752 INDEX
half-space, 37, 86
wave equation, 74, 79
kernel
ball, 41
half-space, 37
polar coordinates, 715
polyconvexity, 480
porous medium equation, 5, 170, 185
Barenblatt–Kompaneetz–Zeldovich
solution of, 186
variant of, 246
potentials, 209, 210
projected characteristics, 98
propagation speed, 187, 203
finite, 78, 84, 89, 164, 416–419,
662–664
infinite, 49, 398
proper function, 562
quasilinear PDE, 2
Rademacher’s Theorem, 127, 298
radial solution, 21, 22, 246
radiation condition
Sommerfeld, 372
Radon transform, 197–203
and Fourier transform, 198–200
applications, 200–203
definition of, 197
inversion formula, 199
properties, 197–199
vanishing of, 197, 200
Rankine–Hugoniot condition, 139, 140,
143, 616
rarefaction
curve, 624
wave, 140, 626, 656, 657
Rauch, J., 698
Rayleigh’s formula, 359
reaction-diffusion equation, 549
bistable, 180
scalar, 5
system, 6, 172, 537
reflection, 69, 86, 271, 560
reflexive space, 725
regularity, 8
heat equation, 59, 61
Laplace’s equation, 28, 29
minimizers, 484–489
second-order elliptic equation,
328–345
second-order hyperbolic equation,
410–416
second-order parabolic equation,
382–390
Rellich–Kondrachov Compactness
Theorem, 288, 324
resolvent, 204
identity, 440
nonlinear, 565
set, 439, 725
response of system to controls, 593
retarded potential, 82, 676
Riemann’s problem, 12, 153–156,
623–636
local solution of, 635
Riemann invariants, 637–643, 657
blow-up, 641
Riesz Representation Theorem, 302,
318, 319
Robin boundary conditions, 368
sandpiles, 578
scalar conservation law, 135–161, 614,
651–655
several variables, 659
Schauder estimates, 489
Schr¨ odinger’s equation, 4, 177, 247
nonlinear, 5, 697–698
second-order elliptic equation, 10,
313–372
bilinear form for, 316
boundedness of inverse, 326
eigenvalues and eigenfunctions,
356–367
existence theorems, 321–326
Harnack’s inequality, 353
maximum principles, 346–353
regularity, 328–345
boundary, 336–345
interior, 329–336
second-order hyperbolic equation, 11,
400–422, 425
as semigroup, 446
finite propagation speed, 416–419
in two variables, 420–422
canonical forms, 422
regularity, 410–416
weak solution of, 403–410
second-order parabolic equation, 11,
373–400
as semigroup, 444
Harnack’s inequality, 393–397
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