Extrapolation for the $L^p$ Dirichlet Problem in Lipschitz domains
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 $$ p_0<p< \frac{2(d-1)}{d-2} +\varepsilon. $$ 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.