ISBN: 3-540-65119-5
TITLE: Differential Equations with Operator Coefficients
AUTHOR: Kozlov, Vladimir; Maz'ya, Vladimir
TOC:

Introduction XV 
Part I. Differential Equations with Constant Operator Coefficients 
1. Power-Exponential Zeros 3 
1.1 Introduction 3 
1.2 Basics on Operator Pencils 4 
1.2.1 Notation 4 
1.2.2 Decomposition of the Resolvent Near the Pole 6 
1.2.3 Two-Term Quadratic Pencils 7 
1.3 Power-Exponential Solutions of the Homogeneous Equation 10 
1.3.1 Notation. Spaces Z(A,lambda_nu) and Z(A*,lambda_nu) 10 
1.3.2 A Biorthogonality Condition 11 
1.3.3 Proof of Proposition 1.3.1 14 
1.3.4 Two-Term Second Order Equations 14 
1.3.5 A Construction of Canonical Systems 
of Jordan Chains 16 
1.4 Power-Exponential Solutions 
of the Nonhomogeneous Equation 16 
1.5 Applications to Elliptic Partial Differential Equations 
with Constant Coefficients 17 
1.5.1 Neumann Problem in a Cylinder 17 
1.5.2 The Dirichlet Problem in a Cone 18 
1.5.3 Properties of the Operator Pencil (1.44) 19 
1.5.4 The Adjoint Pencil 22 
1.5.5 The Dirichlet Problem in a Half-Space 22 
1.5.6 Elliptic Equations in R^n\O 23 
1.6 Comments 25 
2. Differential Operator Equations 
in Weighted Sobolev Spaces 27 
2.1 Introduction 27 
2.2 The Operator Pencil A(lambda) 27 
2.2.1 Conditions on A(lambda) 27 
2.2.2 Examples of Peneils Satisfying Conditions I and II 29 
2.2.3 Notation 30 
2.3 Some Spaces of Vector Valued Functions 31 
2.3.1 Sobolev Spaces 31 
2.3.2 Weighted Sobolev Spaces 32 
2.4 Solvability in W^l_beta (R) 33 
2.5 Application to the Dirichlet Problem in a Cylinder 34 
2.6 Green's Kernel 36 
2.6.1 Definition of Green's Kernel 36 
2.6.2 Properties of G(t) 37 
2.6.3 Integral Representation of Solutions 38 
2.7 Asymptotic Decompositions of Green's Kernel 39 
2.7.1 Representations for G(t) 39 
2.7.2 Representations for G(t-tau) 41 
2.8 Asymptotics of Solutions in W^l_beta (R) 43 
2.8.1 Asymptotic Representations 43 
2.8.2 Solutions of the Homogeneous Equation 44 
2.9 A Local Estimate for Solutions 45 
2.10 Application to the Dirichlet Problem in a Cone 46 
2.11 Comments 48 
3. Solutions in a Local Sobolev Space 49 
3.1 Introduction 49 
3.2 Zeros of A(D_t) 50 
3.2.1 Uniqueness for Homogeneous Equation 50 
3.2.2 Behaviour of Zeros at Infinity 51 
3.3 Unique Solvability of the Nonhomogeneous Equation 52 
3.3.1 An Auxiliary Existence Result 52 
3.3.2 Unique Solvability 54 
3.4 Solutions Corresponding to a Strip 56 
3.5 Comparison Principle 56 
3.5.1 Comparison Equation and Its Green Function 56 
3.5.2 Solvability Criterion for the Comparison Equation 58 
3.5.3 Comparison Principle 59 
3.5.4 A General Asymptotic Representation 
of the (k_{-},k_{+})-Solution 60 
3.6 Estimates for Solutions on a Semiaxis 63 
3.7 The Phragmn-Lindelf Principle 64 
3.8 Asymptotics of Solutions Corresponding to a Strip 65 
3.8.1 A Representation for the Difference of Two Solutions 65 
3.8.2 An Asymptotic Formula 66 
3.8.3 Description of Solutions to Homogeneous Equation 67 
3.9 Applications to Boundary Value Problems 68 
3.9.1 The Dirichlet Problem in a Cylinder 68 
3.9.2 The Neumann Problem in a Cylinder 70 
3.9.3 The Dirichlet Problem in a Cone 71 
3.10 Comments 74 
4. Two-Weight L_2-Estimates 75 
4.1 Introduction 75 
4.2 Weighted Sobolev Spaces 76 
4.3 Uniqueness of Solutions in W^l (R;gamma) 77 
4.4 Existence of Solutions in W^l (R;gamma) 79 
4.4.1 Principal Result 79 
4.4.2 Auxiliary Results on Operators 
of Multiple Integration 80 
4.4.3 Proof of Theorem 4.4.1 84 
4.4.4 Power-Exponential Weight Functions 85 
