Solutions of nonlinear differential equations on a Riemannian manifold and their trace on the Martin boundary

Abstract:

Let $L$ be a second order elliptic differential operator on a Riemannian manifold $E$ with no zero order terms. We say that a function $h$ is $L$-harmonic if $Lh=0$. Every positive $L$-harmonic function has a unique representation \begin{equation*}h(x)=\int _{E’} k(x,y) \nu (dy), \end{equation*} where $k$ is the Martin kernel, $E’$ is the Martin boundary and $\nu$ is a finite measure on $E’$ concentrated on the minimal part $E^{*}$ of $E’$. We call $\nu$ the trace of $h$ on $E’$. Our objective is to investigate positive solutions of a nonlinear equation \begin{equation*}L u=u^{\alpha }\quad \text {on } E \tag {*} \end{equation*} for $1<\alpha \le 2$ [the restriction $\alpha \le 2$ is imposed because our main tool is the $(L,\alpha )$-superdiffusion, which is not defined for $\alpha >2$]. We associate with every solution $u$ of (*) a pair $(\Gamma ,\nu )$, where $\Gamma$ is a closed subset of $E’$ and $\nu$ is a Radon measure on $O=E’\setminus \Gamma$. We call $(\Gamma ,\nu )$ the trace of $u$ on $E’$. $\Gamma$ 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. In an earlier paper, we investigated the case when $L$ is a second order elliptic differential operator in $\mathbb {R}^{d}$ and $E$ is a bounded smooth domain in $\mathbb {R}^{d}$. We obtained necessary and sufficient conditions for a pair $(\Gamma ,\nu )$ to be a trace, and we gave a probabilistic formula for the maximal solution with a given trace. The general theory developed in the present paper is applicable, in particular, to elliptic operators $L$ with bounded coefficients in an arbitrary bounded domain of $\mathbb {R}^{d}$, assuming only that the Martin boundary and the geometric boundary coincide.