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

Hongjie Dong; Dong-ha Kim; Seick Kim

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arXiv · 2504.00190

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

Hongjie Dong·Dong-ha Kim·Seick Kim

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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).

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Hongjie Dong, Dong-ha Kim, Seick Kim. 2025-03-31. The Dirichlet problem for second-order elliptic equations in non-divergence form with continuous coefficients: The two-dimensional case. https://arxiv.org/abs/2504.00190

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