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| Авторизация |
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| Evans L.C. — Partial Differential Equations |
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| Предметный указатель |
A priori estimates 504
Action functional 116
Adjoint 639
Adjoint of elliptic operator 302
Adjoint, bilinear form 302
Admissible states 589
Airy's equation 4 173
Almost everywhere 646
Almost separably valued function 650
Amplitude 207
Analyticity 226—233
Analyticity of harmonic functions 31
Analyticity of solutions to heat equation 62
Analyticity of solutions to PDE 229
Arzela — Ascoli compactness criterion 274 455 540 634
Asymptotics 157 199—221
Averages 616
Backwards uniqueness, heat equation 63
Banach space 241 249 635
Barenblatt's solution 182
Barrier 346
Basis of eigenfunctions 169 335 353 360 363 364 388 390 391 493
Basis, orthonormal 637
Beam equation 4
Bernoulli's law 196
Bessel potential 187
Biharmonic equation 345
Bilinear form 296
Bilinear form, estimates for 299—301
Bilinear form, time-dependent 351 378 403
Bistable equation 175
Blow-up, conservation law 597
Blow-up, nonlinear heat equation 513
Blow-up, porous medium equation 171
Bochner's theorem 650
Boundary 626
Boundary data, admissible 106
Boundary data, compatibility conditions on 104
Boundary data, noncharacteristic 104 106
Boundary,  626
Boundary, analytic 626
Boundary, normal to 626
Boundary, straightening 103 224 228 320 325 627
Boundary-value problem, eigenvalue problem 334
Boundary-value problem, homogenization of 218
Boundary-value problem, Laplace's equation 41
Boundary-value problem, nonlinear boundary conditions 489
Boundary-value problem, nonlinear eigenvalue problem 464
Boundary-value problem, nonlinear Poisson equation 507 518
Boundary-value problem, Poisson's equation 27 42
Boundary-value problem, quasilinear elliptic equation 491 505
Boundary-value problem, second-order elliptic equation 293 296
Boundary-value problem, sernilinear elliptic equation 482 514
Browder — Minty method 497 540
Burgers' equation 5 140 141 143
Burgers' equation, vanishing viscosity 204
Burgers' equation, viscous 195
Calculus of variations 11 43 117 123 431—490
Cauchy data 222
Cauchy problem for boundary data 222
Cauchy problem for heat equation 47
Cauchy sequence 241 249 635
Cauchy — Kovalevskaya Theorem 10 228—233
Chain rule 291
Characteristic equation 398
Characteristics 10 19 97—115 152 200 217 399
Characteristics and boundary conditions 103—106
Characteristics and local solution to PDE 106—110
Characteristics, applications of 110—115
Characteristics, crossing 112
Characteristics, derivation 97
Characteristics, examples of 99—102
Characteristics, projected 98
Characteristics, summary of characteristic ODE 98
Clairaut's equation 93
Closed graph theorem 417 638
Coarea formula 396 628
Coercivity 443 453
Cofactor matrix 411 440
Compact embeddings 271
Compact linear operator 640
Compact mapping 503
Complete integral 92
Complete linear space 635
Condition E 612
Conservation law 5 8 10 136—162
Conservation law, characteristics for 112 143
Conservation law, decay in sup-norm 157
Conservation law, decay to N-wave 158
Conservation law, derivation 567
Conservation law, implicit solution 114
Conservation law, initial-value problem 136
Conservation law, integral solution 137 148
Conservation law, Lax — Oleinik formula 146
Conservation law, Rankine — Hugoniot condition 140
Conservation law, systems 6 12 567—606
Conservation law, viscous 234
Constitutive relation 569 578 595
Constraints, incompressibility 472
Constraints, integral 463
Constraints, one-sided 467
Constraints, pointwise 470
Contraction 499 501 529
Contraction, semigroup 414
Control theory 551
Controls, admissible 551
Controls, optimal 11 560
Converting nonlinear into linear equations 194—199
Convex function 523 621
Convex function, domain 523
Convex function, subdifferential of 163 523
Convex function, supporting hyperplane 621
Convexity 447
Convexity, uniform 131 143 144 621
Corrector problem for Green's function 34
Corrector problem for homogenization 221
Cost 552
Cost, running 552
Cost, terminal 552
Courant minimax principle 347
Crandall, M. — Lions, P.-L. 564
critical exponent 486 514
Critical point 118 432 476—481
Critical value 477
Curvature 201
Curvature, mean 435 489
Curvature, principal curvatures 216
Cutoff function 310
d' Alembert's formula 68 69 78 82 88
Deformation Theorem 477 481
DeGiorgi — Nash estimates 462
Degree 487
Derrick — Pohozaev identity 517
Difference quotients 277—279
Difference quotients and weak derivatives 277
Difference quotients, definition of 277
Differential equations in Banach spaces 413
Differential equations in Hilbert spaces 479 528
Diffusion 200
Diffusion and transport 199—204
Diffusion, equation 4 44
DiPerna, R. 165
Dirac measure 25 35 48 200 201
Dirichlet's boundary condition 294
Dirichlet's principle 42 434
Dirichlet's principle, generalized 434
dispersion 173
Dissipation 173
Divergence form PDE 294 350
Domain of dependence, generalized wave equation 395
| Domain of dependence, wave equation 84
