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