$C^\infty$ regularity of the free boundary for a two-dimensional optimal compliance problem

Abstract

We study the regularizing effect of perimeter penalties for a problem of optimal compliance in two dimensions. In particular, we consider minimizers of $$\mathcal{E}(\Omega) = J(\Omega) + \lambda \vert\Omega\vert + \mu \mathcal{H}^1(\partial \Omega)$$ where $$J(\Omega) = -2 inf \left\{\frac{1}{2} \int_{\Omega} {\bf A} e(u) : e(u)- \int_\Gamma f\cdot u : u\in LD(\Omega), u\equiv 0 \textrm{on} D \right\}. $$ The sets $D\subset \Omega$ , $\Gamma\subset \overline{\Omega}$ , and the force f are given. We show that if we consider only scalar valued u and constant ${\bf A}$ , or if we consider the elastic energy $\vert\nabla u\vert^2$ , then $\partial \Omega$ is $C^\infty$ away from where $\Omega$ is pinned. In the scalar case, we also show that, for any ${\bf A}$ of class $C^{k,\theta}$ , $\partial \Omega$ is $C^{ k+2,\theta}$ . The proofs rely on a notion of weak outward curvature of $\partial \Omega$ , which we can bound without considering properties of the minimizing fields, together with a bootstrap argument.