Maximization of the second non-trivial Neumann eigenvalue

By Dorin Bucur, Antoine Henrot. Acta Mathematica, 2018

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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 P{\'o}lya 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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

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