Curvature-dependent electrostatic field as a principle for modelling membrane MEMS device with fringing field

In a framework of 1D membrane MEMS theory, we consider the MEMS boundary semi-linear elliptic problem with fringing field $$\begin{aligned} u''=-\frac{\lambda ^2(1+\delta |u'|^2)}{(1-u)^2}\;\;\;\text {in}\;\;\varOmega , \;\;u=0\;\;\text {on}\;\;\partial \varOmega , \end{aligned}$$ where $$\lambda ^2$$ and $$\delta $$ are positive parameters, $$\varOmega =[-L,L] \subset {\mathbb {R}}$$ , and u is the deflection of the membrane. In this model, since the electric field $$ {\mathbf {E}} $$ on the membrane is locally orthogonal to the straight line tangent to the membrane at the same point, $$ | {\mathbf {E}} | $$ , proportional to $$\lambda ^2/(1-u)^2$$ , is considered locally proportional to the curvature of the membrane. Thus, we achieve interesting results of existence writing it into its equivalent integral formulation by means of a suitable Green function and applying on it the Schauder–Tychonoff fixed point theory. Therefore, the uniqueness of the solution is proved exploiting both Poincare inequality and Gronwall Lemma. Then once the instability of the only obtained equilibrium position is verified, an interesting limitation for the potential energy dependent on the fringing field capacitance is obtained and studied.