Sharp Morrey-Sobolev inequalities and eigenvalue problems on Riemannian-Finsler manifolds with nonnegative Ricci curvature

TL;DR

This paper establishes sharp Morrey-Sobolev inequalities on Finsler manifolds with nonnegative Ricci curvature, leading to new eigenvalue problem solutions and elliptic PDE results, extending known Riemannian cases.

Contribution

It introduces sharp Morrey-Sobolev inequalities on Finsler manifolds with nonnegative Ricci curvature, combining isoperimetric and symmetrization techniques, and applies these to eigenvalue and PDE problems.

Findings

Proved sharp Morrey-Sobolev inequalities on Finsler manifolds.

Established existence and multiplicity of solutions for eigenvalue problems.

Extended results to the Riemannian setting.

Abstract

Combining the sharp isoperimetric inequality established by Z. Balogh and A. Krist\'aly [Math. Ann., in press, doi:10.1007/s00208-022-02380-1] with an anisotropic symmetrization argument, we establish sharp Morrey-Sobolev inequalities on $n$-dimensional Finsler manifolds having nonnegative $n$-Ricci curvature. A byproduct of this method is a Hardy-Sobolev-type inequality in the same geometric setting. As applications, by using variational arguments, we guarantee the existence/multiplicity of solutions for certain eigenvalue problems and elliptic PDEs involving the Finsler-Laplace operator. Our results are also new in the Riemannian setting.