Rigidity results for variational infinity ground states

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Abstract: We prove two rigidity results for a variational infinity ground state

u

of an open bounded convex domain

O m e g a s u b s e t m a t h b b R^{n}

. They state that

u

coincides with a multiple of the distance from the boundary of

O m e g a

if either

| a b l a u |

is constant on

p a r t i a l O m e g a

, or

u

is of class

C^{1, 1}

outside the high ridge of

O m e g a

. Consequently, in both cases

O m e g a

can be geometrically characterized as a "stadium-like domain".

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