  It is known that the torsional rigidity for a punctured ball, with the
puncture having the shape of a ball, is minimum when the balls are concentric
and the first eigenvalue for the Dirichlet Laplacian for such domains is also a
maximum in this case. These results have been obtained by Ashbaugh and
Chatelain (private communication), Harrell et. al., by Kesavan and, by Ramm and
Shivakumar. In this paper we extend these results to the case of $p$-Laplacian
for $1 < p < \infty$. For proving these results, we follow the same line of
ideas as in the aforementioned articles, namely, study the sign of the shape
derivative using the moving plane method and comparison principles. In the
process, we obtain some interesting new side results such as the Hadamard
perturbation formula for the torsional rigidity functional for the Dirichlet
$p$-Laplacian, the existence and uniqueness result for a nonlinear pde and some
extensions of known comparison results for nonlinear pdes.
