Lp-Solvability of the Dirichlet Problem for Second Order Parabolic Equations

We study the unique Lp-solvability of the homogeneous Dirichlet problem for a second order parabolic equation of divergence form in a bounded cylindrical domain QT 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 W1,0p (QT). Under this condition, the unique Lp-solvability is also established for equations with continuous coefficients in the closure of QT. Similar solvability results are also obtained in the space V1,0p (QT) consisting of functions in W1,0p (QT) that are Lp(D)-continuous on [0, T] and have zero trace on the lower base of the cylinder QT. It is assumed that p ≥ 2 since for 1 < p < 2 the Dirichlet problem can be unsolvable even if D has smooth boundary. Bibliography: 27 titles. © 2011 Springer Science+Business Media, Inc.