Contents ix

16.6. The reduced wave equation 81

16.7. Faddeev’s fundamental solutions for (−Δ −

k2)

84

§17. Existence of a fundamental solution 85

§18. Hypoelliptic equations 87

18.1. Characterization of hypoelliptic polynomials 89

18.2. Examples of hypoelliptic operators 90

§19. The radiation conditions 91

19.1. The Helmholtz equation in

R3

91

19.2. Radiation conditions 93

19.3. The stationary phase lemma 95

19.4. Radiation conditions for n ≥ 2 98

19.5. The limiting amplitude principle 101

§20. Single and double layer potentials 102

20.1. Limiting values of double layers potentials 102

20.2. Limiting values of normal derivatives of single layer

potentials 106

§21. Problems 107

Chapter IV. Second Order Elliptic Equations in Bounded Domains 111

Introduction to Chapter IV 111

§22. Sobolev spaces in domains with smooth boundaries 112

22.1. The spaces

◦

Hs(Ω) and Hs(Ω) 112

22.2. Equivalent norm in Hm(Ω) 113

22.3. The density of C0

∞

in

◦

Hs(Ω) 114

22.4. Restrictions to ∂Ω 115

22.5. Duality of Sobolev spaces in Ω 116

§23. Dirichlet problem for second order elliptic PDEs 117

23.1. The main inequality 118

23.2. Uniqueness and existence theorem in

◦

H1(Ω) 120

23.3. Nonhomogeneous Dirichlet problem 121

§24. Regularity of solutions for elliptic equations 122

24.1. Interior regularity 123

24.2. Boundary regularity 124

§25. Variational approach. The Neumann problem 125

25.1. Weak solution of the Neumann problem 127

25.2. Regularity of weak solution of the Neumann problem 128

§26. Boundary value problems with distribution boundary data 129