  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.
