  In this article we consider the Dirichlet problem on a bounded domain $\Omega
\subset {\bf R}^d$ 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 $H^1_0$-perturbation of a continuous function
on $\overline \Omega$. This is new even for the Laplacian and solves an open
problem.
