Sharp estimates on the first Dirichlet eigenvalue of nonlinear elliptic operators via maximum principle

Abstract

In this paper, we study optimal lower and upper bounds for functionals involving the first Dirichlet eigenvalue λF⁢(p,Ω){\lambda_{F}(p,\Omega)} of the anisotropic p-Laplacian, 1<p<+∞{1<p<+\infty}. Our aim is to enhance, by means of the 𝒫{\mathcal{P}}-function method, how it is possible to get several sharp estimates for λF⁢(p,Ω){\lambda_{F}(p,\Omega)} in terms of several geometric quantities associated to the domain. The 𝒫{\mathcal{P}}-function method is based on a maximum principle for a suitable function involving the eigenfunction and its gradient.