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
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
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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