  Payne-Weinberger showed that \textit{`among the class of membranes with given
area $A$, free along the interior boundaries and fixed along the outer boundary
of given length $L_0$, the annulus $\Omega^\#$ has the highest fundamental
frequency,'} where $\Omega^\#$ is a concentric annulus with the same area as
$\Omega$ and the same outer boundary length as $L_0$. We extend this result for
the higher dimensional domains and $p$-Laplacian with $p\in (1,\infty),$ under
the additional assumption that the outer boundary is a sphere.
  As an application, we prove that the nodal set of the second eigenfunctions
of $p$-Laplacian (with mixed boundary conditions) on a ball and a concentric
annulus cannot be a concentric sphere.
