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