Convergence and local-to-global results for $p$-superminimizers on quasiopen sets

Abstract

In this paper, several convergence results for fine $p$-(super)minimizers on quasiopen sets in metric spaces are obtained. For this purpose, we deduce a Caccioppoli-type inequality and local-to-global principles for fine $p$-(super)minimizers on quasiopen sets. A substantial part of these considerations is to show that the functions belong to a suitable local fine Sobolev space. We prove our results for a complete metric space equipped with a doubling measure supporting a $p$-Poincar\'e inequality with $1<p< \infty$. However, most of the results are new also for unweighted $\mathbf{R}^n$.