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Partial differential equations:

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Bibliographische Detailangaben
Beteilige Person:	Evans, Lawrence C. 1949- (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Providence, RI American Math. Soc. 2010
Ausgabe:	2. ed.
Schriftenreihe:	Graduate studies in mathematics 19
Schlagwörter:	Differential equations, Partial Partielle Differentialgleichung
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018953439&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Abstract:	"This is the second edition of the now definitive text on partial differential equations (PDE). It offers a comprehensive survey of modern techniques in the theoretical study of PDE with particular emphasis on nonlinear equations. Its wide scope and clear exposition make it a great text for a graduate course in PDE. For this edition, the author has made numerous changes, including: a new chapter on nonlinear wave equations, more than 80 new exercises, several new sections, and a significantly expanded bibliography."--Publisher's description.
Beschreibung:	Includes bibliographical references and index
Umfang:	XXI, 749 S. graph. Darst.
ISBN:	9780821849743

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adam_text	CONTENTS Preface to second edition .......................... xvii Preface to first edition ............................ xix 1. Introduction ..................................... 1 1.1. Partial differential equations ..................... 1 1.2. Examples ..................................... 3 1.2.1. Single partial differential equations ............ 3 1.2.2. Systems of partial differential equations ........ 6 1.3. Strategies for studying PDE ...................... 6 1.3.1. Well-posed problems, classical solutions ........ 7 1.3.2. Weak solutions and regularity ................ 7 1.3.3. Typical difficulties ......................... 9 1.4. Overview ..................................... 9 1.5. Problems .................................... 12 1.6. References ................................... 13 PART I: REPRESENTATION FORMULAS FOR SOLUTIONS 2. Four Important Linear PDE ..................... 17 2.1. Transport equation ............................ 18 2.1.1. Initial-value problem ...................... 18 2.1.2. Nonhomogeneous problem .................. 19 2.2. Laplace s equation ............................ 20 2.2.1. Fundamental solution ..................... 21 2.2.2. Mean-value formulas ...................... 25 2.2.3. Properties of harmonic functions ............. 26 2.2.4. Green s function .......................... 33 2.2.5. Energy methods .......................... 41 2.3. Heat equation ................................ 44 2.3.1. Fundamental solution ..................... 45 2.3.2. Mean-value formula ....................... 51 2.3.3. Properties of solutions ..................... 55 2.3.4. Energy methods .......................... 62 2.4. Wave equation ............................... 65 2.4.1. Solution by spherical means ................ 67 2.4.2. Nonhomogeneous problem .................. 80 2.4.3. Energy methods .......................... 82 2.5. Problems .................................... 84 2.6. References ................................... 90 3. Nonlinear First-Order PDE ....................... 91 3.1. Complete integrals, envelopes .................... 92 3.1.1. Complete integrals ........................ 92 3.1.2. New solutions from envelopes ............... 94 3.2. Characteristics ............................... 96 3.2.1. Derivation of characteristic ODE ............. 96 3.2.2. Examples ............................... 99 3.2.3. Boundary conditions ..................... 102 3.2.4. Local solution ........................... 105 3.2.5. Applications ............................ 109 3.3. Introduction to Hamilton-Jacobi equations ........ 114 3.3.1. Calculus of variations, Hamilton s ODE ...... 115 3.3.2. Legendre transform, Hopf-Lax formula ....... 120 3.3.3. Weak solutions, uniqueness ................ 128 3.4. Introduction to conservation laws ............... 135 3.4.1. Shocks, entropy condition ................. 136 3.4.2. Lax-Oleinik formula ..................... 143 3.4.3. Weak solutions, uniqueness ................ 148 3.4.4. Riemann s problem ...................... 153 3.4.5. Long time behavior ...................... 156 3.5. Problems ................................... 