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arXiv · 2106.13738

The Dirichlet problem for p-minimizers on finely open sets in metric spaces

Anders Björn·Jana Björn·Visa Latvala
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Abstract

We initiate the study of fine $p$-(super)minimizers, associated with $p$-harmonic functions, on finely open sets in metric spaces, where $1 < p < \infty$. After having developed their basic theory, we obtain the $p$-fine continuity of the solution of the Dirichlet problem on a finely open set with continuous Sobolev boundary values, as a by-product of similar pointwise results. These results are new also on unweighted $\mathbf{R}^n$. We build this theory in a complete metric space equipped with a doubling measure supporting a $p$-Poincar\'e inequality.

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Anders Björn, Jana Björn, Visa Latvala. 2021-06-25. The Dirichlet problem for p-minimizers on finely open sets in metric spaces. https://doi.org/10.1007/s11118-022-09996-7

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