论文标题
最佳绝缘问题
An optimal insulation problem
论文作者
论文摘要
在本文中，我们考虑了由热绝缘引起的最小化问题。紧凑的连接集合$ k $，代表恒温的导体，例如$ 1 $，是通过用一层热绝缘子围绕它的热绝缘的，开放式套装$ω\ setminus k $，带有$ k \ subset \barΩ$。然后将热分子作为\ [\ inf \ left \ {\int_Ω| \nablaφ|^{2} dx +β\β\ int _ {\ partial^{*}ω}ω}φ^{2}φ^{2} d \ \ Mathcal H^{n-1} r^{n}），\，φ\ ge 1 \ text {in} k \ right \}，\}，对于某些正常数$β$。我们主要将分析限制为恒定厚度绝缘层的情况。我们让集合$ k $在规定的几何约束下变化，我们在热分散方面寻找最佳（或最坏）几何形状。我们表明，在周边约束下，二维磁盘是最糟糕的。在较高维度的球中，球在不同的限制下也是如此。我们终于讨论了一些开放问题。
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.
