An Optimization Problem in Heat Conduction With Volume Constraint and Double Obstacles

Abstract

We consider the optimization problem of minimizing ${\int }_{{\mathbb{R}}^n}|\nabla u{|}^2\mathrm{d}x$ with double obstacles $\varphi \le u\le \psi$ a.e. in $D$ and a constraint on the volume of $\{u>0\}\setminus \overline{D}$, where $D\subset {\mathbb{R}}^n$ is a bounded domain. By studying a penalization problem that achieves the constrained volume for small values of penalization parameter, we prove that every minimizer is $C^{1,1}$ locally in $D$ and Lipschitz continuous in ${\mathbb{R}}^n$ and that the free boundary $\partial \{u>0\}\setminus \overline{D}$ is smooth. Moreover, when the boundary of $D$ has a plane portion, we show that the minimizer is $C^{1,\frac{1}{2}}$ up to the plane portion.

20 pages