Shape optimization for Neumann eigenvalues of euclidean domains

Published in Pavillon André-Aisenstadt, room AA-5183, 2018

In this talk I will review known bounds for the eigenvalues of the Neumann problem on Euclidean domains. My main goal will be to prove an optimal upper bound for the second nonzero eigenvalue mu_2 of planar simply-connected domains of prescribed area. This is joint work from 2009 with Iosif Polterovich and Nikolai Nadirashvili. This talk will be a serve as a preparation for the very recent work of Bucur and Henrot who solved the maximization problem for mu_2 on arbitrary bounded euclidean domains of fixed measure (that is, there is no connectedness assumption, and no constraint on the dimension).