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