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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