An optimization problem in heat conduction with volume constraint and double obstacles

Abstract

<p style='text-indent:20px;'>We consider the optimization problem of minimizing <inline-formula><tex-math id="M1">\begin{document}$ \int_{\mathbb{R}^n}|\nabla u|^2\,{\mathrm{d}}x $\end{document}</tex-math></inline-formula> with double obstacles <inline-formula><tex-math id="M2">\begin{document}$ \phi\leq u\leq\psi $\end{document}</tex-math></inline-formula> a.e. in <inline-formula><tex-math id="M3">\begin{document}$ D $\end{document}</tex-math></inline-formula> and a constraint on the volume of <inline-formula><tex-math id="M4">\begin{document}$ \{u&gt;0\}\setminus\overline{D} $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ D\subset\mathbb{R}^n $\end{document}</tex-math></inline-formula> is a bounded domain. By studying a penalization problem that achieves the constrained volume for small values of penalization parameter, we prove that every minimizer is <inline-formula><tex-math id="M6">\begin{document}$ C^{1,1} $\end{document}</tex-math></inline-formula> locally in <inline-formula><tex-math id="M7">\begin{document}$ D $\end{document}</tex-math></inline-formula> and Lipschitz continuous in <inline-formula><tex-math id="M8">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula> and that the free boundary <inline-formula><tex-math id="M9">\begin{document}$ \partial\{u&gt;0\}\setminus\overline{D} $\end{document}</tex-math></inline-formula> is smooth. Moreover, when the boundary of <inline-formula><tex-math id="M10">\begin{document}$ D $\end{document}</tex-math></inline-formula> has a plane portion, we show that the minimizer is <inline-formula><tex-math id="M11">\begin{document}$ C^{1,\frac{1}{2}} $\end{document}</tex-math></inline-formula> up to the plane portion.</p>