29 citations · 192 across the 45 of their papers we have counts for
5 papers · 1 filter
Shape optimization for low Neumann and Steklov eigenvalues
Alexandre Girouard, Iosif Polterovich
We give an overview of results on shape optimization for low eigenvalues of the Laplacian on bounded planar domains with Neumann and Steklov boundary conditions. These results shar…
On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues
Alexandre Girouard, Iosif Polterovich
We prove that the isoperimetric inequality due to Hersch-Payne-Schiffer for the n-th nonzero Steklov eigenvalue of a bounded simply-connected planar domain is sharp for all n=1,2,.…
Pleijel's nodal domain theorem for free membranes
Iosif Polterovich
We prove an analogue of Pleijel's nodal domain theorem for piecewise analytic planar domains with Neumann boundary conditions. This confirms a conjecture made by Pleijel in 1956. T…
Average growth of the spectral function on a Riemannian manifold
Hugues Lapointe, Iosif Polterovich, Yuri Safarov
We study average growth of the spectral function of the Laplacian on a Riemannian manifold. Two types of averaging are considered: with respect to the spectral parameter and with r…
Maximization of the second positive Neumann eigenvalue for planar domains
Alexandre Girouard, Nikolai Nadirashvili, Iosif Polterovich
We prove that the second positive Neumann eigenvalue of a bounded simply-connected planar domain of a given area does not exceed the first positive Neumann eigenvalue on a disk of…