161 3.6. References .................................. 165 4. Other Ways to Represent Solutions .............. 167 4.1. Separation of variables ........................ 167 4.1.1. Examples .............................. 168 4.1.2. Application: Turing instability ............. 172 4.2. Similarity solutions ........................... 176 4.2.1. Plane and traveling waves, solitons .......... 176 4.2.2. Similarity under scaling ................... 185 4.3. Transform methods ........................... 187 4.3.1. Fourier transform ........................ 187 4.3.2. Radon transform ........................ 196 4.3.3. Laplace transform ....................... 203 4.4. Converting nonlinear into linear PDE ............ 206 4.4.1. Cole-Hopf transformation ................. 206 4.4.2. Potential functions ....................... 208 4.4.3. Hodograph and Legendre transforms ......... 209 4.5. Asymptotics ................................ 211 4.5.1. Singular perturbations .................... 211 4.5.2. Laplace s method ........................ 216 4.5.3. Geometric optics, stationary phase .......... 218 4.5.4. Homogenization ........................ . 229 4.6. Power series ................................. 232 4.6.1. Noncharacteristic surfaces ................. 232 4.6.2. Real analytic functions ................... 237 4.6.3. Cauchy-Kovalevskaya Theorem ............. 239 4.7. Problems ................................... 244 4.8. References .................................. 249 PART II: THEORY FOR LINEAR PARTIAL DIFFERENTIAL EQUATIONS 5. Sobolev Spaces ................................ 253 5.1. Holder spaces ............................... 254 5.2. Sobolev spaces .............................. 255 5.2.1. Weak derivatives ........................ 255 5.2.2. Definition of Sobolev spaces ............... 258 5.2.3. Elementary properties .................... 261 5.3. Approximation .............................. 264 5.3.1. Interior approximation by smooth functions . . . 264 5.3.2. Approximation by smooth functions ......... 265 5.3.3. Global approximation by smooth functions .... 266 5.4. Extensions .................................. 268 5.5. Traces ..................................... 271 5.6. Sobolev inequalities .......................... 275 5.6.1. Gagliardo-Nirenberg-Sobolev inequality ...... 276 5.6.2. Morrey s inequality ...................... 280 5.6.3. General Sobolev inequalities ............... 284 5.7. Compactness ................................ 286 5.8. Additional topics ............................ 289 5.8.1. Poincaré s inequalities .................... 289 5.8.2. Difference quotients ...................... 291 5.8.3. Differentiability a.e ....................... 295 5.8.4. Hardy s inequality ....................... 296 5.8.5. Fourier transform methods ................ 297 5.9. Other spaces of functions ...................... 299 5.9.1. The space H~l .......................... 299 5.9.2. Spaces involving time ..................... 301 5.10. Problems .................................. 305 5.11. References ................................. 309 Second-Order Elliptic Equations ................. 311 6.1. Definitions .................................. 311 6.1.1. Elliptic equations ........................ 311 6.1.2. Weak solutions .......................... 313 6.2. Existence of weak solutions .................... 315 6.2.1. Lax-Milgram Theorem ................... 315 6.2.2. Energy estimates ........................ 317 6.2.3. Fredholm alternative ..................... 320 6.3. Regularity .................................. 326 6.3.1. Interior regularity ........................ 327 6.3.2. Boundary regularity ...................... 334 6.4. Maximum principles .......................... 344 6.4.1. Weak maximum principle ................. 344 6.4.2. Strong maximum principle ................. 347 6.4.3. Harnack s inequality ..................... 351 6.5. Eigenvalues and eigenfunctions .................. 354 6.5.1. Eigenvalues of symmetric elliptic operators .... 354 6.5.2. Eigenvalues of nonsymmetric elliptic operators . 360 6.6. Problems ................................... 365 6.7. References .................................. 370 7. Linear Evolution Equations ..................... 371 7.1. Second-order parabolic equations ................ 371 7.1.1. Definitions ............................. 372 7.1.2. Existence of weak solutions ................ 375 7.1.3. Regularity ............................. 