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
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