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Classification and Probabilistic Representation of the Positive Solutions of a Semilinear Elliptic Equation
 
Front Cover for 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
Front Cover for Classification and Probabilistic Representation of the Positive Solutions of a Semilinear Elliptic Equation
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  • Front Cover for Classification and Probabilistic Representation of the Positive Solutions of a Semilinear Elliptic Equation
  • Back Cover for Classification and Probabilistic Representation of the Positive Solutions of a Semilinear Elliptic Equation
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.

    Readership

    Graduate students and research mathematicians interested in partial differential equations.

  • Table of Contents
     
     
    • 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.

Readership

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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