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 $p$-Poincaré inequality with $1<p<\infty$. However, most of the results are new also for unweighted ${\mathbf{R}}^n$.

arXiv admin note: text overlap with arXiv:2106.13738

Cited by in corpus (1)

• The Perron method associated with finely $p$-harmonic functions on finely open sets