On functionals involving the torsional rigidity related to some classes of nonlinear operators

Abstract

In this paper we study optimal estimates for two functionals involving the anisotropic $p$-torsional rigidity $T_p(\Omega )$, $1<p<+\infty$. More precisely, we study $\Phi (\Omega )=\frac{T_p(\Omega )}{|\Omega |M(\Omega )}$ and $\Psi (\Omega )=\frac{T_p(\Omega )}{|\Omega |[R_F(\Omega ){]}^{\frac{p}{p-1}}}$, where $M(\Omega )$ is the maximum of the torsion function $u_{\Omega }$ and $R_F(\Omega )$ is the anisotropic inradius of $\Omega$.