Maximization of the second non-trivial Neumann eigenvalue

Dorin Bucur (1) , Antoine Henrot (2)

1 LAMA - Laboratoire de Mathématiques
2 IECL - Institut Élie Cartan de Lorraine

In this paper we prove that the second (non-trivial) Neumann eigenvalue of the Laplace operator on smooth domains of 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.

Domaines: Equations aux dérivées partielles [math.AP]

Dorin Bucur, Antoine Henrot. Maximization of the second non-trivial Neumann eigenvalue. Acta Mathematica, 2019, 222 (2), pp.337-361. ⟨10.4310/ACTA.2019.v222.n2.a2⟩. ⟨hal-01690236⟩