Quasi-linear PDEs and low-dimensional sets

Abstract

In this paper we establish new results concerning boundary Harnack inequalities and the Martin boundary problem, for non-negative solutions to equations of $p$-Laplace type with variable coefficients. The key novelty is that we consider solutions which vanish only on a low-dimensional set $\Sigma$ in ${\mathbb{R}}^n$ and this is different compared to the more traditional setting of boundary value problems set in the geometrical situation of a bounded domain in ${\mathbb{R}}^n$ having a boundary with (Hausdorff) dimension in the range $[n-1,n)$. We establish our quantitative and scale-invariant estimates in the context of low-dimensional Reifenberg flat sets.