The Dirichlet problem without the maximum principle

"The maximum principle plays an important role for the solution of the Dirichlet problem. Now 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 sufficiently regular open set $\Omega \subset \mathbb{R}^d$, where $a_{kl}, c_k \in L_\infty(\Omega,\mathbb{R})$ and $b_k,c_0 \in L_\infty(\Omega,\mathbb{C})$. Suppose that the associated operator on $L_2(\Omega)$ with Dirichlet boundary conditions is invertible. Note that in general this operator does not satisfy the maximum principle. Nevertheless, 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 $H^1(\Omega)$. This is joint work with Wolfgang Arendt."