The Dirichlet problem without the maximum principle

TL;DR

This paper proves the existence and uniqueness of continuous solutions to the Dirichlet problem for a class of elliptic operators on Wiener regular domains, even without the maximum principle.

Contribution

It establishes solution existence and uniqueness for the Dirichlet problem under minimal regularity assumptions without relying on the maximum principle.

Findings

Unique continuous solutions exist for all boundary data in C(∂Ω).

Solutions coincide with variational solutions when boundary data is in H^{1/2}.

The results hold for elliptic operators with bounded measurable coefficients.

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 $H^1(\Omega)$.