sharp nash inequalities on manifolds with boundary in the presence of symmetries

in this paper we establish the best constant a_ops(m) for the trace nash inequality on an-dimensional compact riemannian manifold in the presence of symmetries, which is an improvement over the classical case due to the symmetries which arise and reflect the geometry of manifold. this is particularly true when the data of the problem is invariant under the action of an arbitrary compact subgroup g of the isometry group is(m,g), where all the orbits have infinite cardinal.