Phase-field systems with vectorial order parameters including diffusional hysteresis effects

This paper is concerned with phase-field systems of Penrose-Fife type which model the dynamics of a phase transition with non-conserved vectorial order parameter. The main novelty of the model is that the evolution of the order parameter vector is governed by a system consisting of one partial differential equation and one partial differential inclusion, which in the simplest case may be viewed as a diffusive approximation of the so-called multi-dimensional stop operator, which is one of the fundamental hysteresis operators. Results concerning existence, uniqueness and continuous dependence on data are presented which can be viewed as generalizations of recent results by the authors to cases where a diffusive hysteresis occurs

Lagrange multiplier and singular limit of double obstacle problems for Allen-Cahn equation with constraint

We consider an Allen--Cahn equation with a constraint of double obstacle-type. This constraint is a subdifferential of an indicator function on the closed interval, which is a multivalued function. In this paper we study the properties of the Lagrange multiplier to our equation. Also, we consider the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier to our double obstacle problem. Moreover, we give some numerical experiments of our problem by using the Lagrange multiplier