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
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