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