A study on class of elliptic PDEs with measure data

Ratan Kr Giri, Debajyoti Choudhuri

Published 2016-04-30Version 1

We study the following PDE \begin{align*} \begin{split} -\Delta_p u + g\circ u & = \mu\,\,\mbox{in}\,\,\Omega,\\ u & = \nu\,\,\mbox{on}\,\,\partial\Omega, \end{split} \end{align*} where $\Omega$ is a bounded domain $\mathbb{R}^N$ with $\partial\Omega$ being the boundary of $\Omega$, $1<p<N$. The problem investigated here is motivated from the work of Bhakta and Marcus \cite{bhakta} where a problem close to the problem above has been addressed. Suppose $(\mu_n)$, $(\nu_n)$ are a sequence of measures for which the above PDE has a corresponding sequence of solutions say $(u_n)$. In general if $\mu_n$, $\nu_n$ converges weakly to $\mu$, $\nu$ respectively and the corresponding sequence of solutions $u_n$ converges to $u$ in $L^1(\Omega)$, then $u$ need not be a solution to the above equation. However it is found in the literature for $p=2$ that to each pair of measures $(\mu,\nu)$ there exists another pair of measures, say, $(\mu^{\#},\nu^{\#})$, called as the {\it reduced limit} corresponding to which $u$ is a solution to the above equation. We address a similar issue for the case of $p \neq 2$, $p < N$. Further, we also derive a similar relationship between the reduced and the weak limit of the measure as in \cite{bhakta}, but for $p \neq 2$.