In this paper we study a design problem to tune the robustness of a membrane by changing its vulnerability. Consider an energy functional corresponding to solutions of Poisson’s equation with Robin boundary conditions. The aim is to find functions in a rearrangement class such that their energies would be a given specific value. We prove that this design problem has a solution and also we propose a way to find it. Furthermore, we derive some topological and geometrical properties of the configuration of the vulnerability. In addition, some explicit solutions are found analytically when the domain is an $N$-ball. For general domain we develop a numerical algorithm based on rearrangements to find the solution. The algorithm evolves both minimization and maximization processes over two different rearrangement classes. Our algorithm works efficiently for various domains and the numerical results obtained coincide with our analytical findings.

在本文中，我们研究了一个设计问题，以通过更改其脆弱性来调整膜的坚固性。考虑一个具有罗宾边界条件的泊松方程解的能量函数。目的是找到重排类别中的函数，以使它们的能量为给定的特定值。我们证明该设计问题可以解决，并且我们提出了一种找到它的方法。此外，我们得出了漏洞配置的一些拓扑和几何属性。此外，当域为$ñ$-球。对于一般领域，我们开发了一种基于重排的数值算法来找到解决方案。该算法在两个不同的重排类上同时发展了最小化和最大化过程。我们的算法可在各种领域有效运行，并且获得的数值结果与我们的分析结果相吻合。