Symmetry analysis of a canonical MEMS model
In the study of electrostatically actuated MEMS devices the semi-linear elliptic equation \[ \Delta u=\frac{\lambda}{u^2}\;\;\text{in}\;\;\Omega,\quad u=1\;\;\text{on}\;\;\partial\Omega \] arises. Such equation models the deflection of an elastic membrane under the influence of an applied electric field. \(u\) is the position of the membrane. In this article the authors study the one-dimensional version of this generalized equation, i.e. \[ \frac{d^2u}{dx^2}=\frac{\lambda f(x)}{u^2}\quad \text{with } \frac{du}{dx}(0)=0\;\;\text{ and }u(1/2)=1.\tag{1} \] The function \(f(x)\) is assumed to be symmetric about the origin. The authors perform a symmetry analysis of (1) to determine properties of \(f(x)\) that allow invariance under a one-parameter Lie group of transformations. Then the analysis of the problem is made for two particular functions which arise from the found symmetries. It is shown how symmetry is used to convert boundary value problem to initial value problem and to construct bifurcation diagrams.
scientific article; zbMATH DE number 5562039