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Trace on the boundary for solutions of nonlinear differential equations

Authors:	E B Dynkin S E Kuznetsov

Institution:	Department of Mathematics, Cornell University, Ithaca, New York 14853-7901 - ; Department of Mathematics, University of Colorado at Boulder, Boulder, Colorado 80309-0395

Abstract:	Let be a second order elliptic differential operator in $\mathbb{R}^{d}$ with no zero order terms and let be a bounded domain in $\mathbb{R}^{d}$ with smooth boundary $\partial E$ . We say that a function is -harmonic if in . Every positive -harmonic function has a unique representation $\begin{equation}h(x)=\int _{\partial E} k(x,y) \nu (dy), \end{equation}$ where is the Poisson kernel for and $\nu$ is a finite measure on $\partial E$ . We call $\nu$ the trace of on $\partial E$ . Our objective is to investigate positive solutions of a nonlinear equation $\begin{equation}L u=u^{\alpha }\quad \text{in } E \end{equation}$ for $1<\alpha \le 2$ the restriction $\alpha \le 2$ is imposed because our main tool is the $\alpha$ -superdiffusion which is not defined for $\alpha >2$ ]. We associate with every solution a pair $(\Gamma ,\nu )$ , where $\Gamma$ is a closed subset of $\partial E$ and $\nu$ is a Radon measure on $O=\partial E\setminus \Gamma$ . We call $(\Gamma ,\nu )$ the trace of on $\partial E$ . $\Gamma$ is empty if and only if is dominated by an -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 for a pair $(\Gamma ,\nu )$ to be a trace, and we give a probabilistic formula for the maximal solution with a given trace.

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