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

Qualitative Analysis of Rupture Solutions for an MEMS Problem