Perturbation of the eigenvalues of a membrane with a concentrated mass

Abstract

We study a vibrating membrane with a distribution of density depending on ε \varepsilon , which converges, as ε ↘ 0 \varepsilon \searrow 0 , to a uniform density, plus a point mass at the origin. We establish local vibrations at the vicinity of the origin and global vibrations of the membrane. The asymptotic study for ε ↘ 0 \varepsilon \searrow 0 is performed using the method of matched asymptotic expansions.