Partial differential equations:
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| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Society
2008
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| Edition: | Reprinted with corr. |
| Series: | Graduate studies in mathematics
19 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVII, 662 S. Ill., graph. Darst. |
| ISBN: | 9780821807729 |
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| 100 | 1 | |a Evans, Lawrence C. |d 1949- |e Verfasser |0 (DE-588)135567777 |4 aut | |
| 245 | 1 | 0 | |a Partial differential equations |c Lawrence C. Evans |
| 250 | |a Reprinted with corr. | ||
| 264 | 1 | |a Providence, RI |b American Math. Society |c 2008 | |
| 300 | |a XVII, 662 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Graduate studies in mathematics |v 19 | |
| 650 | 4 | |a Differential equations, Partial | |
| 650 | 0 | 7 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |D s |
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| 830 | 0 | |a Graduate studies in mathematics |v 19 |w (DE-604)BV009739289 |9 19 | |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-017002824 | ||
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| _version_ | 1804138446915108864 |
|---|---|
| adam_text | CONTENTS
Preface
........................................... xv
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
...................... 7
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
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
vu
VIU
CONTENTS
2.2.3.
Properties of harmonic functions
............. 27
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
..................... 54
2.3.4.
Energy methods
.......................... 62
2.4.
Wave equation
............................... 65
2.4.1.
Solution by spherical means
................ 67
2.4.2.
Nonhomogeneous problem
.................. 81
2.4.3.
Energy methods
.......................... 83
2.5.
Problems
.................................... 85
2.6.
References
................................... 89
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
............................... 97
3.2.1.
Derivation of characteristic ODE
............. 97
3.2.2.
Examples
............................... 99
3.2.3.
Boundary conditions
..................... 103
3.2.4.
Local solution
........................... 106
3.2.5.
Applications
............................ 110
3.3.
Introduction to Hamilton-Jacobi equations
........ 115
3.3.1.
Calculus of variations, Hamilton s ODE
...... 116
3.3.2.
Legendre transform, Hopf-Lax formula
....... 121
3.3.3.
Weak solutions, uniqueness
................ 129
3.4.
Introduction to conservation laws
............... 136
3.4.1.
Shocks, entropy condition
................. 137
3.4.2.
Lax-Oleinik formula
..................... 144
3.4.3.
Weak solutions, uniqueness
................ 149
3.4.4.
Biemann s problem
...................... 154
3.4.5.
Long time behavior
...................... 157
CONTENTS___________________________________________________ix
3.5. Problems................................... 162
3.6.
References..................................
165
4.
Other Ways to Represent Solutions
.............. 167
4.1.
Separation of variables
........................ 167
4.2.
Similarity solutions
........................... 172
4.2.1.
Plane and traveling waves, solitons
.......... 172
4.2.2.
Similarity under scaling
................... 180
4.3.
Transform methods
.......................... . 182
4.3.1.
Fourier transform
........................ 182
4.3.2.
Laplace transform
....................... 191
4.4.
Converting nonlinear into linear PDE
............ 194
4.4.1.
Hopf-Cole transformation
................. 194
4.4.2.
Potential functions
....................... 196
4.4.3.
Hodograph and Legendre transforms
......... 197
4.5.
Asymptotics
................................ 199
4.5.1.
Singular perturbations
.................... 199
4.5.2.
Laplace s method
........................ 204
4.5.3.
Geometric optics, stationary phase
.......... 206
4.5.4.
Homogenization
......................... 218
4.6.
Power series
................................. 221
4.6.1.
Noncharacteristic surfaces
................. 221
4.6.2.
Real analytic functions
................... 226
4.6.3.
Cauchy-Kovalevskaya Theorem
............. 228
4.7.
Problems
................................... 233
4.8.
References
.................................. 235
PART II: THEORY FOR LINEAR PARTIAL
DIFFERENTIAL EQUATIONS
5.
Sobolev Spaces
................................ 239
5.1.
Holder spaces
............................... 240
5.2.
Sobolev spaces
.............................. 241
5.2.1.
Weak derivatives
........................ 242
5.2.2.
Definition of Sobolev spaces
............... 244
5.2.3.
Elementary properties
.................... 247
5.3.
Approximation
.............................. 250
χ
CONTENTS
5.3.1.
Interior
approximation
by smooth functions
. . . 250
5.3.2.
Approximation by smooth functions
......... 251
5.3.3.
Global approximation by smooth functions
.... 252
5.4.
Extensions
.................................. 254
5.5.
Traces
..................................... 257
5.6.
Sobolev inequalities
.......................... 261
5.6.1.
Gagliardo-Nirenberg-Sobolev inequality
...... 262
5.6.2.
Morrey s inequality
...................... 266
5.6.3.
General Sobolev inequalities
............... 269
5.7.
Compactness
................................ 271
5.8.
