The effective energy in the Allen–Cahn model with deformation

Abstract

The sharp interface limit of a diffuse interface theory of phase transitions is considered in static situations. The diffuse interface model is of the Allen–Cahn type with deformation, with a parameter $\epsilon$ measuring the width of the interface. Equilibrium states of a given elongation and a given interface width are considered and the asymptotics for $\epsilon \to 0$ of the equilibrium energy is determined. The interface energy is defined as the excess energy over the corresponding two phase state with a sharp interface without the interface energy. It is shown that to within the term of order $\mathrm{o}(\epsilon )$ the interface energy is equal to $\sigma \epsilon$ where the coefficient $\sigma$ is given by a new formula that involves the mechanical contribution to the total energy. Also the corresponding equilibrium states are determined and shown to converge to a sharp interface state for $\epsilon \to 0.$