Equivalence of optimal L1-inequalities on Riemannian manifolds

Let (M, g) be a smooth compact Riemannian manifold of dimension n >= 2. This paper concerns the validity of the optimal Riemannian L-1-Entropy inequality Entdv(g) (u) <= n log(A(opt)parallel to Du parallel to Bv(M) + B-opt) for all u is an element of BV(M) with parallel to u parallel to L-1(M) = 1 and existence of extremal functions. In particular, we prove that this optimal inequality is equivalent to an optimal L-1-Sobolev inequality obtained by Druet [3]. (C) 2014 Elsevier Inc. All rights reserved.