### Citation:

Hieber , M & Wood , I 2007 , ' The Dirichlet problem in convex bounded domains for operators with L8-coefficients ' Differential and Integral Equations , vol 20 , no. 7 , pp. 721-734 .

### Abstract:

Consider the Dirichlet problem for elliptic and parabolic equations in non-divergence form with variable coefficients in convex bounded domains of Rn. We prove solvability of the elliptic problem and maximal Lq-Lp-estimates for the solution of the parabolic problem provided the coefficients aij∈L∞ satisfy a Cordes condition and p∈(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∈(1,2] is close to 2, for maximal Lq-Lp-regularity in the parabolic case an additional assumption on the growth of the coefficients is needed.

### Description:

Wood, Ian; Hieber, M., (2007) 'The Dirichlet problem in convex bounded domains for operators with L8-coefficients', Differential and Integral Equations 20 pp.721-734 RAE2008