eBook ISBN:  9781470403966 
Product Code:  MEMO/168/798.E 
List Price:  $66.00 
MAA Member Price:  $59.40 
AMS Member Price:  $39.60 
eBook ISBN:  9781470403966 
Product Code:  MEMO/168/798.E 
List Price:  $66.00 
MAA Member Price:  $59.40 
AMS Member Price:  $39.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 168; 2004; 121 ppMSC: 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.
ReadershipGraduate 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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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