Reduced limit for semilinear boundary value problems with measure data

TL;DR

This paper investigates the behavior of solutions to semilinear elliptic boundary value problems with measure data, focusing on the concept of reduced limits when sequences of measures converge weakly.

Contribution

It introduces the notion of reduced limits for measure data in semilinear elliptic problems and analyzes their relation to weak limits and sequence dependence.

Findings

Reduced limits may differ from weak limits of measures.

The paper characterizes conditions under which solutions converge.

It explores the dependence of reduced limits on measure sequences.

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 $(\mu^*,\tau^*)$ such that $u$ satisfies the equation with $\mu$ replaced by $\mu^*$ and with $u=\tau^*$ on the boundary. The pair $(\mu^*,\tau^*)$ 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.