On the p-torsional rigidity of compact metric graphs: a sharp Kohler--Jobin inequality
We investigate the $p$-torsional rigidity for the $p$-Laplacian, $1<p<\infty$, on compact connected metric graphs equipped with Dirichlet conditions on a nonempty set $\mathcal{V}^D$ of degree-one vertices and nonlinear Kirchhoff conditions at all remaining vertices. We establish the existence, uniqueness, and positivity of the $p$-torsion function, together with a variational characterization of the $p$-torsional rigidity. Our main contribution is the derivation of two sharp isoperimetric inequalities. We first prove a $p$-Saint-Venant inequality, showing that, among all compact metric graphs of prescribed total length, the $p$-torsional rigidity is maximized precisely by the interval with a single Dirichlet endpoint. We then derive a sharp $p$-Kohler--Jobin inequality, providing a scale-invariant lower bound for the first eigenvalue of the $p$-Laplacian in terms of the $p$-torsional rigidity. These results yield nonlinear counterparts, in the setting of compact metric graphs, of the classical Saint-Venant and Kohler--Jobin inequalities, and extend the linear theory, where $p=2$, developed by Mugnolo and Plümer to the full range $1<p<\infty$.