JournalsifbVol. 13, No. 2pp. 255–270

The effective energy in the Allen–Cahn model with deformation

  • M. Šilhavý

    Czech Academy of Sciences, Prague, Czech Republic
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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 ε\varepsilon measuring the width of the interface. Equilibrium states of a given elongation and a given interface width are considered and the asymptotics for ε0\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 o(ε){\rm o}(\varepsilon) the interface energy is equal to σε\sigma\varepsilon where the coefficient σ\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 ε0.\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