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

We consider the optimization problem of minimizing ${\int }_{{\mathbb{R}}^n}\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2\mathrm{d}x$ with double obstacles $\phi\leq u\leq\psi$a\mathrm{.}e\mathrm{.}in$D$andaconstraintonthevolumeof${u>0}\setminus\overline{D}$,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 $\mathrm{\partial }\{u>0\}\setminus \bar{D}$ 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.