In this paper we make a survey of some recent developments of the theory of Sobolev spaces W 1,q (X, d, m), 1 < q < ∞, in metric measure spaces (X, d, m). In the final part of the paper we provide a new proof of the reflexivity of the Sobolev space based on Γ-convergence; this result extends Cheeger's work because no Poincaré inequality is needed and the measure-theoretic doubling property is weakened to the metric doubling property of the support of m. We also discuss the lower semicontinuity of the slope of Lipschitz functions and some open problems.
Abstract. We study a multimarginal optimal transportation problem in one dimension. For a symmetric, repulsive cost function, we show that given a minimizing transport plan, its symmetrization is induced by a cyclical map, and that the symmetric optimal plan is unique. The class of costs that we consider includes, in particular, the Coulomb cost, whose optimal transport problem is strictly related to the strong interaction limit of Density Functional Theory. In this last setting, our result justifies some qualitative properties of the potentials observed in numerical experiments.
Motivated by recent developments on calculus in metric measure spaces (X, d, m), we prove a general duality principle between Fuglede's notion [15] of p-modulus for families of finite Borel measures in (X, d) and probability measures with barycenter in L q (X, m), with q dual exponent of p ∈ (1, ∞). We apply this general duality principle to study null sets for families of parametric and non-parametric curves in X. In the final part of the paper we provide a new proof, independent of optimal transportation, of the equivalence of notions of weak upper gradient based on ppodulus ([21], [23]) and suitable probability measures in the space of curves ([6], [7]).
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