Extrapolation for the $L^p$ Dirichlet Problem in Lipschitz domains

Abstract

Let $\mathcal{L}$ be a second-order linear elliptic operator with complex coefficients. We show that if the $L^p$ Dirichlet problem for the elliptic system $\mathcal{L}(u)=0$ in a fixed Lipschitz domain $\Omega$ in ${\mathbb{R}}^d$ is solvable for some $1<p=p_0<\frac{2(d-1)}{d-2}$, then it is solvable for all $p$ satisfying $$p_0<p<\frac{2(d-1)}{d-2}+\epsilon .$$ The proof is based on a real-variable argument. It only requires that local solutions of $\mathcal{L}(u)=0$ satisfy a boundary Cacciopoli inequality.

References in corpus (1)

• Analyticity of layer potentials and $L^2$ solvability of boundary value problems for divergence form elliptic equations with complex $L^{\infty }$ coefficients

Cited by in corpus (1)

• $L^p$ Neumann Problems in Homogenization of General Elliptic Systems