**Memoirs of the American Mathematical Society**

2004;
121 pp;
Softcover

MSC: Primary 35;
Secondary 60

Print ISBN: 978-0-8218-3509-8

Product Code: MEMO/168/798

List Price: $66.00

AMS Member Price: $39.60

MAA Member Price: $59.40

**Electronic ISBN: 978-1-4704-0396-6
Product Code: MEMO/168/798.E**

List Price: $66.00

AMS Member Price: $39.60

MAA Member Price: $59.40

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

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*Benoît Mselati*

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.

#### Readership

Graduate students and research mathematicians interested in partial differential equations.

#### Table of Contents

# Table of Contents

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

- Contents v6 free
- Introduction and statement of the results vii8 free
- Chapter 1. An analytic approach to the equation Δu = u[sup(2)] 118 free
- Chapter 2. A probabilistic approach to the equation Δu = u[sup(2)] 1734
- Chapter 3. Lower bounds for solutions 4966
- Chapter 4. Upper bounds for solutions 85102
- Chapter 5. The classification and representation of the solutions of Δu = u[sup(2)] in a domain 99116
- Appendix A. Technical results 103120
- Appendix. Bibliography 115132