INDEX 747
critical
exponent, 246, 513, 553, 577,
681–688, 696
point, 117, 456, 503–508
value, 504
curvature, 213
mean, 459, 525
principal curvatures, 228
cutoff function, 330
d’Alembert’s formula, 68, 69, 78, 81, 89
Deformation Theorem, 505, 508
DeGiorgi–Moser–Nash estimates, 489
degree, 523
Derrick–Pohozaev identity, 556, 577
difference quotients, 293–295, 331, 337,
475, 486, 667, 677
and weak derivatives, 294
definition of, 294
differential equations
in Banach spaces, 436
in Hilbert spaces, 506, 567
diffusion, 212, 315
and transport, 212–216
equation, 4, 44
DiPerna, R., 166
Dirac measure, 25, 35, 48, 212, 213
Dirichlet’s
boundary condition, 314
energy functional, 42, 525, 526
principle, 42, 458
generalized, 458
dispersion, 177
dissipation, 177
divergence form PDE, 314, 374, 529
domain of dependence
generalized wave equation, 418
nonlinear wave equation, 662–664
wave equation, 83
domain variations, 371, 513–521, 716
Dominated Convergence Theorem, 133,
153, 223, 435, 476, 548, 650, 674,
735
doubling variables, 589, 652
Douglis, A., 166
duality of Hamiltonian and Lagrangian,
121
dual space, 725
of H0
1,
301
dual variational principle, 250, 525
Duhamel’s principle, 49, 80
dynamic programming, 11, 592, 594,
607, 608
Egoroff’s Theorem, 470, 734
eigenfunction, 169, 325
basis, 357
eigenvalue, 11, 169, 325, 356, 618, 620,
725
nonsymmetric elliptic operator,
362–367
principal, 11, 358–367, 448, 549, 576
symmetric elliptic operator, 356–362
eigenvector, 356, 620, 725
left, 618
right, 618
eikonal equation, 5, 93, 606
generalized, 418
elasticity
linear, 6, 89
nonlinear, 481, 502
ellipticity, 315
elliptic operator
fully nonlinear, 608
linear, 314
divergence form, 314, 315, 329, 356,
367
nondivergence form, 314, 346, 362
quasilinear, 485, 543
elliptic regularization, 523
Emden–Fowler transformation, 246
energy decay
wave equation, 521–522
energy estimates
quasilinear elliptic equation, 532
second-order elliptic equation, 319
second-order hyperbolic equation, 404
second-order parabolic equation, 378
symmetric hyperbolic system, 428
energy flux, 663, 679, 682
energy methods
heat equation, 62–65
Laplace’s equation, 41–43
wave equation, 82–84
entropy/entropy-flux pair, 648, 651,
657, 658
entropy condition, 12, 142, 148, 633, 643
condition E, 658
Lax’s, 633
Liu’s, 646
entropy solution
single conservation law, 149, 165, 658
system, 649, 652
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