The Dirichlet problem for second-order elliptic equations in non-divergence form with continuous coefficients

Abstract

This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^d$. Under certain conditions on the coefficients of $L$, we first establish the existence of a unique Green's function in a ball and derive two-sided pointwise estimates for it. Utilizing these results, we demonstrate the equivalence of regular points for $L$ and those for the Laplace operator, characterized via the Wiener test. This equivalence facilitates the unique solvability of the Dirichlet problem with continuous boundary data in regular domains. Furthermore, we construct the Green's function for $L$ in regular domains and establish pointwise bounds for it. This advancement is significant, as it extends the scope of existing estimates to domains beyond $C^{1,1}$, contributing to our understanding of elliptic operators in non-divergence form.