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 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.