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Calculus Partial Differential Equations by Lawrence C. Evans

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  1. Feb 22, 2013 #1

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    Table of Contents

    Code (Text):
    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. H¨older 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. Poincar´e’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-1 . . . . . . . . . . . . . . . . . . .299
    5.9.2. Spaces involving time . . . . . . . . . . . . . . .301
    5.10. Problems . . . . . . . . . . . . . . . . . . . . . .305
    5.11. References . . . . . . . . . . . . . . . . . . . . .309

    6. 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
    Preview material - http://www.ams.org/bookstore/pspdf/gsm-19-r-prev.pdf
    http://www.ams.org/bookstore-getitem/item=gsm-19-R

    Readership: Graduate students and research mathematicians interested in partial differential equations.
     
    Last edited by a moderator: May 6, 2017
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