Elliptic problems on the ball endowed with Funk-type metrics

TL;DR

This paper investigates Sobolev spaces and elliptic problems on the unit ball with a family of Finsler metrics interpolating between Klein and Funk metrics, establishing conditions for the space's structure and solutions' existence.

Contribution

It characterizes when Sobolev spaces on the Finsler manifold are vector spaces and provides existence and non-existence results for elliptic problems involving the Finsler-Laplace operator.

Findings

Sobolev space is a vector space iff a<1.

Existence of solutions for sublinear elliptic problems when a<1.

Non-existence results under certain conditions.

Abstract

We study Sobolev spaces on the $n-$dimensional unit ball $B^n(1)$ endowed with a parameter-depending Finsler metric $F_a$, $a\in [0,1],$ which interpolates between the Klein metric $(a=0)$ and Funk metric $(a=1)$, respectively. We show that the standard Sobolev space defined on the Finsler manifold $(B^n(1),F_a)$ is a vector space if and only if $a\in [0,1).$ Furthermore, by exploiting variational arguments, we provide non-existence and existence results for sublinear elliptic problems on $(B^n(1),F_a)$ involving the Finsler-Laplace operator whenever $a\in [0,1).$