754 INDEX backwards in time, 63 Cauchy problem for heat equation, 59 conservation law, 149–153, 652–655 Hamilton–Jacobi equation, 131–134 heat equation, 63 in calculus of variations, 473 Poisson’s equation, 27 quasilinear elliptic equation, 535 viscosity solution, 589 value function, 594, 607 vanishing viscosity method, 217, 582, 585, 649 variation first, 456–459, 462 second, 460–461, 524 variational formulation of elliptic equation, 316 variational inequality, 494–497, 526, 576 velocity group, 178 phase, 178, 695 version of a function, 285 viscosity solution, 581–609 definition of, 584 existence, 592 of elliptic equations, 608 uniqueness, 588, 589 wave N-, 157 plane, 176, 424 rarefaction, 140, 626 shock, 142, 617, 627, 633, 644, 649 simple, 623 traveling, 176, 178, 180, 181, 246, 247, 617, 644–647 wave cone, 83, 662 wave equation, 4, 9, 65–84, 177, 200 d’Alembert’s formula, 68 derivation, 66 dimension even, 78–80 odd, 74–78, 201 one, 19, 67–69, 75, 89, 90, 248 three, 71–72, 89, 681, 695 two, 73–74, 695 energy decay, 521 equipartition of energy, 90, 194–196, 694 generalized, 4 Kirchhoff’s formula, 72 method of descent, 74, 78 nonhomogeneous, 65 nonlinear, 5, 12, 248, 518, 622, 661–698 Poisson’s formula, 74 quasilinear, 613, 661, 665–672, 693 semilinear, 661, 672–688 wave map, 696 wave speeds, 176, 424 weak continuity of determinants, 478 weak convergence, 449, 468, 523, 533, 575, 582, 726 weak partial derivative, 257–270 definition, 258 examples, 259–262 properties, 263–264 uniqueness, 259 weak solution conservation law, 149–153, 614, 651 Euler–Lagrange equation, 474 first-order hyperbolic system, 426 Hamilton–Jacobi equation, 128–135 second-order elliptic equation, 315–327 second-order hyperbolic equation, 401–410 second-order parabolic equation, 375–381 transport equation, 19 well-posed problem, 7, 246 yacht race, 607 Yamabe’s equation, 577 Yosida approximation, 565, 569

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