Regularity of elliptic equations in double divergence form and applications to Green's function estimates
Abstract
We investigate the regularity of elliptic equations in double divergence form, where the leading coefficients satisfying the Dini mean oscillation condition. We prove that the solutions are differentiable on the zero level set and derive a pointwise bound for the derivative, which substantially improve a recent result by Leit\~ao, Pimentel, and Santos (Anal. PDE 13(4):1129--1144, 2020). As an application, we establish global pointwise estimates for the Green's function of second-order uniformly elliptic operators in non-divergence form, considering Dini mean oscillation coefficients in bounded domains. This result extends a recent work by Chen and Wang (Electron. J. Probab. 28(36):54 pp, 2023).
Keywords
Cite
@article{arxiv.2401.06621,
title = {Regularity of elliptic equations in double divergence form and applications to Green's function estimates},
author = {Jongkeun Choi and Hongjie Dong and Dong-ha Kim and Seick Kim},
journal= {arXiv preprint arXiv:2401.06621},
year = {2025}
}
Comments
31 pages with no figure. Improved Theorem 1.7 by relaxing the domain assumption from $C^{1,1}$ to $C^{1,\alpha}$