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Classification and Probabilistic Representation of the Positive Solutions of a Semilinear Elliptic Equation
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Memoirs of the American Mathematical Society
2004; 121 pp; softcover
Volume: 168
ISBN-10: 0-8218-3509-2
ISBN-13: 978-0-8218-3509-8
List Price: US$66 Individual Members: US$39.60
Institutional Members: US\$52.80
Order Code: MEMO/168/798

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.

Graduate students and research mathematicians interested in partial differential equations.

• An analytic approach to the equation $$\Delta u = u^2$$
• A probabilistic approach to the equation $$\Delta u = u^2$$
• The classification and representation of the solutions of $$\Delta u = u^2$$ in a domain