The Dirichlet problem without the maximum principle

Abstract: Consider the Dirichlet problem with respect to an elliptic operator [ A = - \sum_{k,l=1}^d \partial_k \, a_{kl} \, \partial_l - \sum_{k=1}^d \partial_k \, b_k + \sum_{k=1}^d c_k \, \partial_k + c_0 ] on a bounded Wiener regular open set $\Omega \subset R^d$, where $a_{kl}, c_k \in L_\infty(\Omega,R)$ and $b_k,c_0 \in L_\infty(\Omega,C)$. Suppose that the associated operator on $L_2(\Omega)$ with Dirichlet boundary conditions is invertible. Then we show that for all $\varphi \in C(\partial \Omega)$ there exists a unique $u \in C(\overline \Omega) \cap H^1_{\rm loc}(\Omega)$ such that $u|_{\partial \Omega} = \varphi$ and $A u = 0$. In the case when $\Omega$ has a Lipschitz boundary and $\varphi \in C(\overline \Omega) \cap H^{1/2}(\overline \Omega)$, then we show that $u$ coincides with the variational solution in $H^1(\Omega)$.