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In this article we consider the Dirichlet problem on a bounded domain $\Omega\subset {\bf R}^d$ with respect to a second-order elliptic differentialoperator in divergence form. We do not assume a divergence condition as in thepioneering work by Stampacchia, but merely assume that $0$ is not a Dirichleteigenvalue. The purpose of this article is to define and investigate a solutionof the Dirichlet problem, which we call Perron solution, in a setting where nomaximum principle is available. We characterise this solution in differentways: by approximating the domain by smooth domains from the interior, byvariational properties, by the pointwise boundary behaviour at regular boundarypoints and by using the approximative trace. We also investigate for whichboundary data the Perron solution has finite energy. Finally we show that thePerron solution is obtained as an $H^1_0$-perturbation of a continuous functionon $\overline \Omega$. This is new even for the Laplacian and solves an openproblem.

The Perron solution for elliptic equations without the maximum principle