Contents
Introduction and statement of the results vii
Chapter 1. An analytic approach to the equation Au = u2 1
1.1. Parametrization near the boundary of the domain 1
1.2. Basic facts about linear elliptic PDE's 2
1.3. The equation Au u2 6
Chapter 2. A probabilistic approach to the equation Au = u2 17
2.1. Linear elliptic PDE's and Brownian motion 17
2.2. The Brownian snake 19
2.3. Stochastic boundary values 26
2.4. Fine topology and fine trace 43
2.5. Two key problems 48
Chapter 3. Lower bounds for solutions 49
Main results proved in this chapter 49
3.1. Upper bounds for UK in terms of the boundary capacity of K 51
3.2. Lower bounds for cr-moderate solutions 74
3.3. Synthesis 80
Chapter 4. Upper bounds for solutions 85
Main results proved in this chapter 85
4.1. The case of a "strongly star-shaped domain" 85
4.2. The case of a general domain 93
4.3. Proof of Proposition 4.1 96
Chapter 5. The classification and representation of the solutions of Au =
u2
in a domain 99
5.1. Representation of solutions 99
5.2. Admissible traces and classification results 101
Appendix A. Technical results 103
A.l. Parametrization near the boundary 103
A.2. Estimates for the Green function 104
A.3. The auxiliary function p 109
A.4. Proof of Lemma 2.20 111
Appendix. Bibliography 115
Notation index 118
Subject index 120
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