An optimization problem in heat conduction with volume constraint and double obstacles
Xiaoliang Li, Cong Wang · Discrete and Continuous Dynamical Systems · 2022
We consider the optimization problem of minimizing \begin{document}$ \int_{\mathbb{R}^n}| abla u|^2\,{\mathrm{d}}x $\end{document} with double obstacles \begin{document}$ \phi\leq u\leq\psi $\end{document} a.e. in \begin{document}$ D $\end{document} and a constraint on the volume of \begin{document}$ \{u>0\}\setminus\overline{D} $\end{document} , where \begin{document}$ D\subset\mathbb{R}^n $\end{document} is a bounded domain. By studying a penalization problem that achieves the constrained volume for small values of penalization parameter, we prove that every minimizer is \begin{document}$ C^{1,1} $\end{document} locally in \begin{document}$ D $\end{document} and Lipschitz continuous in \begin{document}$ \mathbb{R}^n $\end{document} and that the free boundary \begin{document}$ \partial\{u>0\}\setminus\overline{D} $\end{document} is smooth. Moreover, when the boundary of \begin{document}$ D $\end{document} has a plane portion, we show that the minimizer is \begin{document}$ C^{1,\frac{1}{2}} $\end{document} up to the plane portion.