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

  • 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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