An optimization problem in heat conduction with minimal temperature constraint, interior heating and exterior insulation (Q514245)

The author of this interesting paper studies the existence and optimal regularity of the optimal temperature configuration in a heat conduction problem with minimal temperature constraint, interior heating and exterior insulation. The goal of this study is to keep the values of the temperature in a room, above a given temperature profile. The heating sources are situated inside the room, and it is used an insulation material of some volume outside the room. It is known that the optimal configuration is the one that takes the least energy. In other words there is a domain \(D\subset\mathbb{R}^n\) (bounded and smooth, i.e. the room), a smooth nonnegative function \(\varphi : \mathbb{R}^n \to \mathbb{R}\) which is compactly supported in \(D\), that is, the minimum temperature profile. Given a positive number \(m\) which is the volume of the insulation material. The goal is to find a function \(u : \mathbb{R}^n\to \mathbb{R}\) with the Lebesgue measure (denoted by \(|\cdot |\)) \(|{u > 0}\setminus D| = m\), and \(u \geq \phi \). Accept that in the insulation case \(u = 0\) in the set \(\{u > 0\}\setminus D\), and in the case of interior heating \(u \leq 0\) in \(D\). Here one looks for an optimizer which should minimize a certain functional of the energy taken by the interior heating sources. The most natural functional is the total mass of \(-\triangle u\) in \(D\), that is, \(\int_{D}(-\triangle u)dx\). This functional depends on the shape of the set \(\{u > 0\}\) in a highly nonlocal fashion and seems difficult to analyze. Therefore, the author proposes, to study the Dirichlet energy functional \(\int_{D}|\nabla u|^2/2dx\) instead of above stated. Note that both functionals are of the same order. The main ideas here concern solving of a variational problem in Physics, that is, to find a minimizer of the Dirichlet energy functional \(\int_{D}|\nabla u|^2/2dx\) over the class \(K_0 = \bigl\{u\in H_0^1(\mathbb{R}^n): \;u \geq \phi ,|{u > 0}\setminus D| = m, \;\triangle u \leq 0 \;\text{in} \;D \), \( \triangle u = 0 \;\text{in} \;\{u > 0\}\setminus D\bigr\}\). The main result is the existence of a minimizer (suitable function) to the above mentioned variational problem in Physics. This minimizer is Lipschitz continuous in \(\mathbb{R}^n\). Next, concerning the regularity of the interior free boundary the author states the second result, that is, ``for \(\triangle \phi \) uniformly negative in \(\{\phi > 0\}\), the interior free boundary \(\partial (\{u > \phi\} \cap D)\) is smooth except on a set of singular points, which are covered by a countable union of lower-dimensional \(C^1\) manifolds''. The third result includes the theorem: ``The exterior free boundary \(\partial\{u > 0\}\) is smooth except on a \(H^{n-1}\)-null set, where \(H^{n-1}\) is the (\(n-1\))-dimensional Hausdorff measure''.