Tangential approach in the Dirichlet problem for elliptic equations

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.