The Perron solution for elliptic equations without the maximum principle

In this article we consider the Dirichlet problem on a bounded domain $\mathrm{\Omega }\subset {\mathbf{R}}^d$ with respect to a second-order elliptic differential operator in divergence form. We do not assume a divergence condition as in the pioneering work by Stampacchia, but merely assume that $0$ is not a Dirichlet eigenvalue. The purpose of this article is to define and investigate a solution of the Dirichlet problem, which we call Perron solution, in a setting where no maximum principle is available. We characterise this solution in different ways: by approximating the domain by smooth domains from the interior, by variational properties, by the pointwise boundary behaviour at regular boundary points and by using the approximative trace. We also investigate for which boundary data the Perron solution has finite energy. Finally we show that the Perron solution is obtained as an $H_0^1$-perturbation of a continuous function on $\bar{\mathrm{\Omega }}$. This is new even for the Laplacian and solves an open problem.