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