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

Abstract

This paper investigates the Dirichlet problem for a non-divergence form elliptic operator $L$ in a bounded domain of $\mathbb{R}^2$. Assuming that the principal coefficients satisfy the Dini mean oscillation condition, we establish the equivalence between regular points for $L$ and those for the Laplace operator. This result closes a gap left in the authors' recent work on higher-dimensional cases (Math. Ann. 392(1): 573--618, 2025). Furthermore, we construct the Green's function for $L$ in regular two-dimensional domains, extending a result by Dong and Kim (SIAM J. Math. Anal. 53(4): 4637--4656, 2021).