Trace on the boundary for solutions of nonlinear differential equations

Let L be a second order elliptic differential operator in R-d with no zero order terms and let E be a bounded domain in R-d with smooth boundary partial derivative E. We say that a function h is L-harmonic if Lh = 0 in E. Every positive L-harmonic function has a unique representation h(x) = integral(partial derivative E) k(x, y)nu(dy), where k is the Poisson kernel for L and nu is a finite measure on partial derivative E. We call nu the trace of h on partial derivative E. Our objective is to investigate positive solutions of a nonlinear equation Lu = u(alpha) in E for 1 < alpha less than or equal to 2 [the restriction alpha less than or equal to 2 is imposed because our main tool is the alpha-superdiffusion which is not defined for alpha > 2]. We associate with every solution u a pair (Gamma, nu), where Gamma is a closed subset of partial derivative E and nu is a Radon measure on O = partial derivative E \ Gamma. We call (Gamma, nu) the trace of u on partial derivative E. Gamma is empty if and only if u. is dominated by an L-harmonic function. We call such solutions moderate. A moderate solution is determined uniquely by its trace. In general, many solutions can have the same trace. We establish necessary and sufficient conditions fur a pair (Gamma, nu) to be a trace, and we give a probabilistic formula for the maximal solution with a given trace.