The family of Zolotarev distances between two probabilities naturally extends the Wasserstein-1 metric to higher orders: one bounds the Lipschitz constant of the relevant derivatives of the potential. So far, however, the optimal transport perspective has been available only for the first order. In my talk I will demonstrate how a PDE motivation revolving around optimal elastic structures has led us to a new (OT) framework for the second-order Zolotarev distance. The new duality theory paves a way to efficient computational method, including a variant of the famous Sinkhorn algorithm. The talk is partially based on a joint work with Guy Bouchitté (Université de Toulon).