Phase-field model with concentrating-potential terms on the boundary

Abstract

In this paper we analyze a generalization of the semilinear phase field model from G. Caginalp (1986, 1991) and A. Jiménez-Casas-A. Rodriguez-Bernal (1996, 2005), where we consider a singular term concentrated in a neighborhood of $\Gamma$, the boundary of domain. The neighborhood shrinks to $\Gamma$ as a parameter $ϵ$ approaches zero.

We prove that this family of solutions, of the new semilinear phase field model, converges in suitable spaces when this parameter tends to zero, to the solutions of a semilinear phase field problem where the concentrating potential are transformed into an extra flux condition on $\Gamma$.