Estimates in Morrey Spaces and Hölder Continuity for Weak Solutions to Degenerate Elliptic Systems

Abstract

In this paper, we establish gradient estimates in Morrey spaces and Holder continuity for weak solutions of the following degenerate elliptic system $$-X_{\alpha}^{\ast}(a_{ij}^{\alpha\beta}(x)X_{\beta}u^{j})=g_{i}-X_{\alpha}^{\ast}f_{i}^{\alpha}(x),$$ where X1, . . . , Xq are real smooth vector fields satisfying Hormander’s condition, coefficients \({a_{ij}^{\alpha \beta }\in VMO_X \cap L^\infty (\Omega ), \alpha,\beta=1,2, \,.\,.\,.\, ,q, i,j=1,2, \,.\,.\,.\, ,N, X_{\alpha}^{\ast}}\) is the transposed vector field of Xα.