INDEX 749 uniqueness, 131 weak solution, 131 Hamiltonian, 118, 598 duality with Lagrangian, 121 harmonic function, 20 mapping, 497, 526 heat ball, 53 heat equation, 4, 9, 44–65, 176 backwards uniqueness, 63 derivation, 44 energy methods, 62–65 fundamental solution, 46 infinite propagation speed, 49 initial-value problem, 47 maximum principle, 57 mean-value formula, 53 nonhomogeneous problem, 44, 49 nonlinear, 246, 247, 249, 549, 577, 579 regularity, 59, 61 strong maximum principle, 55 uniqueness, 57, 59 Helmholtz’s equation, 3, 325 Hilbert space, 723 Hille–Yosida Theorem, 441, 444, 446 hodograph transform, 209 H¨ older spaces, 256–257 homogeneous PDE, 2 homogenization, 230–233, 249 Hopf’s Lemma, 349, 352, 363, 365, 367, 558 Hopf–Lax formula, 10, 120–135, 164, 165, 581, 602–604 horizon finite, 594 infinite, 607 Huygens’ principle, 80, 203 hyperbolicity, 421, 423, 424 for symmetric system, 424 strict, 424, 617, 622, 639 uniform, 426 hyperbolic system constant coeﬃcients, 201–203, 431–435 nonlinear, 617, 656 symmetric, 425–431 hyperelastic materials, 481 Implicit Function Theorem, 104, 225, 229, 492, 557, 620, 621, 629, 720 incompressibility, 499–503 inequality Cauchy’s, 708 with , 708 Cauchy–Schwarz, 43, 84, 419, 447, 710, 723, 731 Gagliardo–Nirenberg–Sobolev, 10, 278–282 general Sobolev-type, 286 Gronwall’s, 379, 381, 383, 386, 406, 410, 411, 413, 419, 429, 431, 539, 597, 664, 671, 678 differential form, 711 integral form, 712 Hardy’s, 298–299, 310, 679 Harnack’s, 32, 86, 353, 393 H¨ older’s, 709 interpolation, 308, 309, 687 Jensen’s, 707 Minkowski’s, 265, 709 Morrey’s, 10, 282–286 Poincar´ e’s, 282, 291–293, 472, 533 Young’s, 708 with , 709 inhibitor, 175 initial-value problem Burgers’ equation, 139 conservation law, 135, 143, 149, 612, 649, 651 Hamilton–Jacobi equation, 114, 123, 128, 134, 581, 584, 588, 592, 602 heat equation, 47, 192, 203 nonlinear wave equation, 665, 669, 672, 676, 679, 685, 689, 691 quasilinear parabolic equation, 206, 582 Schr¨ odinger’s equation, 193 telegraph equation, 196 transport equation, 18, 85 viscous Burgers’ equation, 207 wave equation, 67, 194, 201, 204, 220 initial/boundary-value problem beam equation, 450 heat equation, 57, 88, 168 parabolic equation, 549 quasilinear parabolic equation, 574 reaction-diffusion system, 172, 537 second-order hyperbolic equation, 401 second-order parabolic equation, 374 wave equation, 69, 82, 521 inner product, 723 integral solution, 136, 147, 614, 652 integration by parts formula, 714

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