750 INDEX interior ball condition, 350, 558 invariance, 513–522 Galilean, 697 in time, 517 scaling, 45, 185, 246, 516, 518, 682 translation, 516, 527 Inverse Function Theorem, 105, 210, 211, 635, 636, 719 inversion hyperbolic, 519 through sphere, 39 irreversibility, 194, 578 irrotational fluid equations, 209 J¨ orgens, K., 698 k-contact discontinuity, 632 k-rarefaction wave, 626 k-shock wave, 633 k-simple wave, 624 Kelvin transform Laplace’s equation, 87 wave equation, 519, 520, 695 Kirchhoff’s formula, 72 Klein–Gordon equation, 4, 177, 661, 694 Kolmogorov’s equation, 4 Korteweg–deVries (KdV) equation, 5, 178, 248 Kovalevskaya’s example, 250 Kru˘ zkov, S., 659 Lagrange multiplier as eigenvalue, 491 as function, 498 integral constraints, 491 pressure as, 499 Lagrangian, 115, 456, 461 duality with Hamiltonian, 121 null, 463, 465, 523, 524 Lambertian surface, 606 Laplace’s equation, 3, 9, 20–43, 315 analyticity, 31 derivation, 20 Green’s function, 33–41 Harnack’s inequality, 32 Liouville’s Theorem, 30 mean-value formulas, 25 regularity, 28, 29 Laplace’s method, 217 Laplace transform, 203–206, 440 applications, 203–206 definition of, 203 Lax–Milgram Theorem, 317, 319, 321, 367, 510, 731 Lax–Oleinik formula, 10, 143–161, 218 Lax pairs, 248 Lebesgue integral, 734 measure, 732 point, 736 Lebesgue’s Differentiation Theorem, 298, 717, 735 Legendre transform, 120, 121, 163, 164, 603 classical, 210 Leibniz’s formula, 13, 263 linear degeneracy, 625, 631 linear operator, 724 bounded, 724 closed, 725 symmetric, 357, 726 linear PDE definition of, 2 Liouville’s equations, 3, 248 Theorem, 30 Lipschitz continuity, 702 and differentiability a.e., 296 and weak partial derivatives, 296 lower semicontinuity, 562 weak, 469, 470, 564 majorants, 239, 243 maximum principle, 10 Cauchy problem for heat equation, 57 heat equation, 55, 88 Laplace’s equation, 27, 85 strong, 362, 363 heat equation, 55 Laplace’s equation, 27 second-order elliptic equation, 349–353 second-order parabolic equation, 398–400 weak second-order elliptic equation, 346–349 second-order parabolic equation, 391–393 Maxwell’s equations, 6, 89 nonlinear, 657 Mazur’s Theorem, 473, 564, 726 mean-value formulas heat equation, 53

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