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

Abstract

We consider the problem of minimizing J(u,E)=\int_{E}f_{h}^{+}(\cdot,\varepsilon(u))dx+\int_{\Omega-E}f^{-}(\cdot,\varepsilon(u))dx+\sigma\left|\partial E\cap\Omega\right| among functions u:\mathbb{R}^{d}\supset\Omega\rightarrow\mathbb{R}^{d}, u_{\mid\partial\Omega}=0, and measurable subsets E of \Omega. Here f_{h}^{+}, f^{-} denote quadratic potentials defined on \overline{\Omega}x{symmetric d x d matrices}, h is the minimum energy of f_{h}^{+} and \varepsilon(u) is the symmetric gradient of the displacement field u. An equilibrium state \hat{u}, \hat{E} of J(u,E) is called one-phase if E=\emptyset or E=\Omega, two-phase otherwise. For two-phase states \sigma\left|\partial E\cap\Omega\right| measures the effect of the separating surface, and we investigate in which way the distribution of phases is affected by the choice of the parameters h\in\mathbb{R}, \sigma > 0. Additional results concern the smoothness of two-phase equilibrium states and the behaviour of inf J(u,E) in the limit \sigma\downarrow0. Moreover, we discuss the case of additional volume force potentials, and extend the previous results to non-zero boundary values.