INDEX 751 Laplace’s equation, 25, 26, 85 measurable function, 734 strongly, 736 weakly, 736 sets, 733 method of descent, 74, 78, 694 minimal surface equation, 5, 210, 459 minimax methods, 370, 507–513 minimizer, 457, 462 existence of, 467–473, 477, 481 local, 482–484 regularity, 484–489 uniqueness of, 473 mixed boundary conditions, 368 mollifier, 266, 716 momentum, generalized, 118 Monge–Amp` ere equation, 5 Monotone Convergence Theorem, 471, 735 monotonicity, 11, 529–535, 563 strict, 534 monotonicity formula p-Laplacian, 517 Almgren’s, 527 Laplace’s equation, 527 Morawetz identity, 520, 683 Morse Lemma, 224 Mountain Pass Theorem, 507, 512 moving plane method, 560–561 multiindices, 13, 703 Multinomial Theorem, 13, 239 multipliers, 513–521 multivalued equation, 525, 567, 568 N-wave, 157 Navier–Stokes equations, 6 Neumann boundary conditions, 368 Newton’s law, 66, 118, 613 Noether’s Theorem, 513–522 noncharacteristic boundary data, 104, 105, 163 surface, 235, 236, 238 nonconvex flux function, 658–659 nondivergence form PDE, 314, 374 nonexistence of solutions, 549–556, 641–643, 688–693, 697 nonhomogeneous problem heat equation, 49, 51 transport equation, 19 wave equation, 80 nonlinear eigenvalue problem, 491 nonlinear Schr¨odinger (NLS) equation, 697 norm, 257, 722, 723 normal derivative, 713 to surface, 713 notation for estimates, 706 for functions, 701–705 for matrices, 699–700 geometric, 700–701 null Lagrangian, 463, 465, 523, 524 O, o notation, 706 obstacle problem, 494 Ohm’s law, 21 Oleinik, O., 166 optimality conditions, 594 orthogonality, 724 p-Laplacian, 5, 516 monotonicity formula, 517 p-system, 612, 651 generalization of, 656 traveling waves for, 646 Palais–Smale condition, 504, 505, 507, 511 parabolic approximation, 216 boundary, 52 cylinder, 52 parabolicity, 374 partial derivatives, 13, 703 count of, 12 partial differential equation definition of, 1 partition of unity, 267, 269, 272, 276, 308 pattern formation, 175 penalty method, 576 periodic solution, 230–233, 576 Perron–Frobenius Theorem, 356 phase, 219 plane, 181 velocity, 178, 695 Plancherel’s Theorem, 187 plane wave, 176, 178, 197, 424 Poisson’s equation, 20, 23, 315, 448 nonlinear, 5, 459 formula ball, 41, 86, 87

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