Abstract

We study thermal insulating of a bounded body $\Omega \subset {\mathbb{R}}^n.$ Under a prescribed heat source $f\ge 0,$ we consider a model of heat transfer between Ω and the environment determined by convection; this corresponds, before insulation, to Robin boundary conditions. The body is then surrounded by a layer of insulating material of thickness of size $\epsilon >0,$ and whose conductivity is also proportional to ε. This corresponds to the case of a small amount of insulating material, with excellent insulating properties. We then compute the Γ-limit of the energy functional $F_{\epsilon }$ and prove that this is a functional F whose minimizers still satisfy an elliptic PDEs system with a non uniform Robin boundary condition depending on the distribution of insulating layer around Ω. In a second step we study the maximization of heat content (which measures the goodness of the insulation) among all the possible distributions of insulating material with fixed mass, and prove an optimal upper bound in terms of geometric properties. Eventually we prove a conjecture in [6 Friedman, A. (1980). Reinforcement of the principal eigenvalue of an elliptic operator. Arch. Rational Mech. Anal. 73(1):1–17. DOI: https://doi.org/10.1007/BF00283252.[Crossref], [Web of Science ®] , [Google Scholar]] which states that the ball surrounded by a uniform distribution of insulating material maximizes the heat content.

摘要

我们研究有界体的热绝缘 $\Omega \subset {\mathbb{电阻}}^n.$ 在规定的热源下 $F\ge 0,$我们考虑由对流确定的 Ω 和环境之间的热传递模型；在绝缘之前，这对应于 Robin 边界条件。然后主体被一层厚度为$\epsilon >0,$并且其电导率也与ε成正比。这对应于少量绝缘材料的情况，具有优良的绝缘性能。然后我们计算能量泛函的Γ-limit$F_{\epsilon }$并证明这是一个泛函F，其极小值仍然满足椭圆偏微分方程系统，该系统具有非均匀罗宾边界条件，取决于 Ω 周围绝缘层的分布。在第二步中，我们研究具有固定质量的绝缘材料的所有可能分布中热含量的最大化（测量绝缘的好坏），并证明几何特性方面的最佳上限。最终我们证明了 [ 6] 中的一个猜想 弗里德曼，A.（1980 年）。椭圆算子的主特征值的强化。拱。理性机械。肛门。73(1): 1 – 17。DOI：https://doi.org/10.1007/BF00283252。[Crossref]、[Web of Science®]、[  Google Scholar] ] 中指出，被均匀分布的绝缘材料包围的球可使热含量最大化。