Additional topics
............................ 275
5.8.1.
Poincaré s
inequalities
.................... 275
5.8.2.
Difference quotients
...................... 277
5.8.3.
Differentiability a.e
....................... 280
5.8.4.
Fourier transform methods
................ 282
5.9.
Other spaces of functions
...................... 283
5.9.1.
The space
Я 1
.......................... 283
5.9.2.
Spaces involving time
..................... 285
5.10.
Problems
.................................. 289
5.11.
References
................................. 292
6.
Second-Order Elliptic Equations
................. 293
6.1.
Definitions
.................................. 293
6.1.1.
Elliptic equations
........................ 293
6.1.2.
Weak solutions
.......................... 295
6.2.
Existence of weak solutions
.................... 297
6.2.1.
Lax-Milgram Theorem
................... 297
6.2.2.
Energy estimates
........................ 299
6.2.3.
Fredholm
alternative
..................... 302
6.3.
Regularity
.................................. 308
6.3.1.
Interior regularity
........................ 309
6.3.2.
Boundary regularity
...................... 316
6.4.
Maximum principles
.......................... 326
6.4.1.
Weak maximum principle
................. 327
6.4.2.
Strong maximum principle
................. 330
CONTENTS____________________________________________________xi
6.4.3.
Harnack s inequality
..................... 333
6.5.
Eigenvalues and eigenfunctions
.................. 334
6.5.1.
Eigenvalues of symmetric elliptic operators
.... 334
6.5.2.
Eigenvalues of nonsymmetric elliptic operators
. 340
6.6.
Problems
................................... 345
6.7.
References
.................................. 347
7.
Linear Evolution Equations
..................... 349
7.1.
Second-order parabolic equations
................ 349
7.1.1.
Definitions
............................. 350
7.1.2.
Existence of weak solutions
................ 353
7.1.3.
Regularity
............................. 358
7.1.4.
Maximum principles
...................... 367
7.2.
Second-order hyperbolic equations
............... 377
7.2.1.
Definitions
............................. 377
7.2.2.
Existence of weak solutions
................ 380
7.2.3.
Regularity
............................. 387
7.2.4.
Propagation of disturbances
............... 394
7.2.5.
Equations in two variables
................. 397
7.3.
Hyperbolic systems of first-order equations
........ 400
7.3.1.
Definitions
............................. 400
7.3.2.
Symmetric hyperbolic systems
.............. 402
7.3.3.
Systems with constant coefficients
........... 408
7.4.
Semigroup theory
............................ 412
7.4.1.
Definitions, elementary properties
........... 413
7.4.2.
Generating contraction semigroups
.......... 418
7.4.3.
Applications
............................ 420
7.5.
Problems
................................... 425
7.6.
References
.................................. 427
PART III: THEORY FOR NONLINEAR PARTIAL
DIFFERENTIAL EQUATIONS
8.
The Calculus of Variations
...................... 431
8.1.
Introduction
................................ 431
8.1.1.
Basic ideas
............................. 431
8.1.2.
First variation, Euler-Lagrange equation
..... 432
XU
CONTENTS
8.1.3.
Second variation
......................... 436
8.1.4. Systems............................... 437
8.2.
Existence
of minimizers
....................... 443
8.2.1.
Coercivity, lower semicontinuity
............ 443
8.2.2.
Convexity
.............................. 445
8.2.3.
Weak solutions of Euler-Lagrange equation
. . . 450
8.2.4.
Systems
............................... 453
8.3.
Regularity
.................................. 458
8.3.1.
Second derivative estimates
................ 458
8.3.2.
Remarks on higher regularity
.............. 461
8.4.
Constraints
................................. 463
8.4.1.
Nonlinear eigenvalue problems
.............. 463
8.4.2.
Unilateral constraints, variational inequalities
. 467
8.4.3.
Harmonic maps
......................... 470
8.4.4.
Incompressibility
........................ 472
8.5.
Critical points
............................... 476
8.5.1.
Mountain Pass Theorem
.................. 476
8.5.2.
Application to
semilinear
elliptic PDE
....... 482
8.6.
Problems
................................... 486
8.7.
References
.................................. 490
9.
Nonvariational Techniques
...................... 491
9.1.
Monotonicity methods
........................ 491
9.2.
Fixed point methods
.......................... 498
9.2.1.
Banach s Fixed Point Theorem
............. 498
9.2.2.
Schauder s, Schaefer s Fixed Point Theorems
. . 502
9.3.
Method of
subsolutions
and
supersolutions
........ 507
9.4.
Nonexistence
................................ 511
9.4.1.
Blow-up
............................... 511
9.4.2.
Derrick-Pohozaev identity
................. 514
9.5.
Geometric properties of solutions
................ 517
9.5.1.
Star-shaped level sets
..................... 517
9.5.2.
