The authors introduce a method for solving parabolic problems with nonhomogeneous boundary values in non-cylindrical domains. Their starting point is a result of Arendt and Bénilan which says that if \(\Omega\) is a bounded open subset of \(\mathbb R^n\) and if \(A\) is a second-order, uniformly elliptic operator in divergence form then the Dirichlet problem \(Au=0\) in \(\Omega\), \(u=\varphi\) on \(\partial\Omega\) has a unique solution for any continuous \(\varphi\) if and only if \(A_0\) (the restriction of \(A\) to \(\{f\in D(A)\cap C_0(\Omega): Af\in C_0(\Omega)\}\)) generates a semigroup. On the other hand, such a result need not be true if \(A\) is degenerate elliptic. The authors develop a general theory of identifying domains and operators for which this equivalence is valid. By recasting the procedure in an abstract setting, they are able to prove corresponding results for parabolic problems as well. Their condition for parabolic problems is a barrier condition, which is similar to the usual barrier condition (i.e., that there is a positive supersolution to the problem).