Tuning the total displacement of membranes

•A design problem to tune the robustness of a membrane by changing its vulnerability is studied.•Determine functions in a rearrangement class such that their energies would be a given specific value.•Topological and geometrical properties of the configuration of the vulnerability are given.•Explicit solutions are found analytically when the domain is an N-ball.•For a general domain, a numerical algorithm based on rearrangements is developed to find the solution. 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.