An abstract approach to domain perturbation for parabolic equations and parabolic variational inequalities

Parinya Sa Ngiamsunthorn

Abstract

We study the behaviour of solutions of linear non-autonomous parabolic equations subject to Dirichlet or Neumann boundary conditions under perturbation of the domain. We prove that Mosco convergence of function spaces for non-autonomous parabolic problems is equivalent to Mosco convergence of function spaces for the corresponding elliptic problems. As a consequence, we obtain convergence of solutions of non-autonomous parabolic equations under domain perturbation by variational methods using the same characterisation of domains as in elliptic case. A similar technique can be applied to obtain convergence of weak solutions of parabolic variational inequalities when the underlying convex set is perturbed.

Keywords: boundary value problems, domain perturbation, Mosco convergence, non-autonomous parabolic equations, parabolic variational inequalities.

AMS Subject Classification: Primary 35B20; secondary 35K90, 49J40.