solvability of the dirichlet problem for second order parabolic equations

We study the unique [L.sup.p]-solvability of the homogeneous Dirichlet problem for a second order parabolic equation of divergence form in a bounded cylindrical domain [Q.sub.T] in the case where the boundary of the base D is irregular. For the heat operator we find a necessary and sufficient condition on the boundary of D for the solvability of the problem for all p > 1 and obtain the corresponding estimate in the space | [[*.W].sub.p.sup.1,0] ([Q.sub.T]). Under this condition, the unique [L.sub.p]-solvability is also established for equations with continuous coefficients in the closure of [Q.sub.T]. Similar solvability results are also obtained in the space [[*.V].sub.p.sup.1,0]([Q.sub.T]) consisting of functions in [[*.W].sub.p.sup.1,0]([Q.sub.T]) that are [L.sub.p](D)-continuous on [0, T] and have zero trace on the lower base of the cylinder [Q.sub.T]. It is assumed that p [greater than or equal to] 2 since for 1 < p < 2 the Dirichlet problem can be unsolvable even if D has smooth boundary. Bibliography: 27 titles.