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 boun dary partial derivative E. We say that a function h is L-harmonic if L h = 0 in E. Every positive L-harmonic function has a unique representa tion h(x) = integral(partial derivative E) k(x, y)nu(dy), where k is t he Poisson kernel for L and nu is a finite measure on partial derivati ve 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(a lpha) in E for 1 < alpha less than or equal to 2 [the restriction alph a less than or equal to 2 is imposed because our main tool is the alph a-superdiffusion which is not defined for alpha > 2]. We associate wit h 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 deriv ative E \ Gamma. We call (Gamma, nu) the trace of u on partial derivat ive 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 dete rmined uniquely by its trace. In general, many solutions can have the same trace. We establish necessary and sufficient conditions fur a pai r (Gamma, nu) to be a trace, and we give a probabilistic formula for t he maximal solution with a given trace.