In this paper we consider a minimization problem which arises from thermal insulation. A compact connected set $K$, which represents a conductor of constant temperature, say $1$, is thermally insulated by surrounding it with a layer of thermal insulator, the open set $\Omega\setminus K$ with $K\subset\bar\Omega$. The heat dispersion is then obtained as \[ \inf\left\{ \int_{\Omega}|\nabla \varphi|^{2}dx +\beta\int_{\partial^{*}\Omega}\varphi^{2}d\mathcal H^{n-1} ,\;\varphi\in H^{1}(\mathbb R^{n}), \, \varphi\ge 1\text{ in } K\right\}, \] for some positive constant $\beta$. We mostly restrict our analysis to the case of an insulating layer of constant thickness. We let the set $K$ vary, under prescribed geometrical constraints, and we look for the best (or worst) geometry in terms of heat dispersion. We show that under perimeter constraint the disk in two dimensions is the worst one. The same is true for the ball in higher dimension but under different constraints. We finally discuss few open problems.

中文翻译：

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一个最优绝缘问题

在本文中，我们考虑了由隔热引起的最小化问题。一个紧凑的连通集 $K$，代表一个恒定温度的导体，比如 $1$，通过用一层热绝缘体围绕它来隔热，开集 $\Omega\setminus K$ with $K\subset\bar \欧米茄$。然后得到热分散为 \[ \inf\left\{ \int_{\Omega}|\nabla \varphi|^{2}dx +\beta\int_{\partial^{*}\Omega}\varphi^ {2}d\mathcal H^{n-1} ,\;\varphi\in H^{1}(\mathbb R^{n}), \, \varphi\ge 1\text{ in } K\right \}, \] 为一些正常数 $\beta$。我们主要将我们的分析限制在恒定厚度的绝缘层的情况下。在规定的几何约束下，我们让集合 $K$ 变化，并在散热方面寻找最佳（或最差）几何。我们表明，在周长约束下，二维磁盘是最差的。对于更高维度但在不同约束下的球也是如此。我们最后讨论了几个未解决的问题。