Some overdetermined problems related to the anisotropic capacity

We characterize the Wulff shape of an anisotropic norm in terms of solutions to overdetermined problems for the Finsler p-capacity of a convex set Omega subset of R-N, with 1 < p < N. In particular we show that if the Finsler p-capacitary potential u associated to 0 has two homothetic level sets then 0 is Wulff shape. Moreover, we show that the concavity exponent of u is q = -(p - 1)/(N - p) if and only if Omega is Wulff shape.