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 $C^{1,\alpha}$ domains. This result extends a recent work by Chen and Wang (Electron. J. Probab. 28(36):54 pp, 2023).