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

Abstract

We consider the problem of minimizing I\left[u,\chi,h,\sigma\right]=\int_{\Omega}(\chi f_{h}^{+}(\varepsilon(u))+(1-\chi)f^{-}(\varepsilon(u)))dx+\sigma(\int_{\Omega}\left|\bigtriangleup u\right|^{2}dx)^{p/2}, 0<p<1, h\in\mathbb{R}, \sigma>0, among functions u:\mathbb{R}^{d}\supset\Omega\rightarrow\mathbb{R}^{d}, u_{\mid\partial\Omega}=0, and measurable characteristic functions \chi:\Omega\rightarrow\mathbb{R}. Here f_{h}^{+}, f^{-} denote quadratic potentials defined on the space of all symmetric d x d matrices, h is the minimum energy of f_{h}^{+} and \varepsilon(u) denotes the symmetric gradient of the displacement field. An equilibrium state \hat{u}, \hat{\chi} of I[\cdot,\cdot,h,\sigma] is termed one-phase if \hat{\chi}\equiv0 or \hat{\chi}\equiv1, two-phase otherweise. We investigate in which way the distribution of phases is affected by the parameters h and \sigma.