Journal of the European Mathematical Society
Full-Text PDF (367 KB) | Metadata | Table of Contents | JEMS summary
Published online: 2018-05-22
Quasi-linear PDEs and low-dimensional setsJohn L. Lewis and Kaj Nyström (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