*Submitted Paper*

**Inserted:** 22 jan 2018

**Last Updated:** 22 jan 2018

**Year:** 2018

**Abstract:**

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 $\frac m2$. As a consequence, the P\'olya 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.

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