Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation
著者
書誌事項
Classification and probabilistic representation of the positive solutions of a semilinear elliptic equation
（Memoirs of the American Mathematical Society, no. 798）
American Mathematical Society, 2004
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注記
"March 2004, volume 168, number 798 (third of 4 numbers)."
Includes bibliographical references (p. 115-117) and indexes
内容説明・目次
内容説明
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 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.
目次
An analytic approach to the equation $\Delta u = u^2$ A probabilistic approach to the equation $\Delta u = u^2$ Lower bounds for solutions Upper bounds for solutions The classification and representation of the solutions of $\Delta u = u^2$ in a domain Appendix A. Technical results Appendix. Bibliography Notation index Subject index.
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