p-Laplacian first eigenvalue 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 := g + β), where g is a Riemannian metric on M and β an appropriate 1-form on M , it is shown that the first eigenvalue λ 1,p (M, F ) of the Finslerian p-Laplacian defined by the Finsler metric F is controled by the first eigenvalue λ 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.