  In this paper for all $p>1$ we prove that the pull-in voltage of the $p$-MEMS
(micro-electro mechanical systems) problems on a smooth bounded domain of
$\mathbb R^{d}, d\geq1,$ is minimized by symmetrizing the domain and the
permittivity profile. The proofs rely on some suitable version of Talenti's
comparison principle. We also demonstrate our method to the multidimensional
MEMS type problems on the whole space $\mathbb R^{d}, d\geq3,$ and the
Dirichlet boundary value problems of second order uniformly elliptic
differential operators.
