Calculus of Variations and Geometric Measure Theory

D. Mazzoleni

Recent existence results for spectral problems

created by mazzoleni on 07 Nov 2014



Inserted: 7 nov 2014
Last Updated: 7 nov 2014

Year: 2014


In this survey we present the new techniques developed for proving existence of optimal sets when one minimizes functionals depending on the eigenvalues of the Dirichlet Laplacian with a measure constraint, the most important being:\[ \min{\left\{\lambda_k(\Omega)\;:\Omega\subset\mathbb{R}^N,\;
=1\right\}}. \] In particular we sketch the main ideas of some recent works, which allow to extend the now classic result by Buttazzo and Dal Maso to $\mathbb{R}^N$.