The effective energy in the Allen–Cahn model with deformation

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 e measuring the width of the interface. Equilibrium states of a given elongation and a given interface width are considered and the asymptotics as e → 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 o(e) the interface energy is equal to σe where the coefficient σ 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 as e→ 0.