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

2025/03/31 by Hongjie Dong, Dong, Hongjie, Dong-ha Kim +3 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2504.00190

Abstract

This paper investigates the Dirichlet problem for a non-divergence form elliptic operator L in a bounded domain of ℝ². 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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• Decay Properties of Invariant Measure and Application to Elliptic Homogenization of Non-divergence Form with an Interface
• Regular boundary points and the Dirichlet problem for elliptic equations in double divergence form

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