Elliptic Partial Differential Equations

by Volpert, Vitaly

• ISBN13: 9783034605366
• ISBN10: 3034605366
• Format: Hardcover
• Copyright: 2011-03-07
• Publisher: Birkhauser

Summary

The theory of elliptic partial differential equations has developed during about two centuries together with electrostatics, heat and mass diffusion, hydrodynamics and many other applications. It remains one of the most rapidly developing fields of mathematics. The theory of general elliptic problems is presented in the present first volume of the book. A priori estimates, normal solvability and Fredholm property, index, operators with a parameter, nonlinear Fredholm operators are discussed. Particular attention is paid to elliptic problems in unbounded domains which are not yet sufficiently well presented in the existing literature and which require some special approaches. The second volume will be devoted to reaction-diffusion equations. Existence and bifurcations of solutions, travelling waves, spectral properties and other questions are studied in relation with numerous applications in chemical physics, biology and medicine.

Table of Contents

Preface	p. xv
Introduction
Elliptic problems in applications	p. 1
Heat and mass transfer	p. 1
Electrostatic and gravitational fields	p. 4
Hydrodynamics	p. 5
Biological applications	p. 7
Classical theory of linear elliptic problems	p. 10
Function spaces	p. 10
Elliptic problems	p. 13
A priori estimates	p. 20
Fredholm operators	p. 25
Elliptic problems in unbounded domains	p. 30
Function spaces	p. 31
Limiting problems	p. 32
Fredholm property and solvability conditions	p. 36
Index	p. 39
A priori estimates	p. 42
Non-Fredholm operators	p. 42
Nonlinear Fredholm operators	p. 43
Topological degree	p. 43
Existence and bifurcations of solutions	p. 45
Function Spaces and Operators
The space E	p. 48
Systems of functions	p. 49
The space Ep	p. 50
The space E&infty;	p. 55
Completeness of the space E&infty;	p. 57
Other definitions of the space E&infty;	p. 59
Bounded sequences in E&infty;	p. 62
The space Ep(¿)	p. 65
Spaces in unbounded domains	p. 73
Dual spaces	p. 78
Dual spaces in domains	p. 89
Spaces Wqs,p(Rn)	p. 93
Local operators	p. 100
A Priori Estimates
Model problems in a half-space	p. 106
Formulation of the main result	p. 106
Auxiliary results	p. 107
Reduction to homogeneous systems	p. 112
Proof of Theorem 1.2	p. 113
Proof of Theorem 1.1	p. 116
A priori estimates in the spaces W&infty;s,p	p. 116
A priori estimates for adjoint operators. Model systems	p. 120
The general problem in a half-space	p. 123
The general problem in unbounded domains	p. 130
Normal Solvability
Limiting domains	p. 137
General properties	p. 137
Examples	p. 142
Sobolev spaces	p. 146
A priori estimates in the spaces W&infty;k,p	p. 146
Limiting problems	p. 147
A priori estimates with condition NS	p. 154
Normal solvability	p. 163
Hölder spaces	p. 170
Operators and spaces	p. 170
Normal solvability	p. 171
Dual spaces. Invertibility of limiting operators	p. 176
Weighted spaces	p. 180
Fredholm Property
Estimates with Condition Ns*	p. 183
Abstract operators	p. 193
Elliptic problems in spaces W&infty;s,p(¿)	p. 198
Exponential decay	p. 203
The space Eq	p. 205
The space Eq. Continuation	p. 210
Weighted spaces	p. 215
Holder spaces	p. 217
Examples	p. 220
Bounded domains	p. 220
Unbounded domains	p. 222
Formally Adjoint Problems
Definitions	p. 227
Spaces V2m (¿)	p. 231
Fredholm theorems	p. 237
Spaces Eq	p. 243
Smoothness of solutions	p. 246
Examples	p. 247
Elliptic Problems with a Parameter
Parameter-elliptic boundary value problems	p. 252
Spaces	p. 254
Model problem in Rn	p. 256
Model problem in a half-space	p. 258
Problem in Rn	p. 272
Problem in R+n	p. 275
Problem in ¿	p. 276
