On the Uniqueness of the Solution for a Semi-Linear Elliptic Boundary Value Problem of the Membrane MEMS Device for Reconstructing the Membrane Profile in Absence of Ghost Solutions

In this paper, the authors present a new condition of the uniqueness of the solution for a previous $1D$ semi-linear elliptic boundary value problem of membrane MEMS devices, where the amplitude of the electric field is considered proportional to the curvature of the membrane. The existence of the solution (membrane deflection) depends on the material of the membrane, which is obtained by Shauder-Tychonoff's fixed point approach. Thus, in this paper, the result of uniqueness has been completely reformulated to obtain a condition depending on the material of the membrane achieving a new result of existence and uniqueness, depending on both the material of the membrane and the geometrical characteristics of the device. Then, by shooting numerical method, more realistic conditions for detecting eventual ghost solutions and new ranges of both operational parameters and mechanical tension of the membrane ensuring convergence have been achieved confirming the useful information on the industrial applicability of the model under study.