  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.