4.5 Application to the Dirichlet Problem in a Cone 85 
4.6 Comments 87 
Part II. Differential Equations with Variable Operator Coefficients 
5. Existence, Uniqueness and "Pointwise" Estimates 91 
5.1 Introduction 91 
5.2 Auxiliary Information on the Comparison Equation 93 
5.2.1 Green's Function 93 
5.2.2 Existence and Uniqueness Results 
for the Comparison Equation 96 
5.3 Existence 98 
5.3.1 Assumptions on the Operator L 98 
5.3.2 Construction of a (k_{-},k_{+})-Solution 99 
5.4 Uniqueness Theorems 101 
5.4.1 A Class of Uniqueness 101 
5.4.2 Another Class of Uniqueness 102 
5.4.3 The Case m_{} <= 2 104 
5.4.4 An Explicit Uniqueness Condition in Terms of omega_0 104 
5.5 Behaviour of Zeros at Infinity 104 
5.5.1 Zeros of L 104 
5.5.2 Zeros of the Adjoint Operator 105 
5.6 Estimates for Solutions on the Semiaxis t > t_0 107 
5.7 Applications to Partial Differential Equations 
with Variable Coefficients 109 
5.7.1 The Dirichlet Problem in a Cylinder 109 
5.7.2 The Dirichlet Problem in a Cone 110 
6. Corollaries of Previous Results 
Under Special Assumptions on L(t, D_t) 113 
6.1 Introduction 113 
6.2 General Perturbations 114 
6.2.1 Existence and Uniqueness Theorems 114 
6.2.2 Estimates for Solutions at Infinity 116 
6.3 The Case m_{+} = m_{-} = 1 117 
6.3.1 Equation (5.1) on R 117 
6.3.2 Equation (6.8) on a Semiaxis 120 
6.4 Estimates of the Phragmn-Lindelf Type 
for Solutions of (6.8) when rho Dominates Either t^{-m_{+}} 
or t^{-m_{-}} for Large t > 0 121 
6.5 Applications to Partial Differential Equations in a Cylinder 124 
6.6 Other Applications 126 
6.6.1 Isolated Singularities 126 
6.6.2 The Neumann Problem in a Cylinder 127 
6.6.3 The Dirichlet Problem 
for Other Nonsmooth Domains 128 
6.7 Comments 131 
7. Two-Weight L_2-Estimates for Equations 
with Variable Coefficients 133 
7.1 Introduction 133 
7.2 Uniqueness Theorems in Weighted Sobolev Spaces 135 
7.3 Existence Theorems for Solutions in Weighted Sobolev 
Spaces and Two-Weight Estimates 136 
7.4 Unique Solvability in W^l_{beta_{-},beta_{+}} (R) 138 
7.5 The Case m_{} = 1 140 
7.6 Two-Weight Estimates when rho Dominates t^{-m_{+}} 144 
7.7 Comments 146 
8. Connection of Solutions Corresponding to Different Strips 147 
8.1 Introduction 147 
8.2 Auxiliary Information 147 
8.2.1 Notation 147 
8.2.2 Estimates for Green's Functions 
of the Comparison Equations 148 
8.2.3 An Auxiliary Existence Result 150 
8.3 Zeros of L(t, D_t) 151 
8.3.1 The Class X(L) 151 
8.3.2 The Dimension of X(L) 152 
8.3.3 The Norm in X(L) 154 
8.4 Solutions of (5.1) Corresponding to Different Strips 155 
8.4.1 The Auxiliary Dual Space 155 
8.4.2 The Subspace X*(L) 156 
8.4.3 The Difference of Two Solutions Belongs to X(L) 157 
8.4.4 A Sesquilinear Form and the Dimension of X*(L) 158 
8.4.5 Main Result 159 
8.5 Structure of Solutions of (6.8) at Infinity 160 
8.5.1 The Spaces Y_1(L) and Y_2(L) 160 
8.5.2 (Y_1,Y_2)-Spaces 161 
8.5.3 An Asymptotic Representation 162 
8.6 Comments 164 
9. Applications to the Case of Perturbations 
Vanishing at Infinity 165 
9.1 Introduction 165 
9.2 The Case rho(t) --> 0 as t -->  Infinity 166 
9.2.1 Description of the Class X(L) 167 
9.2.2 Description of the Class X*(L) 167 
9.2.3 Characteristic Exponents 168 
9.2.4 The Difference of Two Solutions 168 
9.3 The Case rho(t) --> 0 as t --> + Infinity 169 
9.3.1 Structure of Solutions at + Infinity 169 
9.3.2 Existence of Characteristic Exponents 
for Solutions of (8.53) 171 
9.3.3 Asymptotic Equivalence of Two Equations 172 
9.4 The Case of Absence of Generalized Eigenvectors 173 
