$C^{0}$-regularity for solutions of elliptic equations with distributional coefficients

Abstract

In this paper, the continuity of solutions for elliptic equations in divergence form with distributional coefficients is considered. Inspired by the discussion on necessary and sufficient conditions for the form boundedness of elliptic operators by Maz'ya and Verbitsky (Acta Math., 188, 263-302, 2002 and Comm. Pure Appl. Math., 59, 1286-1329, 2006), we propose two kinds of sufficient conditions, which are some Dini decay conditions and some integrable conditions named Kato class or $K^{1}$ class, to show that the weak solution of the Schr\"{o}dinger type elliptic equation with distributional coefficients is continuous and give an almost optimal priori estimate. These estimates can clearly show that how the coefficients and nonhomogeneous terms influence the regularity of solutions. The $\ln$-Lipschitz regularity and H\"{o}lder regularity are also obtained as corollaries which cover the classical De Giorgi's H\"{o}lder estimates.