arXiv 2005.119342020-05math.APmath.OC

一个最优绝热问题

An optimal insulation problem

Francesco Della Pietra, Carlo Nitsch, Cristina Trombetti

摘要：

在本文中，我们考虑一个最小化问题，这是由热绝缘。一个紧凑的连通集$K$，它代表一个恒定温度的导体，比如说$1$，通过用一层热绝缘体包围它来热绝缘，开集$Ω\setminus K$与$K\subset\barΩ$。然后，对于某个正常数$β$，得到热分散为\[ \inf\left\{ \int_Ω|\nabla φ|^{2}dx +β\int_{\partial^{*}Ω}φ^{2}d\mathcal H^{n-1} ,\;φ\in H^{1}(\mathbb R^{n}), \, φ\ge 1\text{ in } K\right\}, \]。我们的分析主要限于绝热层厚度不变的情况。我们让集合$K$在规定的几何约束下变化，并且我们在热分散方面寻找最佳（或最差）几何。我们表明，周长约束下的磁盘在两个维度是最坏的。对于更高维度的球也是如此，但在不同的约束下。最后，我们讨论几个开放的问题。

英文摘要：

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 $Ω\setminus K$ with $K\subset\barΩ$. The heat dispersion is then obtained as \[ \inf\left\{ \int_Ω|\nabla φ|^{2}dx +β\int_{\partial^{*}Ω}φ^{2}d\mathcal H^{n-1} ,\;φ\in H^{1}(\mathbb R^{n}), \, φ\ge 1\text{ in } K\right\}, \] for some positive constant $β$. 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.

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