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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