SECOND-ORDER DIFFERENTIABILITY FOR SOLUTIONS OF ELLIPTIC EQUATIONS IN THE PLANE

Abstract. For a second-order elliptic equation of nondivergence form in the plane, we in-vestigate conditions on the coefficients which imply that all strong solutions have first-order derivatives that are Lipschitz continuous or differentiable at a given point. We assume the coefficients have modulus of continuity satisfying the square-Dini condition, and obtain addi-tional conditions associated with a dynamical system that is derived from the coefficients of the elliptic equation. Our results extend those of previous authors who assume the modulus of continuity satisfies the Dini condition