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Journal of the European Mathematical Society

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Volume 20, Issue 7, 2018, pp. 1689–1746
DOI: 10.4171/JEMS/797

Published online: 2018-05-22

Quasi-linear PDEs and low-dimensional sets

John L. Lewis[1] and Kaj Nyström[2]

(1) University of Kentucky, Lexington, USA
(2) Uppsala University, Sweden

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.

Keywords: Boundary Harnack inequality, $p$-harmonic function, $A$-harmonic function, variable coefficients, Reifenberg flat domain, low-dimensional sets, Martin boundary

Lewis John, Nyström Kaj: Quasi-linear PDEs and low-dimensional sets. J. Eur. Math. Soc. 20 (2018), 1689-1746. doi: 10.4171/JEMS/797