Partial differential equations:
"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 g...
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|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc.
2010
|
| Edition: | 2. ed. |
| Series: | Graduate studies in mathematics
19 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Summary: | "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. |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XXI, 749 S. graph. Darst. |
| ISBN: | 9780821849743 |
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MARC
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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 2. ed. | ||
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2010 | |
| 300 | |a XXI, 749 S. |b 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 | |
| 500 | |a Includes bibliographical references and index | ||
| 520 | 3 | |a "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. | |
| 650 | 4 | |a Differential equations, Partial | |
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Record in the Search Index
| _version_ | 1804141106825265152 |
|---|---|
| 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
|
| any_adam_object | 1 |
| author | Evans, Lawrence C. 1949- |
| author_GND | (DE-588)135567777 |
| author_facet | Evans, Lawrence C. 1949- |
| author_role | aut |
| author_sort | Evans, Lawrence C. 1949- |
| author_variant | l c e lc lce |
| building | Verbundindex |
| bvnumber | BV036061981 |
| callnumber-first | Q - Science |
| callnumber-label | QA377 |
| callnumber-raw | QA377 |
| callnumber-search | QA377 |
| callnumber-sort | QA 3377 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 500 SK 540 |
| classification_tum | MAT 350f |
| ctrlnum | (OCoLC)465190110 (DE-599)BVBBV036061981 |
| dewey-full | 515/.353 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.353 |
| dewey-search | 515/.353 |
| dewey-sort | 3515 3353 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV036061981 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:10:36Z |
| institution | BVB |
| isbn | 9780821849743 |
| language | English |
| lccn | 2009044716 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-018953439 |
| oclc_num | 465190110 |
| open_access_boolean | |
| owner | DE-91G DE-BY-TUM DE-739 DE-83 DE-634 DE-355 DE-BY-UBR DE-824 DE-19 DE-BY-UBM DE-29T DE-20 DE-703 DE-706 DE-188 DE-11 DE-384 DE-M347 DE-92 |
| owner_facet | DE-91G DE-BY-TUM DE-739 DE-83 DE-634 DE-355 DE-BY-UBR DE-824 DE-19 DE-BY-UBM DE-29T DE-20 DE-703 DE-706 DE-188 DE-11 DE-384 DE-M347 DE-92 |
| physical | XXI, 749 S. graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | American Math. Soc. |
| record_format | marc |
| 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 2. ed. Providence, RI American Math. Soc. 2010 XXI, 749 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 19 Includes bibliographical references and index "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. 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 Passau application/pdf 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 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=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 |