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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