380 7.1.4. Maximum principles ...................... 389 7.2. Second-order hyperbolic equations ............... 398 7.2.1. Definitions ............................. 398 7.2.2. Existence of weak solutions ................ 401 7.2.3. Regularity ............................. 408 7.2.4. Propagation of disturbances ............... 414 7.2.5. Equations in two variables ................. 418 7.3. Hyperbolic systems of first-order equations ........ 421 7.3.1. Definitions ............................. 421 7.3.2. Symmetric hyperbolic systems .............. 423 7.3.3. Systems with constant coefficients ........... 429 7.4. Semigroup theory ............................ 433 7.4.1. Definitions, elementary properties ........... 434 7.4.2. Generating contraction semigroups .......... 439 7.4.3. Applications ............................ 441 7.5. Problems ................................... 446 7.6. References .................................. 449 PART III: THEORY FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS 8. The Calculus of Variations ...................... 453 8.1. Introduction ................................ 453 8.1.1. Basic ideas ............................. 453 8.1.2. First variation, Euler-Lagrange equation ..... 454 8.1.3. Second variation ......................... 458 8.1.4. Systems ............................... 459 8.2. Existence of minimizers ....................... 465 8.2.1. Coercivity, lower semicontinuity ............ 465 8.2.2. Convexity .............................. 467 8.2.3. Weak solutions of Euler-Lagrange equation . . . 472 8.2.4. Systems ............................... 475 8.2.5. Local minimizers ........................ 480 8.3. Regularity .................................. 482 8.3.1. Second derivative estimates ................ 483 8.3.2. Remarks on higher regularity .............. 486 8.4. Constraints ................................. 488 8.4.1. Nonlinear eigenvalue problems .............. 488 8.4.2. Unilateral constraints, variational inequalities . 492 8.4.3. Harmonic maps ......................... 495 8.4.4. Incompressibility ........................ 497 8.5. Critical points ............................... 501 8.5.1. Mountain Pass Theorem .................. 501 8.5.2. Application to semilinear elliptic PDE ....... 507 8.6. Invariance, Noether s Theorem .................. 511 8.6.1. Invariant variational problems .............. 512 8.6.2. Noether s Theorem ...................... 513 8.7. Problems ................................... 520 8.8. References .................................. 525 9. Nonvariational Techniques ...................... 527 9.1. Monotonicity methods ........................ 527 9.2. Fixed point methods .......................... 533 9.2.1. Banach s Fixed Point Theorem ............. 534 9.2.2. Schauder s, Schaefer s Fixed Point Theorems . . 538 9.3. Method of subsolutions and supersolutions ........ 543 9.4. Nonexistence of solutions ...................... 547 9.4.1. Blow-up ............................... 547 9.4.2. Derrick-Pohozaev identity ................. 551 9.5. Geometric properties of solutions ................ 554 9.5.1. Star-shaped level sets ..................... 554 9.5.2. Radial symmetry ........................ 555 9.6. Gradient flows ............................... 560 9.6.1. Convex functions on Hubert spaces .......... 560 9.6.2. Subdifferentials and nonlinear semigroups .... 565 9.6.3. Applications ............................ 571 9.7. Problems ................................... 573 9.8. References .................................. 577 10. Hamilton—Jacobi Equations .................... 579 10.1. Introduction, viscosity solutions ................ 579 10.1.1. Definitions ............................ 581 10.1.2. Consistency ........................... 583 10.2. Uniqueness ................................ 586 10.3. Control theory, dynamic programming ........... 590 10.3.1. Introduction to optimal control theory ...... 591 10.3.2. Dynamic programming ................... 592 10.3.3. Hamilton-Jacobi-Bellman equation ......... 594 10.3.4. Hopf-Lax formula revisited ............... 600 10.4. Problems .................................. 603 10.5. References ................................. 606 11. Systems of Conservation Laws ................. 609 11.1. Introduction ............................... 609 11.1.1. Integral solutions ....................... 