The Dirichlet problem in convex bounded domains for operators in non-divergence form with L?-coefficients

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 $a_{ij} \in L^\infty$ 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.