  In this paper, we establish a Kantorovich duality for weak optimal total
variation transport problems. As consequences, we recover a version of duality
formula for partial optimal transports established by Caffarelli and McCann;
and we also get another proof of Kantorovich-Rubinstein Theorem for generalized
Wasserstein distance $\widetilde{W}_1^{a,b}$ proved before by Piccoli and
Rossi. Then we apply our duality formula to study generalized Wasserstein
barycenters. We show the existence of these barycenters for measures with
compact supports. Finally, we prove the consistency of our barycenters.
