Tangential approach in the Dirichlet problem for elliptic equations

TL;DR

This paper shows that for elliptic equations on Lipschitz domains, the boundary convergence of solutions improves from nontangential to tangential under certain regularity conditions, with precise estimates on failure sets.

Contribution

It extends classical harmonic analysis results to elliptic equations with measurable coefficients, establishing tangential convergence from $A_$ absolute continuity.

Findings

Tangential convergence holds under $A_$ condition.

Sharp Hausdorff dimension estimates for failure sets.

Extension of classical harmonic function results.

Abstract

It is well-known that solvability of the $\mathrm{L}^{p}$-Dirichlet problem for elliptic equations $Lu:=-\mathrm{div}(A\nabla u)=0$ with real-valued, bounded and measurable coefficients $A$ on Lipschitz domains $\Omega\subset\mathbb{R}^{1+n}$ is characterised by a quantitative absolute continuity of the associated $L$-harmonic measure. We prove that this local $A_{\infty}$ property is sufficient to guarantee that the nontangential convergence afforded to $\mathrm{L}^{p}$ boundary data actually improves to a certain \emph{tangential} convergence when the data has additional (Sobolev) regularity. Moreover, we obtain sharp estimates on the Hausdorff dimension of the set on which such convergence can fail. This extends results obtained by Dorronsoro, Nagel, Rudin, Shapiro and Stein for classical harmonic functions in the upper half-space.