The effect of a surface energy term on the distribution of phases in an elastic medium with a two‐well elastic potential

We consider the problem of minimizing$${J(u,E)}=\int\nolimits_{E} {f_{h}^{+}} {(\cdot , \epsilon(u)){\rm d}x}+\int\nolimits_{\Omega-E} f^{-}(\cdot, \epsilon(u)) {\rm d} x + \sigma {| \partial E{\cap} \Omega|}$$ among functions u :ℝ d ⊃Ω→ℝ d , u ∣∂Ω =0, and measurable subsets E of Ω. Here f h + , f − denote quadratic potentials defined on Ω¯×{symmetric d × d matrices}, h is the minimum energy of f h + and ε ( u ) is the symmetric gradient of the displacement field u . An equilibrium state û , Ê of J ( u , E ) is called one‐phase if E =∅ or E =Ω, two‐phase otherwise. For two‐phase states, σ ∣ ∂ E ∩Ω∣ measures the effect of the separating surface, and we investigate the way in which the distribution of phases is affected by the choice of the parameters h ϵℝ, σ >0. Additional results concern the smoothness of two‐phase equilibrium states and the behaviour of inf J ( u , E ) in the limit σ ↓0. Moreover, we discuss the case of additional volume force potentials, and extend the previous results to non‐zero boundary values. Copyright © 2002 John Wiley & Sons, Ltd.