# The effective energy in the Allen–Cahn model with deformation

### M. Šilhavý

Czech Academy of Sciences, Prague, Czech Republic

## 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 $\varepsilon$ measuring the width of the interface. Equilibrium states of a given elongation and a given interface width are considered and the asymptotics for $\varepsilon\to0$ 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 ${\rm o}(\varepsilon)$ the interface energy is equal to $\sigma\varepsilon$ 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 $\varepsilon\to0.$

## Cite this article

M. Šilhavý, The effective energy in the Allen–Cahn model with deformation. Interfaces Free Bound. 13 (2011), no. 2, pp. 255–270

DOI 10.4171/IFB/257