In this paper we show that, given a planar Reifenberg flat domain with small constant and a divergence form operator associated to a real (not necessarily symmetric) uniformly elliptic matrix with Lipschitz coefficients, the Hausdorff dimension of its elliptic measure is at most 1. More precisely, we prove that there exists a subset of the boundary with full elliptic measure and with σ-finite one-dimensional Hausdorff measure. For Reifenberg flat domains, this result extends a previous work of Thomas H. Wolff for the harmonic measure.

We give sufficient conditions for quasiconformal mappings between simply connected Lipschitz domains to have Hölder, Sobolev and Triebel-Lizorkin regularity in terms of the regularity of the boundary of the domains and the regularity of the Beltrami coefficients of the mappings. The results can be understood as a counterpart for the Kellogg-Warchawski Theorem in the context of quasiconformal mappings.

We study the stability of Triebel-Lizorkin regularity of bounded functions and Lipschitz functions under bi-Lipschitz changes of variables and the regularity of the inverse function of a Triebel-Lizorkin bi-Lipschitz map in Lipschitz domains. To obtain our results we provide an equivalent norm for the Triebel-Lizorkin spaces with fractional smoothness in uniform domains in terms of the first-order difference of the last weak derivative available averaged on balls.

We consider the “thin one-phase” free boundary problem, associated to minimizing a weighted Dirichlet energy of the function in \(\mathbb{R}^{n+1}_+\) plus the area of the positivity set of that function in \(\mathbb{R}^{n}\). We establish full regularity of the free boundary for dimensions \(n \leq 2\), prove almost everywhere regularity of the free boundary in arbitrary dimension and provide content and structure estimates on the singular set of the free boundary when it exists. All of these results hold for the full range of the relevant weight.

While our results are typical for the calculus of variations, our approach does not follow the standard one first introduced by Alt and Caffarelli. Instead, the nonlocal nature of the distributional measure associated to a minimizer necessitates arguments which are less reliant on the underlying PDE.

Let \(\Omega^+\subset\mathbb R^{n+1}\) be an NTA domain and let \(\Omega^-= \mathbb R^{n+1}\setminus \overline{\Omega^+}\) be an NTA domain as well. Denote by \(\omega^+\) and \(\omega^-\) their respective harmonic measures. Assume that \(\Omega^+\) is a \(\delta\)-Reifenberg flat domain, for some \(\delta>0\) small enough. In this paper we show that \(\log\frac{d\omega^-}{d\omega^+}\in VMO(\omega^+)\) if and only if \(\Omega^+\) is vanishing Reifenberg flat, \(\omega^+\) has big pieces of uniformly rectifiable measures, and the inner unit normal of \(\Omega^+\) has vanishing oscillation with respect to the approximate normal. This result can be considered as a two-phase counterpart of a more well known related one-phase problem for harmonic measure solved by Kenig and Toro.