Dominated Convergence Theorem 134 154 211 412 452 510 606 648
Doubling variables 547 608
Doughs, A. 165
Dual space 638
Dual space of  283
Duality of Hamiltonian and Lagrangian 122
Duhamel's principle 49 81
Dynamic programming 11 551 552
Egoroff's theorem 446 647
Eigenfunction 169 305
Eigenfunction, basis 335
Eigenvalue 11 169 305 334 574 575 638
Eigenvalue, nonsymmetric elliptic operator 340—344
Eigenvalue, principal 11 336—344 512 537
Eigenvalue, symmetric elliptic operator 334—340
Eigenvector 334 575 638
Eigenvector, left 574
Eigenvector, right 574
Eikonal equation 5 93 96 564
Eikonal equation, generalized 395
Elasticity, linear 6
Elasticity, nonlinear 457 475
Elliptic regularization 487
Ellipticity 294
Energy estimates, quasilinear elliptic equation 494
Energy estimates, second-order elliptic equation 299
Energy estimates, second-order hyperbolic equation 381
Energy estimates, second-order parabolic equation 354
Energy estimates, symmetric hyperbolic system 405
Energy methods, heat equation 62—65
Energy methods, Laplace's equation 41—43
Energy methods, wave equation 83—85
Entropy condition 12 143 149 589 599
Entropy condition, Lax's 589
Entropy condition, Liu's 602
Entropy solution, single conservation law 150
Entropy solution, system 604 607
Entropy/entropy-flux pair 604 607
Envelopes 94
Envelopes, construction of solutions using 94
Equipartition of energy 88
Essential supremum (ess sup) 648
Euler equations 569 578
Euler equations, barotropic 595 611
Euler equations, incompressible 6 196
Euler — Lagrange equation 434 450
Euler — Lagrange equation system 454
Euler — Lagrange equation systems 438
Euler — Lagrange equation, harmonic maps 470
Euler — Lagrange equation, one-dimensional 117
Euler — Poisson — Darboux equation 70 75
Euler's formula for Jacobians 476
Exponential solutions 172
Extensions 254—257
Factoring PDE 67 398
Fatou's lemma 532 648
Feedback control 553
First variation 433
First-order equation 9 91—165
First-order equation, complete integral 92
First-order equation, notation for 91
First-order hyperbolic system 11 400—412
Fixed point theorem, Banach's 498 500 502 537
Fixed point theorem, Brouwer's 441 493
Fixed point theorem, Schaefer's 342 503 507
Fixed point theorem, Schauder's 502
Fixed-point methods 11 498—507
Fokker — Planck equation 4
Fourier transform 10 182—190 282—283 292 408
Fourier transform and Sobolev spaces 282
Fourier transform, applications 186—190
Fourier transform, definition of 183 184
Fourier transform, properties 183 184
Fredholm alternative 220 641 643
Free boundary problem 470
Fully nonlinear PDE 2
Function spaces  618
Function spaces 618
Function spaces  241
Function spaces  245
Function spaces  246
Function spaces  283
Function spaces  283
Function spaces  618 636
Function spaces  244
Function spaces  245
Function spaces BMO 276
Function spaces, Banach space-valued 285—289
Fundamental solution, heat equation 46 180 188
Fundamental solution, Laplace's equation 22 25
Fundamental solution, Schroedinger's equation 188
Galerkin's method 169 353 380 493
Gauss — Bonnet theorem 488
Gauss — Green Theorem 20 627
General integral 96
Generator of semigroup 414—418
Genuine nonlinearity 581 588 594
Geometric notation 614—615
Geometric optics 207—217 395
Gidas, B. — Ni, W. — Nirenberg, L. 538
Gradient flow 529
Green's formulas 33 34 628
Green's function 33—41
Green's function for ball 40
Green's function for half-space 37
Green's function, derivation of 33—35
Green's function, symmetry of 35
Hadamard's example 234
Hamilton — Jacobi equation 5 10 11 94 96 115—136 171 395 539
Hamilton — Jacobi equation, characteristics for 114 116 123
Hamilton — Jacobi equation, Hopf — Lax formula 124
Hamilton — Jacobi equation, semiconcavity condition 130
Hamilton — Jacobi equation, solution a.e. 128
Hamilton — Jacobi equation, uniqueness 132
Hamilton — Jacobi equation, weak solution 132
Hamilton — Jacobi — Bellman equation 557
Hamilton — Jacobi — Bellman equation and optimal controls 560
Hamilton's differential equations 115 116 120
Hamiltonian 119 557
Hamiltonian duality with Lagrangian 122
Harmonic function 20
Harmonic mapping 470
Heat ball 52
Heat equation 4 9 44—65 172
Heat equation, backwards uniqueness 63
Heat equation, derivation 44
Heat equation, energy methods 62—65
Heat equation, fundamental solution 46
Heat equation, infinite propagation speed 49
Heat equation, initial-value problem 47
Heat equation, maximum principle 57
Heat equation, mean-value formula 52
Heat equation, nonhomogeneous problem 44 49
Heat equation, nonlinear 511
Heat equation, regularity 59 61
Heat equation, strong maximum principle 54
Heat equation, uniqueness 56 58
Helmholtz's equation 3 305
Hilbert space 636
Hille — Yosida theorem 418 421 423
Hodograph transform 197
Hoelder spaces 240—241
Homogeneous PDE 2
Homogenization 218—221 235
Hopf — Cole transformation 195
Hopf — Lax formula 10 121—136 539 560—563
Hopf's Lemma 330 333 341 343 344 519
Huygens' principle 80
Hyperbolic system, constant coefficients 408—412
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