Capacity and principal eigenvalues: the method of enlargement of obstacles revisited

Abstract

We describe a coarse graining method, which provides lower bounds on the principal Dirichlet eigenvalue of the Laplacian in regions receiving small obstacles, and sharpens the previous method of enlargement of obstacles. Based on a quantitative Wiener criterion, one replaces the actual obstacles by obstacles of a much larger size. Controls on the shift of principal eigenvalues and capacity estimates on the locus where the Wiener criterion breaks down are derived. The results are written in a self-contained fashion.