Classification and Probabilistic Representation of the Positive Solutions of a Semilinear Elliptic Equation

Classification and Probabilistic Representation of the Positive Solutions of a Semilinear Elliptic Equation

by Benoit Mselati
ISBN-10:
0821835092
ISBN-13:
9780821835098
Pub. Date:
03/31/2004
Publisher:
American Mathematical Society

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Overview

Classification and Probabilistic Representation of the Positive Solutions of a Semilinear Elliptic Equation

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]. A probabilistic formula for a solution in terms of its fine trace and of the Brownian snake is also provided. A major role is played by the solutions which are dominated by a harmonic function in $D$. The latters are called moderate in Dynkin's terminology. We show that every nonnegative solution of $\Delta u = u^2$ in $D$ is the increasing limit of moderate solutions.

Product Details

ISBN-13: 9780821835098
Publisher: American Mathematical Society
Publication date: 03/31/2004
Series: Memoirs of the American Mathematical Society Series , #168
Pages: 121
Product dimensions: 6.00(w) x 1.25(h) x 9.00(d)

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