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