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

TL;DR

This paper studies a heat conduction optimization problem with volume constraints and obstacles, proving regularity and smoothness of solutions and free boundaries, including special regularity near flat boundary portions.

Contribution

It introduces a penalization approach to handle volume constraints and establishes regularity results for minimizers and free boundaries in obstacle problems.

Findings

Minimizers are locally $C^{1,1}$ in the domain and Lipschitz continuous in $\,\mathbb{R}^n$.

The free boundary outside the domain is smooth.

Regularity up to flat boundary portions is $C^{1,1/2}$.

Abstract

We consider the optimization problem of minimizing $\int_{\mathbb{R}^n}|\nabla u|^2\,\mathrm{d}x$ with double obstacles $\phi\leq u\leq\psi$ 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.