Elliptic and Parabolic Problems in Non-Smooth Domains

Regularity of solutions is an important part of the theory of partial differential equations. In this text, the regularity of solutions to elliptic and parabolic problems in Lipschitz domains is investigated. Maximal regularity estimates are useful when dealing with nonlinear parabolic problems. However, the known maximal regularity results for smooth domains no longer hold in Lp-spaces over Lipschitz domains for the whole range of exponents p. Here, maximal regularity estimates are shown for the Laplacian with suitable domain in Lp-spaces for a restricted range of p. Operators with L-coefficients in convex domains and Ornstein-Uhlenbeck operators in exterior Lipschitz domains are also discussed.