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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