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

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Electronic ISBN: 978-1-4704-0396-6
Product Code: MEMO/168/798.E
List Price: $66.00 MAA Member Price:$59.40
AMS Member Price: $39.60 Click above image for expanded view Classification and Probabilistic Representation of the Positive Solutions of a Semilinear Elliptic Equation Available Formats:  Electronic ISBN: 978-1-4704-0396-6 Product Code: MEMO/168/798.E  List Price:$66.00 MAA Member Price: $59.40 AMS Member Price:$39.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 1682004; 121 pp
MSC: Primary 35; Secondary 60;

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.

• Chapters
• 1. An analytic approach to the equation $\Delta u = u^2$
• 2. A probabilistic approach to the equation $\Delta u = u^2$
• 3. Lower bounds for solutions
• 4. Upper bounds for solutions
• 5. The classification and representation of the solutions of $\Delta u = u^2$ in a domain
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Volume: 1682004; 121 pp
MSC: Primary 35; Secondary 60;

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.

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