Memoirs of the American Mathematical Society 2004; 121 pp; softcover Volume: 168 ISBN10: 0821835092 ISBN13: 9780821835098 List Price: US$66 Individual Members: US$39.60 Institutional Members: US$52.80 Order Code: MEMO/168/798
 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
