Nonuniqueness of infinity ground states

TL;DR

This paper constructs a specific domain demonstrating that the principal infinity eigenvalue can be non-unique, challenging prior assumptions and highlighting open questions about eigenvalue simplicity in convex domains.

Contribution

It provides a counterexample showing non-uniqueness of infinity ground states, addressing a longstanding open problem in the field.

Findings

Constructed a dumbbell domain with non-unique infinity eigenvalues

Negatively answered the question of eigenvalue simplicity in certain domains

Highlights open problem for convex domains

Abstract

In this paper, we construct a dumbbell domain for which the associated principle $\infty$-eigenvalue is not simple. This gives a negative answer to the outstanding problem posed by Juutinen-Lindquivst-Manfredi ("The $\infty$-eigenvalue problem", Arch. Ration. Mech. Anal. 148, 1999, no.2, 89-105). It remains a challenge to determine whether simplicity holds for convex domains.