Reduced limit for semilinear boundary value problems with measure data

Abstract

We study boundary value problems for semilinear elliptic equations of the form $-\Delta u+g\circ u=\mu$ in a smooth bounded domain $\Omega \subset R^N$. Let $\{{\mu }_n\}$ and $\{{\tau }_n\}$ be sequences of measure in $\Omega$ and $\partial \Omega$ respectively. Assume that there exists a solution $u_n$ of the equation with $\mu ={\mu }_n$ subject to boundary data ${\tau }_n$. Further assume that the sequences of measures converge in a weak sense to $\mu$ and $\tau$ respectively and $\{u_n\}$ converges to $u$ in $L^1(\Omega )$. In general $u$ is not a solution of the boundary value problem with data $(\mu ,\tau )$. However there exist measures $({\mu }^{\ast },{\tau }^{\ast })$ such that $u$ satisfies the equation with $\mu$ replaced by ${\mu }^{\ast }$ and with $u={\tau }^{\ast }$ on the boundary. The pair $({\mu }^{\ast },{\tau }^{\ast })$ is called the reduced limit of the sequence $\{({\mu }_n,{\tau }_n)\}$. We investigate the relation between the weak limit and the reduced limit and the dependence of the latter on the sequence.

arXiv comment17 pages