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On strict convexity and continuous differentiability of potential functions in optimal transportation. (English) Zbl 1214.49038

Summary: This note concerns the relationship between conditions on cost functions and domains, the convexity properties of potentials in optimal transportation and the continuity of the associated optimal mappings. In particular, we prove that if the cost function satisfies the condition: for any \((x,y)\in{\mathcal U}\), and \(\xi,\eta\in\mathbb R^n\) with \(\xi\bot\eta\),
\[ \sum_{i,j,k,l,p,q,r,s} (c^{p,q}c_{ij,p}c_{q,rs}- c_{ij,rs}) c^{r,k} c^{s,l} \xi_i \xi_j \eta_k \eta_l\geq c_0|\xi|^2|\eta|^2, \]
where \(c_0\) is a positive constant, and \((c^{i,j})\) is the inverse matrix of \((c_{i,j})\), introduced in our previous work with Xinan Ma, the densities and their reciprocals are bounded and the target domain is convex with respect to the cost function, then the potential is continuously differentiable and its dual potential is strictly concave with respect to the cost function. Our results extend, by different and more direct proof, similar results of Loeper proved by approximation from our earlier work on regularity.

MSC:

49N60 Regularity of solutions in optimal control
49Q20 Variational problems in a geometric measure-theoretic setting
35J25 Boundary value problems for second-order elliptic equations
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