Rough boundary trace for solutions of $Lu=\psi(u)$

Let $L$ be a second order elliptic differential operator in $\mathbf{R}^d$ and let $E$ be a bounded domain in $\mathbf{R}^d$ 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 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 Véron. In this paper we cover a wide class of functions $\psi$ and simplify substantially the proofs contained in our earlier papers.