9.4.1 Zeros of L and L* 173 
9.4.2 Relation of Solutions Corresponding 
to Different Strips 174 
9.4.3 An Asymptotic Representation of Solutions at + Infinity 174 
9.4.4 The Asymptotic Equivalence of Two Equations 177 
9.5 Application to the Local Regularity of Solutions 
to Elliptic Equations 178 
9.6 Comments 180 
10. Variants and Extensions of the Previous Theory 181 
10.1 Introduction 181 
10.2 Estimates of Solutions to Operator Differential Inequalities 182 
10.3 Perturbation of a Differential Operator 
with Variable Coefficients 184 
10.3.1 General Case 184 
10.3.2 A Second Order Differential Operator 185 
10.3.3 Perturbations of the Second Order 
Differential Operator 188 
10.4 Applications to Partial Differential Equations 191 
10.5 "Parabolic" First Order Operators 
with a Variable Dissipative Term 194 
10.5.1 Unperturbed Operator 194 
10.5.2 Perturbed Operator 196 
10.6 "Hyperbolic" Operator Equations 
with Constant Coefficients 200 
10.6.1 Assumptions on the Pencil A(lambda) 200 
10.6.2 Example of a Second Order Differential Operator 201 
10.6.3 Solutions in Weighted Sobolev Spaces 202 
10.6.4 Comparison Principle and Solvability in W^{l-1}_{loc} (R) 205 
10.7 "Hyperbolic" Operator Equation with Variable Coefficients 208 
10.8 The Operator L in Variational Form 209 
10.8.1 Function Spaces 209 
10.8.2 An Equivalent Norm in W^{-q}_{beta} (R) 214 
10.8.3 Assumptions on the Pencil A(lambda) 216 
10.8.4 Equation with Constant Coefficients 218 
10.8.5 Equation with Variable Coefficients 220 
10.8.6 The Variational Form of the Dirichlet Problem 
in a Cone 222 
10.9 Ordinary Differential Equations in Banach Spaces 226 
10.9.1 Assumptions on the Operator 
with Constant Coefficients 226 
10.9.2 Solvability of the Equation 
with Constant Coefficients 229 
10.9.3 Uniqueness for the Equation 
with Constant Coefficients 232 
10.9.4 Comparison Principle for the Equation 
with Constant Coefficients 234 
10.9.5 Equation with Variable Coefficients 234 
10.9.6 Application to the Dirichlet Problem in a Cylinder 235 
10.10 Applications to Elliptic Boundary Value Problems in a Cone 237 
10.10.1 The Comparison Principle for the Model Problem 237 
10.10.2 The Comparison Principle for the Boundary Value 
Problem with Variable Coefficients 240 
Part III. Asymptotic Theory of Operator Differential Equations 
11. Complete Asymptotic Expansions Under Exponential 
and Power Perturbations of A(D_t) 245 
11.1 Introduction 245 
11.2 Perturbation with Exponential Decay 
(Homogeneous Equation) 246 
11.3 Perturbation with Exponential Decay 
(Nonhomogeneous Equation) 249 
11.4 Perturbation in the Form of Laurent Series 
(Homogeneous Equation) 251 
11.5 Perturbation in the Form of Laurent Series 
(Nonhomogeneous Equation) 255 
11.6 The Dirichlet Problem for a Cuspidal Domain 259 
11.7 Comments 262 
12. Reduction to a First Order System 265 
12.1 Introduction 265 
12.2 Prerequisites for the Subsequent Asymptotic Theory 265 
12.3 Linearization of the Pencil A( ) 268 
12.4 Canonical Sets of Jordan Chains of I + A 271 
12.5 The Riesz Projector and Its Properties 273 
12.6 The Vector Function Spaces S_{loc}(R), X_{loc}(R) and Y_{loc}(R) 275 
12.7 Existence of Solutions in X_{loc}(R) 277 
12.8 Uniqueness of Solutions in X_{loc}(R) 279 
12.9 From the Equation with Variable Coefficients to a First 
Order System 281 
12.10 Comments 283 
13. General Asymptotic Representation 285 
13.1 Introduction 285 
13.2 Spectral Decomposition of the First Order System 286 
13.3 Solvability of the Infinite-Dimensional Part 
