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$${I[u,\chi,h,\sigma]=\int\nolimits_{\Omega} (\chi f_{h}^{+}({\epsilon}(u))+(1-\chi) f^{-}({\epsilon}(u))){\rm d}x+\sigma \left(\int\nolimits_{\Omega} {| \Delta u|}^{2} {\rm d}x \right)^{p/2}}$$ 0< p <1, h ∈ℝ, σ >0, among functions u :ℝ d ⊃Ω→ℝ d , u ∣∂Ω =0, and measurable characteristic functions χ:Ω→ℝ. Here ƒ + h , ƒ − , denote quadratic potentials defined on the space of all symmetric d × d matrices, h is the minimum energy of ƒ + h and ε ( u ) denotes the symmetric gradient of the displacement field. An equilibrium state û , χˆ , of I [·,·, h , σ] is termed one‐phase if χˆ ≡0 or χˆ ≡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.