Generation of analytic semigroups	p. 284
Elliptic problems with a parameter at infinity	p. 285
Examples	p. 286
Invertibility of second-order operators	p. 286
Operators with a parameter	p. 288
Index of Elliptic Operators
Definitions	p. 292
Limiting domains	p. 293
Limiting operators and problems	p. 295
Conditions Ns and estimates	p. 296
Results and examples	p. 298
Stabilization of the index	p. 302
Formally adjoint problems	p. 313
Convergence of solutions	p. 319
Cauchy-Riemann system	p. 324
Laplace operator with oblique derivative	p. 331
Cauchy-Riemann system and Laplace operator	p. 336
Reduction of the problems to each other	p. 336
Formula for index	p. 345
General first-order systems on the plane	p. 346
Reduction to canonical systems	p. 346
Index in bounded domains	p. 350
Index in unbounded domains	p. 352
Examples	p. 355
Problems in Cylinders
Ordinary differential operators on the real line	p. 360
Representation of solutions	p. 360
Calculation of the index	p. 362
Second-order equations	p. 365
Reduction to first-order equations	p. 365
Solvability conditions	p. 368
Second-order problems in cylinders	p. 375
Normal solvability	p. 375
Calculation of the index	p. 380
Lower estimates of elliptic operators	p. 388
Operators with constant coefficients	p. 389
Variable coefficients	p. 393
Weighted spaces	p. 397
Predholm property and applications	p. 399
Non-Fredholm Operators
Weakly non-Fredholm operators	p. 402
Some properties of operators	p. 402
Solvability conditions	p. 408
Examples of weakly non-Fredholm operators	p. 418
Non-Fredholm solvability conditions	p. 426
Example of first-order Ode	p. 426
Ordinary differential systems on the real line	p. 428
Second-order equations	p. 432
Space decomposition of operators	p. 434
Normal solvability on a subspace	p. 436
Scalar equation	p. 440
Strongly non-Fredholm operators	p. 442
Nonlinear Fredholm Operators
Nonlinear elliptic problems	p. 446
Properness	p. 448
Lemma on properness of operators in Banach spaces	p. 448
Properness of elliptic operators in Holder spaces	p. 449
Operators depending on a parameter	p. 453
Example of non-proper operators	p. 454
Properness in Sobolev spaces	p. 455
Topological degree	p. 468
Definition and main properties of the degree	p. 468
Orientation of operators	p. 470
Degree for Fredholm and proper operators	p. 474
Uniqueness of the degree	p. 480
Degree for elliptic operators	p. 484
Non-existence of the degree	p. 487
Existence and bifurcations of solutions	p. 489
Methods of nonlinear analysis	p. 489
Existence of solutions	p. 493
Elliptic problems in unbounded domains	p. 501
Bifurcations	p. 513
Supplement. Discrete operators	p. 527
One-parameter equations	p. 527
Limiting operators and normal solvability	p. 527
Solvability conditions	p. 533
Spectrum of difference and differential operators	p. 534
First-order systems	p. 535
Solvability conditions	p. 536
Higher-order equations	p. 540
Principal eigenvalue	p. 541
Auxiliary results	p. 543
Location of the spectrum	p. 545
Stability of finite difference schemes	p. 549
Historical and Bibliographical Comments	p. 553
Historical Notes	p. 554
Beginning of the theory	p. 554
Existence of solutions of boundary value problems	p. 556
Other elliptic equations	p. 559
Analyticity	p. 561
Eigenvalues	p. 563
Fredholm theory	p. 564
Linear equations	p. 568
A priori estimates	p. 568
Normal solvability and Fredholm property	p. 570
Index	p. 576
Elliptic problems with a parameter	p. 581
Decay and growth of solutions	p. 583
Topological degree	p. 586
Existence and bifurcation of solutions	p. 591
Concluding remarks	p. 593
Acknowledgement	p. 595
Bibliography	p. 597
Notation of Function Spaces	p. 635
Index	p. 637
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