The Dirichlet problem in convex bounded domains for operators with L^\infty-coefficients

Abstract

Consider the Dirichlet problem for elliptic and parabolic equations in nondivergence form with variable coefficients in convex bounded domains of R^n. We prove solvability of the elliptic problem and maximal L^q-L^p-estimates for the solution of the parabolic problem provided the coefficients are bounded, satisfy a Cordes condition and p in (1,2] is close to 2. This implies that in two dimensions, i.e. n=2, the elliptic Dirichlet problem is always solvable if the associated operator is uniformly strongly elliptic, and p in (1,2] is close to 2, for maximal L^q-L^p-regularity in the parabolic case an additional assumption on the growth of the coefficients is needed.