612 11.1.2. Traveling waves, hyperbolic systems ........ 615 11.2. Riemann s problem .......................... 621 11.2.1. Simple waves .......................... 621 11.2.2. Rarefaction waves ....................... 624 11.2.3. Shock waves, contact discontinuities ........ 625 11.2.4. Local solution of Riemann s problem ........ 632 11.3. Systems of two conservation laws ............... 635 11.3.1. Riemann invariants ..................... 635 11.3.2. Nonexistence of smooth solutions .......... 639 11.4. Entropy criteria ............................. 641 11.4.1. Vanishing viscosity, traveling waves ......... 642 11.4.2. Entropy/entropy-flux pairs ............... 646 11.4.3. Uniqueness for scalar conservation laws ..... 649 11.5. Problems .................................. 654 11.6. References ................................. 657 12. Nonlinear Wave Equations ..................... 659 12.1. Introduction ............................... 659 12.1.1. Conservation of energy ................... 660 12.1.2. Finite propagation speed ................. 660 12.2. Existence of solutions ........................ 663 12.2.1. Lipschitz nonlinearities .................. 663 12.2.2. Short time existence ..................... 666 12.3. Semilinear wave equations .................... 670 12.3.1. Sign conditions ......................... 670 12.3.2. Three space dimensions .................. 674 12.3.3. Subcritical power nonlinearities ............ 676 12.4. Critical power nonlinearity .................... 679 12.5. Nonexistence of solutions ..................... 686 12.5.1. Nonexistence for negative energy ........... 687 12.5.2. Nonexistence for small initial data ......... 689 12.6. Problems .................................. 691 12.7. References ................................. 696 APPENDICES Appendix A: Notation ............................ 697 A.I. Notation for matrices ......................... 697 A. 2. Geometric notation .......................... 698 A.3. Notation for functions ........................ 699 A. 4. Vector-valued functions ....................... 703 A.5. Notation for estimates ........................ 703 Α. 6. Some comments about notation ................ 704 Appendix B: Inequalities .......................... 705 B.I. Convex functions ............................ 705 B.2. Useful inequalities ........................... 706 Appendix C: Calculus ............................. 710 C.I. Boundaries ................................. 710 C.2. Gauss-Green Theorem ........................ 711 C.3. Polar coordinates, coarea formula ............... 712 C.4. Moving regions .............................. 713 C.5. Convolution and smoothing .................... 713 C.6. Inverse Function Theorem ..................... 716 C.7. Implicit Function Theorem .................... 717 C.8. Uniform convergence ......................... 718 Appendix D: Functional Analysis .................. 719 D.I. Banach spaces .............................. 719 D.2. Hubert spaces .............................. 720 D.3. Bounded linear operators ...................... 721 D.4. Weak convergence ........................... 723 D.5. Compact operators, Fredholm theory ............ 724 D.6. Symmetric operators ......................... 728 Appendix E: Measure Theory ..................... 729 E.I. Lebesgue measure ............................ 729 E.2. Measurable functions and integration ............ 730 E.3. Convergence theorems for integrals .............. 731 E.4. Differentiation .............................. 732 E.5. Banach space-valued functions .................. 733 Bibliography ..................................... 735 Index ............................................ 741
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title_full	Partial differential equations Lawrence C. Evans
title_fullStr	Partial differential equations Lawrence C. Evans
title_full_unstemmed	Partial differential equations Lawrence C. Evans
title_short	Partial differential equations
title_sort	partial differential equations
topic	Differential equations, Partial Partielle Differentialgleichung (DE-588)4044779-0 gnd
topic_facet	Differential equations, Partial Partielle Differentialgleichung
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018953439&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
volume_link	(DE-604)BV009739289
work_keys_str_mv	AT evanslawrencec partialdifferentialequations

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