Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation

• Main Author: Mselati, Benoîti
• Format: eBook Book
• Language: English
• Published: Providence, R.I American Mathematical Society 2004
• Edition: 1
• Series: Memoirs of the American Mathematical Society
• Subjects: Differential equations, Elliptic
• ISBN: 9780821835098; 0821835092

We are concerned with the nonnegative solutions of $\Delta u = u^2$ in a bounded and smooth domain in $\mathbb{R}^d$. We prove that they are uniquely determined by their fine trace on the boundary as defined in [DK98a], thus answering a major open question of [Dy02]. In this title, a probabilistic f...

Table of Contents:

• Intro -- Contents -- Introduction and statement of the results -- Chapter 1. An analytic approach to the equation Δu = u[sup(2)] -- 1.1. Parametrization near the boundary of the domain -- 1.2. Basic facts about linear elliptic PDE's -- 1.3. The equation Δu = u[sup(2)] -- Chapter 2. A probabilistic approach to the equation Δu = u[sup(2)] -- 2.1. Linear elliptic PDE's and Brownian motion -- 2.2. The Brownian snake -- 2.3. Stochastic boundary values -- 2.4. Fine topology and fine trace -- 2.5. Two key problems -- Chapter 3. Lower bounds for solutions -- Main results proved in this chapter -- 3.1. Upper bounds for u[sub(K)] in terms of the boundary capacity of K -- 3.2. Lower bounds for cr-moderate solutions -- 3.3. Synthesis -- Chapter 4. Upper bounds for solutions -- Main results proved in this chapter -- 4.1. The case of a "strongly star-shaped domain -- 4.2. The case of a general domain -- 4.3. Proof of Proposition 4.1 -- Chapter 5. The classification and representation of the solutions of Δu = u[sup(2)] in a domain -- 5.1. Representation of solutions -- 5.2. Admissible traces and classification results -- Appendix A. Technical results -- A.1. Parametrization near the boundary -- A.2. Estimates for the Green function -- A.3. The auxiliary function p -- A.4. Proof of Lemma 2.20 -- Appendix. Bibliography -- Notation index -- Subject index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- L -- M -- N -- P -- R -- S -- T