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

Abstract

Consider the Dirichlet problem for elliptic and parabolic equations in non-divergence form with variable coefficients in convex bounded domains of $\mathbb R^n$. We prove solvability of the elliptic problem and maximal $L^q$-$L^p$-estimates for the solution of the parabolic problem provided the coefficients $a_{ij} \in L^\infty$ 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.