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

TL;DR

This work extends the Dirichlet problem for non-divergence elliptic operators in two dimensions to coefficients with Dini mean oscillation, proving that regular boundary points for L coincide with those for the Laplacian and establishing unique solvability for continuous boundary data in regular domains. A Green's function with a logarithmic-type bound is constructed in 2D via a perturbation of the principal part, and a Wiener-type criterion is developed using relative capacity and potentials. The results bridge 2D potential theory with higher-dimensional theory, providing boundary Hölder continuity estimates and capacity-based regularity criteria, thus extending Green's function techniques to include lower-order terms in 2D. These findings resolve a gap left in prior higher-dimensional work and offer a robust framework for 2D Dirichlet problems in non-divergence form.

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).

Theorems & Definitions (53)

• Theorem 2.1: Theorem \ref{['thm0800sat']}
• Theorem 2.2: Theorem \ref{['thm0802sat']}
• Theorem 2.3: Theorem \ref{['thm1127sat']}
• Theorem 2.4: Theorem \ref{['thm0738']}
• Theorem 3.3
• proof
• Lemma 3.13
• proof
• Theorem 3.15
• proof