Radial symmetry
........................ 518
9.6.
Gradient flows
....................·........... 523
9.6.1.
Convex functions on Hubert spaces
.......... 523
CONTENTS xiii
9.6.2. Subdifferentials
and nonlinear semigroups
. ... 528
9.6.3. Applications............................ 534
9.7. Problems................................... 536
9.8.
References
.................................. 538
10. Hamilton-Jacobi
Equations
.................... 539
10.1.
Introduction, viscosity solutions
................ 539
10.1.1.
Definitions
............................ 541
10.1.2.
Consistency
........................... 543
10.2.
Uniqueness
................................ 546
10.3.
Control theory, dynamic programming
........... 550
10.3.1.
Introduction to control theory
............. 551
10.3.2.
Dynamic programming
................... 552
10.3.3.
Hamilton-Jacobi-Bellman equation
......... 554
10.3.4.
Hopf-Lax formula revisited
............... 560
10.4.
Problems
.................................. 563
10.5.
References
................................. 564
11·
Systems of Conservation Laws
................. 567
11.1.
Introduction
............................... 567
11.1.1.
Integral solutions
....................... 570
11.1.2.
Traveling waves, hyperbolic systems
........ 572
11.2.
Riemann s problem
.......................... 579
11.2.1.
Simple waves
.......................... 579
11.2.2.
Rarefaction waves
....................... 582
11.2.3.
Shock waves, contact discontinuities
........ 583
11.2.4.
Local solution of Riemann s problem
........ 590
11.3.
Systems of two conservation laws
............... 593
11.3.1.
Riemann invariants
..................... 593
11.3.2.
Nonexistence of smooth solutions
.......... 597
11.4.
Entropy criteria
............................. 599
11.4.1.
Vanishing viscosity, traveling waves
......... 600
11.4.2.
Entropy/entropy-flux pairs
............... 604
11.4.3.
Uniqueness for a scalar conservation law
..... 607
11.5.
Problems
.................................. 611
11.6.
References
................................. 612
CONTENTS
APPENDICES
Appendix
A:
Notation............................ 613
A.l. Notation
for matrices
......................... 613
A.2. Geometric notation
.......................... 614
A.3. Notation for functions
........................ 615
A.4. Vector-valued functions
....................... 619
A.5. Notation for estimates
........................ 619
A.6. Some comments about notation
................ 620
Appendix B: Inequalities
.......................... 621
B.I. Convex functions
............................ 621
B.2. Elementary inequalities
....................... 622
Appendix C: Calculus Facts
....................... 626
C.I. Boundaries
................................. 626
C.2. Gauss-Green Theorem
........................ 627
C.3. Polar coordinates,
coarea
formula
............... 628
C.4. Convolution and smoothing
.................... 629
C.5. Inverse Function Theorem
..................... 632
C.6. Implicit Function Theorem
.................... 633
C.7. Uniform convergence
......................... 634
Appendix D: Linear Functional Analysis
............ 635
D.I.
Banach spaces
.............................. 635
D.2. Hubert spaces
.............................. 636
D.3. Bounded linear operators
...................... 637
D.4. Weak convergence
........................... 639
D.5. Compact operators,
Fredholm
theory
............ 640
D.6. Symmetric operators
......................... 644
Appendix E: Measure Theory
..................... 645
E.I. Lebesgue measure
............................ 645
E.2. Measurable functions and integration
............ 647
E.3. Convergence theorems for integrals
.............. 648
E.4. Differentiation
.............................. 648
E.5. Banach space-valued functions
.................. 649
Bibliography
..................................... 651
Index
............................................ 655
|
| any_adam_object | 1 |
| author | Evans, Lawrence C. 1949- |
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| discipline | Mathematik |
| edition | Reprinted with corr. |
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| id | DE-604.BV035196305 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:28:19Z |
| institution | BVB |
| isbn | 9780821807729 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017002824 |
| oclc_num | 271446672 |
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| physical | XVII, 662 S. Ill., graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
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| publisher | American Math. Society |
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| series | Graduate studies in mathematics |
| series2 | Graduate studies in mathematics |
| spelling | Evans, Lawrence C. 1949- Verfasser (DE-588)135567777 aut Partial differential equations Lawrence C. Evans Reprinted with corr. Providence, RI American Math. Society 2008 XVII, 662 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 19 Differential equations, Partial Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Graduate studies in mathematics 19 (DE-604)BV009739289 19 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017002824&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Evans, Lawrence C. 1949- Partial differential equations Graduate studies in mathematics Differential equations, Partial Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4044779-0 |
| title | Partial differential equations |
| title_auth | Partial differential equations |
| title_exact_search | Partial differential equations |
| 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=017002824&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009739289 |
| work_keys_str_mv | AT evanslawrencec partialdifferentialequations |