  For $d\geq 2$ and $\frac{2d+2}{d+2} < p < \infty $, we prove a strict
Faber-Krahn type inequality for the first eigenvalue $\lambda _1(\Omega )$ of
the $p$-Laplace operator on a bounded Lipschitz domain $\Omega \subset
\mathbb{R}^d$ (with mixed boundary conditions) under the polarizations. We
apply this inequality to the obstacle problems on the domains of the form
$\Omega \setminus \mathscr{O}$, where $\mathscr{O}\subset \subset \Omega $ is
an obstacle. Under some geometric assumptions on $\Omega $ and $\mathscr{O}$,
we prove the strict monotonicity of $\lambda _1 (\Omega \setminus \mathscr{O})$
with respect to certain translations and rotations of $\mathscr{O}$ in $\Omega
$.
