A canonical model arising in MEMS and modeling the electrostatic deflections of an elastic membrane with spatially varying dielectric properties is considered. Symmetry methods are used to determine the form of the spatially varying dielectric properties that allow for invariance under a one-parameter Lie group of transformations. In these cases it is shown how symmetry may be used to analyze the boundary value problem and construct a bifurcation diagram for the system. This bifurcation diagram relates back to the MEMS model; a non-dimensional version of the applied voltage is plotted versus the maximum deflection of the membrane.
"Symmetry Analysis of a Canonical MEMS Model." Methods Appl. Anal. 15 (3) 327 - 340, September 2008.