Nonuniqueness of infinity ground states

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