In this paper we establish new results concerning boundary Harnack inequalities and the Martin boundary problem, for non-negative solutions to equations of -Laplace type with variable coefficients. The key novelty is that we consider solutions which vanish only on a low-dimensional set in and this is different compared to the more traditional setting of boundary value problems set in the geometrical situation of a bounded domain in having a boundary with (Hausdorff) dimension in the range . We establish our quantitative and scale-invariant estimates in the context of low-dimensional Reifenberg flat sets.
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John L. Lewis, Kaj Nyström, Quasi-linear PDEs and low-dimensional sets. J. Eur. Math. Soc. 20 (2018), no. 7, pp. 1689–1746DOI 10.4171/JEMS/797