Stable foliations with respect to the Fuglede p-modulus and level sets of q-harmonic functions

We consider the p-modulus of a foliation on a Riemannian manifold. We study the second variation of the p-modulus, and we derive a stability condition for this variation, which we formulate as a type of the Hardy inequality with the weight depending on the geometry of the foliation. In particular, we show that foliations defined by the distance function are p  -stable for any p≥2p≥2. We examine the critical points of the p-modulus of the level sets of a smooth function. This leads, via a new approach, to generalizations of the well-known result that the q-capacity of a condenser is the reciprocal of the p-modulus of the family of all hyper-surfaces separating the plates of the condenser.