Trace on the boundary for solutions of nonlinear differential equations

Abstract:

Let $L$ be a second order elliptic differential operator in $\mathbb {R}^{d}$ with no zero order terms and let $E$ be a bounded domain in $\mathbb {R}^{d}$ with smooth boundary $\partial E$. We say that a function $h$ is $L$-harmonic if $Lh=0$ in $E$. Every positive $L$-harmonic function has a unique representation \begin{equation*}h(x)=\int _{\partial E} k(x,y) \nu (dy), \end{equation*} where $k$ is the Poisson kernel for $L$ and $\nu$ is a finite measure on $\partial E$. We call $\nu$ the trace of $h$ 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 $u$ 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 $u$ on $\partial E$. $\Gamma$ 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 for a pair $(\Gamma ,\nu )$ to be a trace, and we give a probabilistic formula for the maximal solution with a given trace.