Acta Mathematica

Volume 222 (2019)

Number 2

Maximization of the second non-trivial Neumann eigenvalue

Pages: 337 – 361



Dorin Bucur (Université Grenoble Alpes, Université Savoie Mont Blanc, Chambéry, France)

Antoine Henrot (Institut Elie Cartan, Université de Lorraine, Vandoeuvre-lès-Nancy, France)


In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of $\mathbb{R}^N$ with prescribed measure $m$ attains its maximum on the union of two disjoint balls of measure $m/2$. As a consequence, the Pólya conjecture for the Neumann eigenvalues holds for the second eigenvalue and for arbitrary domains. We moreover prove that a relaxed form of the same inequality holds in the context of non-smooth domains and densities.

2010 Mathematics Subject Classification

35P15, 49Q10

Received 22 January 2018

Accepted 2 January 2019

Published 7 June 2019