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
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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