748 INDEX
envelopes, 94, 162
construction of solutions using, 94
equipartition of energy, 90, 194–196, 694
essential supremum (ess sup), 735
Euler’s formula for Jacobians, 162, 503
Euler–Lagrange equation, 458, 474, 513,
515
harmonic maps, 498
one-dimensional, 116, 163
systems, 462, 478
Euler–Poisson–Darboux equation, 70,
75
Euler equations, 613, 622
barotropic, 639, 657
incompressible, 6, 208
exponential solutions, 176
extensions, 270–273
factoring PDE, 67, 421
Fatou’s Lemma, 571, 675, 735
feedback control, 594
Fick’s law, 21
first-order equation, 9, 91–166
complete integral, 92
notation for, 91
first-order hyperbolic system, 11, 201,
423–435
fixed point methods, 11, 535–545
fixed point theorem
Banach’s, 536, 538, 539, 576, 666,
672, 677
Brouwer’s, 465, 531
Schaefer’s, 541, 544
Schauder’s, 540
Fokker–Planck equation, 4
Fourier’s law, 21
Fourier transform, 10, 187–196,
299–300, 311, 431
and Radon transform, 198–200
and Sobolev spaces, 299
applications, 190–196
definition of, 187, 189
inversion formula, 189
properties, 187, 189
Fredholm alternative, 232, 728, 730
free boundary problem, 497
frequency function, 527
Fujita, H., 579
fully nonlinear PDE, 2
elliptic, 608
function spaces
BMO, 293
Ck, 704
Ck,γ, 257
C1 2, 704
Hk, 260
Hs, 300
H−1,
301
H0
k, 261
Lp,
704, 723
W
k,p,
260
W0
k,p
, 261
Banach space-valued, 303–307
function variations, 513–521
fundamental solution
heat equation, 46, 87, 185, 192
Laplace’s equation, 22, 25
Schr¨ odinger’s equation, 193
Galerkin’s method, 170, 377, 403, 448,
531
Gauss–Bonnet Theorem, 523
Gauss–Green Theorem, 20, 308, 484,
714
general integral, 95
generator of semigroup, 437–441
genuine nonlinearity, 625, 632, 638
geometric optics, 219–229, 417
Gidas, B.–Ni, W.–Nirenberg, L., 579
gradient flow, 568, 578
Green’s
formulas, 33, 34, 715
function, 33–41
derivation of, 33–35
for ball, 40
for half-space, 37
symmetry of, 35
Grillakis, M., 698
group velocity, 178
Hadamard’s
example, 246
variational formula, 371
Hamilton’s differential equations, 114,
115, 119
Hamilton–Jacobi–Bellman equation,
598
and optimal controls, 602
Hamilton–Jacobi equation, 5, 10, 11,
94, 96, 114–135, 171, 417, 581
characteristics for, 113, 115, 122
Hopf–Lax formula, 123
semiconcavity condition, 129
solution a.e., 127
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