$p$-Laplacian first eigenvalues controls on Finsler manifolds

Abstract

Given a Finsler manifold $(M,F)$, it is proved that the first eigenvalue of the Finslerian $p$-Laplacian is bounded above by a constant depending on $\ p$, the dimension of $M$, the Busemann-Hausdorff volume and the reversibility constant of $(M,F)$. For a Randers manifold $(M,F:=\sqrt{g}+\beta)$, where $g$ is a Riemannian metric on $M$ and $\beta$ an appropriate $1$-form on $M$, it is shown that the first eigenvalue $\lambda_{1,p}(M,F)$ of the Finslerian $p$-Laplacian defined by the Finsler metric $F$ is controled by the first eigenvalue $\lambda_{1,p}(M,g)$ of the Riemannian $p$-Laplacian defined on $(M,g)$. Finally, the Cheeger's inequality for Finsler Laplacian is extended for $p$-Laplacian, with $p > 1$.