The effect of a penalty term involving higher order derivatives on the distribution of phases in an elastic medium with a two‐well elastic potential

We consider the problem of minimizing 0<p<1, h∈?, σ>0, among functions u:?^d?Ω→?^d, u_∣?Ω=0, and measurable characteristic functions χ:Ω→?. Here ?⁺_h, ?^?, denote quadratic potentials defined on the space of all symmetric d×d matrices, h is the minimum energy of ?⁺_h and ε(u) denotes the symmetric gradient of the displacement field. An equilibrium state û, χ?, of I ·,·,h, σ] is termed one‐phase if χ?≡0 or χ?≡1, two‐phase otherwise. We investigate the way in which the distribution of phases is affected by the choice of the parameters h and σ. Copyright 2002 John Wiley & Sons, Ltd.