Index

a priori estimates, 4

adjoint differential operators, 39, 268

analytic functions, 105, 261

auxiliary functions, 121

Bernstein method, 121

Burgers’ equation, 22

Cauchy problems, 11, 48, 251, 256

Cauchy values, 11, 48, 251, 256

Cauchy-Kovalevskaya theorem, 263

characteristic cones, 57

characteristic curves, 14, 50, 253

characteristic hypersurfaces, 13, 14, 16,

50, 253, 256

noncharacteristic hypersurfaces, 13,

14, 16, 50, 253, 256

characteristic ODEs, 19, 21, 26

characteristic triangle, 202

compact supports, 41

comparison principles, 114, 119, 177

compatibility conditions, 25, 79, 83,

207, 210

conservation laws, 24, 282

conservation of energies, 64, 237

convergence of series, 105, 260

absolute convergence, 260

convolutions, 150

d’Alembert’s formula, 204

decay estimates, 230

degenerate differential equations, 51

diameters, 60

differential Harnack inequalities

heat equations, 191

Laplace equations, 109, 122

Dirichlet energy, 142

Dirichlet problems, 58, 93, 111

Green’s function, 94

domains, 1

domains of dependence, 19, 35, 204, 220

doubling condition, 145

Duhamel’s principle, 235

eigenvalue problems, 75, 85

Einstein field equations, 286

elliptic differential equations, 51, 254,

279

energy estimates

first-order PDEs, 37

heat equations, 62

wave equations, 63, 238, 241

Euclidean norms, 1

Euler equations, 284

Euler-Poisson-Darboux equation, 214

exterior sphere condition, 132

finite-speed propagation, 35, 221

first-order linear differential systems,

281

first-order linear PDEs, 11

initial-value problems, 31

first-order quasilinear PDEs, 14

Fourier series, 76

Fourier transforms, 148

inverse Fourier transforms, 153

frequency, 145

291