Construction of Solutions to Heat-Type Problems with Volume Constraint via the Discrete Morse Flow

Abstract

A volume-constrained parabolic problem is investigated by means of the discrete Morse flow. We construct a weak solution as the limit of approximate weak solutions, each defined in terms of minimizers of time-discretized functionals, and the uniqueness of the weak solution is discussed. Basic properties (Hölder continuity) of the weak solution are studied and the results of applying this method to models of physical phenomena are shown.