1 = imP*, = kerQ*, y*1 = imQ*. Let X and y be Hilbert spaces, and let the operator M be strongly (L,p)-sectorial. Then by Lemma 7.1.5, Corollaries 3.2.1, 3.4.2 and Theorem 3.4.3 problem (7.1.1), (7.1.2) will be split into two problems RW0 = y*° + y*l=T*Hy*'+x*\

y*°(0)=yS°, y*1(0)=y*0\ 0

(7.1.6) (7.1.7) 1

where operators R*H = € ¿(y* ), T*H = {L\)~lM{ G ¿(y* ), operators LQ, MQ, L*, M* are the restrictions of operators L* and M* into the subspaces y ° and respectively, and the vectors y*° — (I — Q*)y*, y*1 = Q*y\ x*° = (Mí)" 1 (J - P*)x*, χ*1 = (L\)~1P*x*. Obviously, problem (7.1.7) is solved using the standard procedure, because from Lemma 7.1.5, Corollary 3.4.4 and definition of a linear operator there follows its sectoriality. Therefore, there holds Lemma 7.1.6. Let X and y be Hilbert spaces, and let the operator M be strongly (L,p)-sectorial, and function x* 1 from class W%+l{X*1). Then for every y^1 G J' 1 there exists a unique strong solution of problem (7.1.7), of the form y*\t) =

ftY*{t~s)x*1(s)ds,

+ Jo

1

where {Y* \ t G M+} is the restriction of the semigroup {Y*1 | t G K+} into y*1. Lemma 7.1.7. Let X and y be Hilbert spaces, and let the operator M be (L,p)-sectorial. Let also the vector-function χ G

188

G. A. Sviridyuk and V. E. Fedorov

1 < q < oo. Then if y*0° = -J2(RH)qz*0(q)( g=0

0),

there exists a unique strong solution of problem (7.1.6), of the form y*°(i) = - E ( Â î r ) V 0 W ( t ) >

(7.1.8)

Proof. By Lemma 7.1.5 and Corollary 3.2.3 the operator R*H is nilpotent. By substituting (7.1.8) into system (7.1.6) we obtain the desired. • Let us denote =

[ (/iL* - M*)~leßt άμ, 2m Jr

Vi ε Κ.

The natural corollary of Lemmas 7.1.6 and 7.1.7 is the following Theorem 7.1.3. Let X and y be Hilbert spaces, the operator M be (L,p)-sectorial, and function x* e w¡+l{x*), ι < q < oo. Then for every y* e A^ = {y*

e

y

I (I-Q*)y*

= - ¿ ( ¿ ^ ( M o T ^ - n ^ O ) } 9=0

there exists a unique strong solution of problem (7.1.1), (7.1.2), of the form y*(t) = - ¿ W W o T H l - P*)x*{q)(t) + Y*Vo + Γ R*{t~s)x*(s) ds.

7.2.

PROBLEM OF OPTIMAL CONTROL FOR A N EQUATION WITH RELATIVELY σ - B O U N D E D OPERATOR

Let us consider the Cauchy problem x{0) = i 0

(7.2.1)

Lx = Mx + y + Bu,

(7.2.2)

for equation

Chapter 7. Optimal control

189

where the functions x, y and u lie in Hilbert spaces X, y and U respectively; operator L G (kerL φ {0}), M 6 Cl(X-,y), Β G £(W;}>). Moreover, the operator M will be (L, a)-bounded and oo will be a nonessential singularity of the order ρ G No = {0} U Ν of the ¿-resolvent of the operator M. Let us introduce into consideration a control space HP+1{U)

= {u Ε

L2{0,t;U)

J

U CP+I) €

= 0, k = 0 , 1 , . . . ,p}.

¿ 2 ( 0 , Τ ; Μ ) , UW(0)

O The space

HP+1(U)

product in the space

is a Hilbert space because H

P+1

(U)

U

is Hilbert. The scalar

will be

[«.«] = Σ

fT(^k)Mk))udt.