of the Split System 287 
13.4 Uniqueness for the Infinite-Dimensional Part 
of the Split System 290 
13.5 The Function alpha 291 
13.6 Reduction of the Split System 
to a Finite Dimensional System 294 
13.7 Estimates for the Operator K and the Function f 297 
13.8 Estimates for the Function beta 298 
13.9 The Finite Dimensional System in the Matrix Form 302 
13.10 Operator Differential Equations on a Semiaxis 305 
13.10.1 Preliminaries 305 
13.10.2 An Existence Result 306 
13.10.3 A Representation for Solutions to (13.92) 307 
13.10.4 The Matrix Form of System (13.95) 308 
13.11 Properties of Zeros of the Finite Dimensional System 309 
13.11.1 The System on R 309 
13.11.2 The System on a Semiaxis 311 
13.12 Estimate for a Neumann Series 312 
14. Power-Exponential Asymptotics 317 
14.1 Introduction 317 
14.2 Special Solutions of the Finite Dimensional System 320 
14.2.1 The Functions alpha, beta, gamma and chi 320 
14.2.2 Lemma on Special Solutions 321 
14.3 Special Solutions of (14.1) 325 
14.3.1 Construction of Solutions 
with Prescribed Asymptotics 325 
14.3.2 Main Result 327 
14.4 Asymptotics of Arbitrary Solutions of (14.1) 330 
14.4.1 Main Result 330 
14.4.2 Refinement of the Asymptotics 331 
14.5 Nonhomogeneous Finite Dimensional System 334 
14.6 A Special Solution of the Nonhomogeneous System (14.2) 336 
14.7 Asymptotics of Solutions 
to the Nonhomogeneous System (14.2) 338 
14.8 Asymptotics of Solutions in a Weighted Sobolev Space 340 
14.9 Asymptotic Behaviour of Solutions to Elliptic Equations 
near an Interior Point 342 
14.10 Comments 344 
15. The Case of One Simple Eigenvalue on the Line 345 
15.1 Introduction 345 
15.2 A Comparison Principle for Integral Inequalities 346 
15.3 Estimate for Solutions of the Scalar Equation (13.108) 347 
15.4 Zeros with Prescribed Asymptotics at Infinity 353 
15.5 Asymptotics of Solutions of the Homogeneous 
Higher Order Equation at + Infinity 356 
15.6 Asymptotics of Solutions to the Nonhomogeneous Equation 361 
15.7 Comments 366 
16. Several Simple Eigenvalues on the Line 369 
16.1 Introduction 369 
16.2 Special Solutions of the Finite Dimensional System 369 
16.2.1 Functions alpha and beta 369 
16.2.2 Dichotomy Condition 370 
16.2.3 Special Solutions of the Finite Dimensional System
(16.3) 371 
16.2.4 Proof of Lemma 16.2.2 372 
16.3 Asymptotic Formulae for Solutions 376 
16.4 A Second Order Equation 379 
16.5 Application to the Schrdinger Equation in a Cylinder 382 
16.6 Comments 383 
17. The Case of a Single Multiple Eigenvalue 385 
17.1 Introduction 385 
17.2 Solution of an Auxiliary Matrix Equation 386 
17.3 Special Solutions of the Finite Dimensional System 390 
17.3.1 Prerequisites 390 
17.3.2 Lemma on Special Solutions 392 
17.4 Asymptotics of Solutions 397 
17.5 An Example 399 
17.6 Comments 402 
A. Holomorphic Operator Functions 403 
A.1 Introduction 403 
A.2 Prerequisites on Fredholm Operators 404 
A.3 Basic Notions of the Spectral Theory 
of Holomorphic Operator Functions 405 
A.4 Canonical Generating System in S(F, lambda_0) 
and Canonical Set of Jordan Chains 407 
A.5 The Local Equivalence of Holomorphic Operator Functions 411 
A.6 The Smith Form of a Holomorphic Matrix Function 413 
A.7 The Resolvent of a Holomorphic Matrix Function 417 
A.8 Fredholm Holomorphic Operator Functions 419 
A.9 The Adjoint Holomorphic Operator Function 422 
A.10 The Structure of F(lambda)^{-1} Near the Pole 425 
References 431 
Index of Notation 435 
Index 439 
Index of Names 441 
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