Rough Boundary Trace for Solutions of $Lu=\psi{u}$

Abstract

Let L be a second order elliptic differential operator in ${\bf R}^d$ and let E be a bounded domain in ${\bf R}^d$ with smooth boundary $\partial E$. A pair $(\Gamma,\nu)$ is associated with every positive solution of a semilinear differential equation $Lu=\psi(u)$ in E, where $\Gamma$ is a closed subset of~$\partial E$ and~$\nu$ is a Radon measure on $O=\partial E\setminus \Gamma$. We call this pair the {\it rough trace of u on~$\partial E$}. (In [E. B. Dynkin and S. E. Kuznetsov, Comm. Pure Appl. Math., 51 (1998), pp. 897--936], we introduced a fine trace allowing us to distinguish solutions with identical rough traces.) The case of $\psi(u)=u^\alpha$ with $\alpha>1$ was investigated using various methods by Le Gall, Dynkin, and Kuznetsov and by Marcus and V\'eron. In this paper we cover a wide class of functions~$\psi$ and simplify substantially the proofs contained in our earlier papers.