O o Let us separate in space HP+1{U) a closed and convex subset HPD {U) (a set of admissible controls). Let further Ζ be a certain Hilbert observation space and a self-adjoint operator C G C{X\Z) specifies an observation z(t) = Cx{t). Note that if χ G Hl{X), then ζ G Η1 (Ζ). Finally, let us specify positively defined and self-adjoint operators Nk G £(U), k = 0 , 1 , . . . ,p + 1. I i

O ,,

Our objective is to prove the existence of a unique control no G Hg {U), minimizing the cost functional l _T p+l -T zik) ( k] 2 j(u) = j 2 \\ (t)-z 0 m zdt+Y/ (NkuW(t),uW(t))udt k=0J° k=0 (7.2.3) for equation (7.2.2). o ,. Vector function uq G Ha (U) such that J(«o) =

min J(u), ueHpa+1(u)

is called an optimal control for equation (7.2.2). By Theorem 5.4.1 for every y G e Ay and u G H^+l{U) l there exists a unique strong solution χ G H {X) of problem (7.2.1), (7.2.2), of the form x(t) = (Α ι + A2)(y + Bu)(t) + Χιχ0, (7.2.4)

190

G. A. Sviridyuk and V. E. Fedorov

where operators Αι and A¿ are given by relations (5.4.7) and (5.4.8) respectively. Let us fix y G Hp+1(y), xo G Ay, where ρ Ay = {xex\(i-P)x

HqM

= -J2

ô\i

Q)y{q\o)}.

-

9=0

Let us consider (7.2.4) as the mapping D:u

-> x(u).

(7.2.5)

Lemma 7.2.1. Let X, y and li be Hilbert spaces, let the operator M be (L, o)-bounded, and y G Hp+l(y) and XQ € Ay be fixed. Then the mapping D : HP+1{U) ->• Hl{X),

given by formula (7.2.5), is continuous.

Proof. As is readily seen, the operator Β G C{Hp+l{U)\Hp+l{y)). Therefore, the statement of the lemma follows from (7.2.4) and Lemma 5.4.3. • Using (7.2.5), let us rewrite the cost functional (7.2.3) in the form J{u) = \\Cx(t;u) - z0\\2Hl{2) + [v,u], where v^(t)

= Nku^k\t),

k = 0 , . . . ,p + 1. Hence

J(u) = τr(u, u) — 2λ(«) + ||2b — Cx(t; 0)

{z),

where τr(u,u) = IIC{x(t;u) - x{t\0))||^i(2) O is a bilinear continuous coercive form in Hp+l{U)\ \{u) - {z0 - Cx(t·, 0), C(x(t; u) - x{t-

+ [i;,u]

0))}ffl(z)

O is a linear form continuous on H P + 1 (U). Thus, there holds Theorem 7.2.1. Let an operator M be (L,a)-bounded, while oo is a nonessential singularity of the order ρ of the L-resolvent of the operator M. Then for every y G Hp+l(y) and XQ G Ay there exists a unique optimal control u0 G Hpa+1{U).

Chapter 7. Optimal control O ,,

Control uq E Ha

191

{U) is optimal precisely when

J'(tt 0 ) · (u - u0) > 0

Vit e

Hpd+\U),

i.e., for functional (7.2.3) there holds a relation {Cx{t; u0) - z0,C{x(t]u)

- x{t]Uo))}Hi(Z)

+ [«o,« - uo] > 0,

VueH p d + \U), where

P+i

η

[«0,u - uo] = Σ íiN^WM'Ht) q=óJ° is a bilinear continuous coercive form on

(7.2.6)

- U{0q\t))u dt

O

HP+1{U).

Let us introduce in consideration the canonical isomorphisms A: Ζ

Z*,

Ku-U-^U*.

Then inequality (7.2.6) can be rewritten as (C*A(Cx(t; u0) - zQ), x(t; u) - x(t\ u0))Hi(z) p+1 rr + T / (Nqu^(t)Mq)(t)-u^(t))udt>0, q—0 J0

Vu£Hpd+1(U).

(7.2.7)

Now let us define adjoint state of system (7.2.1), (7.2.2) x{t\u) H (y*) as a solution of equation

€

p+1

-L*x

= M*x + C*A{Cx{t-u)-z0)

(7.2.8)

in the interval (0, r), under final condition X(r;u)

= 0

(7.2.9)

It follows from Theorem 7.1.1 that Theorem 7.2.2. Let X and y be Hilbert spaces, let the operator M be (L, σ)-bounded and oo b e a removable singularity of the L-resolvent of the operator M. Then there exists a unique solution x(t;u) G H1 (y*) of problem (7.2.8), (7.2.9).

