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 $p$-{\it ellipticity}. Specifically, let $\Omega$ be a chord-arc domain in ${\mathbb{R}}^n$ and the operator $\mathcal{L}={\partial }_i\left(A_{ij}(x){\partial }_j\right)+B_i(x){\partial }_i$ be elliptic, with $|B_i(x)|\le K\delta (x{)}^{-1}$ for a small $K$. Let $p_0 = \sup\{p>1: A \,\,\text{is}\,\, \text{$p$-elliptic}\}$. We establish that if the $L^q$ Dirichlet problem is solvable for $\mathcal{L}$ for some $1<q<\frac{p_0(n-1)}{(n-2)}$, then the $L^p$ Dirichlet problem is solvable for all $p$ in the range $[q,\frac{p_0(n-1)}{(n-2)})$. In particular, if the matrix $A$ is real, or $n=2$, the $L^p$ Dirichlet problem is solvable for $p$ in the range $[q,\infty )$.

16 pages. To appear in JFA