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arXiv · 2008.00883

Resolutivity and invariance for the Perron method for degenerate equations of divergence type

Anders Björn·Jana Björn·Abubakar Mwasa
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Abstract

We consider Perron solutions to the Dirichlet problem for the quasilinear elliptic equation $\mathop{\rm div}\mathcal{A}(x,\nabla u) = 0$ in a bounded open set $\Omega\subset\mathbf{R}^n$. The vector-valued function $\mathcal{A}$ satisfies the standard ellipticity assumptions with a parameter $1<p<\infty$ and a $p$-admissible weight $w$. We show that arbitrary perturbations on sets of $(p,w)$-capacity zero of continuous (and certain quasicontinuous) boundary data $f$ are resolutive and that the Perron solutions for $f$ and such perturbations coincide. As a consequence, we prove that the Perron solution with continuous boundary data is the unique bounded solution that takes the required boundary data outside a set of $(p,w)$-capacity zero.

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Anders Björn, Jana Björn, Abubakar Mwasa. 2020-08-03. Resolutivity and invariance for the Perron method for degenerate equations of divergence type. https://doi.org/10.1016/j.jmaa.2021.125937

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