Higher integrability of the gradient for the thermal insulation problem (Q6166034)

In this paper, the authors prove higher integrability of the gradient for local minimizers of the thermal insulation problem. Given a bounded connected domain \(\Omega\subset \mathbb{R}^n\), the thermal insulation problem consists of finding a minimizing pair \((A, u)\) of the energy \[ \mathcal{I}(A,u):=\int_A|\nabla u|^2 d\mathcal{L}^n+ \int_{\partial A} |u|^2 d\mathcal{H}^{n-1}+ \mathcal{L}^n(A), \] where \(A\) is a set containing \(\Omega\) and \(u\geq 0\) in \(A\) with \(u=1\) on \(\Omega\). The existence of minimizers in the class of SBV functions for a suitably relaxed energy and certain geometric properties of free boundaries (boundary of the minimizing set) have been established in [\textit{L. Caffarelli} and \textit{D. Kriventsov}, Commun. Partial Differ. Equations 41, No. 7, 1149--1182 (2016; Zbl 1351.35268); \textit{D. Bucur} and \textit{S. Luckhaus}, Arch. Ration. Mech. Anal. 211, No. 2, 489--511 (2014; Zbl 1283.49056)]. In this paper, the authors show that if \(u\in \text{SBV}(\mathbb{R}^n)\) is a local minimizer, then \(|\nabla u|^2\in L^p_{loc}(\mathbb{R}^n\setminus \overline{\Omega})\) for some \(p>1\). As a consequence, the dimension of the singular set of the free boundary is strictly less than \(n-1\). The proof is inspired by the techniques of [\textit{G. De Philippis} and \textit{A. Figalli}, Arch. Ration. Mech. Anal. 213, No. 2, 491--502 (2014; Zbl 1316.49044)], in which a parallel property for minimizers for the Mumford-Shah functional was established.