An interior regularity result for the MEMS problem

Abstract: In this talk we present an interior regularity result for the class of stable solutions to a semilinear elliptic equation with a singular nonlinearity. The class of nonlinearities that we consider are real-valued functions defined on [0,1) which are positive, nondecreasing, and whose integral on [0,1) is infinite. This equation is a generalization of a model for the deflection of a dielectric elastic membrane in a microelectromechanical system (MEMS).

Solutions to this equation are critical points of an associated energy functional. We say that a solution is stable when the second variation of the energy at the solution is nonnegative. Under a growth assumption on the nonlinearity, we are able to prove that every stable solution is regular up to the optimal dimension, n=6.

Mathematics

Audience: advanced learners

Barcelona Mathematics Informal Seminar (SIMBa)

Series comments: SIMBa is a youth mathematics seminar organized by graduate students of the Barcelona area. It is aimed towards graduate and last course undergraduate students. Our goals are divulging the knowledge from different branches of mathematics for those interested and promote networking between the attendants.

This seminar is backed by the Faculty of Mathematics and Computer Science at Universitat de Barcelona, Faculty of Mathematics and Statistics at Universitat Politècnica de Catalunya, the Department of Mathematics from Universitat Autònoma de Barcelona, CRM, IMUB and BGSMath.