192

G. A. Sviridyuk and V. E. Fedorov

Proof. The proof follows from Theorem 7.1.2 taking into account that the removable singularity implies that ρ = 0. • Inequality (7.2.7) for arbitrary ρ G Ν may be written in the following form P+1 r £ / (A^B*x^(t-,uo) Jo g =o

+

Nquiq)(t),u(q)(t)-u^(t))u>0,

VueHpd+1(U). Therefore, we have proved Theorem 7.2.3. Let an operator M be (L,a)-bounded, while oo is a nonessential singularity of the order ρ of the L-resolvent of the operator M. Then for every y G Hp+1(y) and χ G Ay the optimal control for system (7.2.1), (7.2.2) is characterised by relations Lx = Mx+ y + BUQ, RR(0) -L*x = Μ*χ + C*A(Cx(t·, uo) - zo), p+1 . T ^

/ (A-1B*x^(t-,uo)

+

=

XO,

χ(τ) = 0,

Nqu{0q)(t)Mq)(i)-4q\t))u>0,

VuEHpd+1(U), where x{t\UQ) G H L { X ) , X(UQ) G Hp+l(y*), while for Ρ = 0 these conditions are necessary and sufficient, and for ρ > 0 they are only sufficient. 7.3.

PROBLEM OF OPTIMAL CONTROL FOR EQUATION WITH A RELATIVELY p-SECTORIAL OPERATOR

Let us consider the Cauchy problem z(0) = x0

(7.3.1)

Lx = Mx + y + Bu,

(7.3.2)

for equation where the functions χ G X, y G y and u EU, X, y and U are Hilbert spaces, the operator L G (kerL φ {0}), M G Cl(X-,y), Β G C(U-,y). In addition, the operator M is strongly (L,p)-sectorial.

Chapter 7. Optimal control

193

As in Section 7.2, let us introduce into consideration a control space HP+L{U)

= {u E

L2(0,T;U)

e I 2 ( 0 , r ; W ) , «(*)(0) = 0, fc = 0 , l , . . . , p } . o The space HP+1{U) is a Hilbert space because continuity of the embedding HP+1{U)

^

U

is Hilbert and due to the

CP([0-,t];U).

O ,1 o Let us single out a closed and convex subset HQ (K) in the space Hp+l{U) (the set of admissible controls). Let us fix y G Hp+1(y), XO E Ay and U 6 HG+1{U)· In the conditions of Theorem 5.4.2 there exists a unique solution χ € Η 1 (Λ') of problem (7.3.1), (7.3.2) of the form x = {Al + A2){y + Bu) + Xt x0.

(7.3.3)

Let Ζ be some Hilbert observation space and let the operator C 6 C{X\ Ζ) specify the observation ζ = Cx. Similarly to Section 7.2, let us introduce into the consideration self-adjoint and positively defined operators Nq E C(U), q — 0,1,... ,p + 1, and let us construct the cost functional

(7.3.4) Pursuing the line of reasoning similar to the previous section, we obtain the following theorems. Theorem 7.3.1. Let an operator M be strongly (L,p)-sectorial. Then for every y G Hp+1(y) and xq 6 Ay there exists a unique optimal control uq G Hpa+1{U) for problem (7.3.1), (7.3.2). Theorem 7.3.2. Let X and y be Hilbert spaces, and let the operator M be strongly (L,0)-sectorial. Then there exists a unique solution x(t;u) 6 Hl{y*) of problem (7.2.8), (7.2.9).

194

G. A. Sviridyuk and V. E. Fedorov

Theorem 7.3.3. Let an operator M be strongly (L,p)-sectorial. Then for any y G Hp+1(y) and χ G Ay the optimal control for system (7.3.1), (7.3.2) is characterised by the following relations Lx = Mx + y + Buo, —L*x = Μ*χ + C*A(Cx(tP+1 r ^ / (A^B*x^(f,uo) J( q=o >

ζ(0) = xo,

u 0 ) - z 0 ),

+

χ ( τ ) = 0,

Nqu^(t),u^(t)-u¡q)(t))u>0,

VueHpd+1(U), where x{t]Uo) E Hl{X), x(uq) G Hp+l(y*), while for ρ = 0 these conditions are necessary and sufficient, and for ρ > 0 they are only sufficient.

7.4.

BARENBLATT—ZHELTOV—KOCHINA

EQUATION

Let us consider the problem of optimal control for the Barenblatt—Zheltov— Kochina equation (λ - A)xt = aAx + y. (7.4.1) Let Ω C be a bounded domain with the boundary