Hölder regularity for double divergence form elliptic equations with Morrey data

Abstract

We establish the regularity properties of adjoint solutions to elliptic equations in double divergence form with variable coefficients and rough forcing terms. More precisely, we consider Dirichlet problems associated with the formal adjoint of second-order elliptic operators in non-divergence form, assuming that the coefficient matrix is uniformly elliptic and Hölder continuous, and that the forcing term belongs to a Morrey space $L^{1,\lambda }(\Omega )$.
The existence and uniqueness of adjoint solutions are established via a duality argument. Under the additional assumption that the coefficients are Hölder continuous, we prove that solutions are locally Hölder continuous. Our results parallel classical regularity theory to adjoint problems in double divergence form with non-constant coefficients and merely integrable data, highlighting the natural role of Morrey spaces in this framework.