The Dirichlet problem in convex bounded domains for operators with L8-coefficients

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

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.

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