Extrapolation of the Dirichlet problem for elliptic equations with complex coefficients

Abstract

In this paper, we prove an extrapolation result for complex coefficient divergence form operators that satisfy a strong ellipticity condition known as p-ellipticity. Specifically, let Ω be a chord-arc domain in Rn and the operator L=∂i(Aij(x)∂j)+Bi(x)∂i be elliptic, with |Bi(x)|≤Kδ(x)−1 for a small K. Let p0=sup⁡{p>1:Aisp-elliptic}.We establish that if the Lq Dirichlet problem is solvable for L for some 1<q<p0(n−1)(n−2), then the Lp Dirichlet problem is solvable for all p in the range [q,p0(n−1)(n−2)). In particular, if the matrix A is real, or n=2, the Lp Dirichlet problem is solvable for p in the range [q,∞).