Qualitative Analysis of Rupture Solutions for an MEMS Problem

TL;DR

This paper establishes sharp regularity estimates for rupture solutions of a nonlinear PDE with negative exponent, proving existence of solutions with isolated ruptures in convex domains.

Contribution

It provides the first sharp Hölder continuity estimates for rupture sets and demonstrates the existence of solutions with isolated ruptures in bounded convex domains.

Findings

Sharp Hölder continuity estimates for rupture sets

Existence of rupture solutions with isolated ruptures

Applicability to bounded convex domains in 

Abstract

We prove a sharp H\"older continuity estimates of rupture sets for sequences of solutions of the following nonlinear problem with negative exponent $$ \Delta u= \frac{1}{u^p}, \ p>1, \ \mbox{in} \ \Omega .$$ As a consequence, we prove the existence of rupture solutions with isolated ruptures in a bounded convex domain in $\R^2$.