Domain Perturbation for Linear and Nonlinear Parabolic Equations

We shall develop a general theory for domain perturbation for linear and nonlinear parabolic equations with measurable coefficients subject to Dirichlet boundary conditions. We show how solutions of linear and nonlinear parabolic equations behave as a sequence of domains approaches an open set. Convergence of domains is understood in a very general sense which allows that certain parts of the domains degenerate and are deleted in the limit, or that small sets are removed. We also consider the periodic problem and establish existence and uniqueness results for periodic solutions for the perturbed problem in a neighborhood of a periodic solution of the original equation. The theory can be used for instance to construct domains where a given nonlinear periodic–parabolic equation has many periodic solutions.