Nonuniqueness of infinity ground states
  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. 
Ryan Hynd, Charles K. Smart, Yifeng Yu
math.AP