746 INDEX
semilinear elliptic equation, 509, 553
boundary condition
Dirichlet, 314
mixed, 368
Neumann, 368, 448
nonlinear, 525
Robin, 368
boundary data
admissible, 105
compatibility conditions on, 103
noncharacteristic, 104, 105, 163
Browder–Minty method, 534, 582
Burgers’ equation, 5, 139, 140, 142
vanishing viscosity, 217
viscous, 207
calculus of variations, 11, 43, 116, 123,
455–528
Cauchy–Kovalevskaya Theorem, 10,
240–245
Cauchy data, 234
Cauchy problem
for boundary data, 234
for heat equation, 47
for Laplace’s equation, 246
Cauchy sequence, 257, 265, 675, 722
chain rule, 310
characteristic equation, 421
characteristics, 10, 19, 96–114, 151, 162,
212, 229, 422
and boundary conditions, 102–105
and local solution to PDE, 105–108
applications of, 109–114
crossing, 111
derivation, 97
examples of, 99–102
projected, 98
summary of characteristic ODE, 98
Clairaut’s equation, 93
Closed Graph Theorem, 439, 725
coarea formula, 418, 715
coercivity, 467, 477
cofactor matrix, 434, 464
Cole–Hopf transformation, 207
compact
embedding, 288
linear operator, 727
mapping, 541
complete
integral, 92, 161
linear space, 257, 722
condition E, 658
conformal
energy, 520
transformation, 87
conservation law, 5, 8, 10, 135–161, 163,
611–659
characteristics for, 111, 142
decay in sup-norm, 156
decay to N-wave, 157
derivation, 611
implicit solution, 113
initial-value problem, 135
integral solution, 136, 147
Lax–Oleinik formula, 145
Rankine–Hugoniot condition, 139
systems, 6, 12, 611–651
uniqueness, 149–153, 652–655
viscous, 247
conservation of
energy, 517, 613, 662, 672, 689, 694,
697
mass, 613, 614, 697
momentum, 527, 613, 614, 697
constitutive relation, 613, 622, 639
constraints
incompressibility, 499
integral, 490
one-sided, 494
pointwise, 497
contraction, 536, 538, 568, 655, 659
semigroup, 437
controls
admissible, 593
optimal, 11, 602
control theory, 592
converting nonlinear into linear
equations, 206–211, 247
convex function, 163, 530, 562, 707
domain, 562
subdifferential of, 164, 562
supporting hyperplane, 707
convexity, 471
uniform, 130, 142, 143, 707
corrector problem
for Green’s function, 34
for homogenization, 233, 250
cost
functional, 594
running, 594
terminal, 594
Courant minimax principle, 370
Crandall, M.–Lions, P.-L., 609
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