Free boundary regularity in the optimal partial transport problem

2012/10/11 by Emanuel Indrei, Indrei, Emanuel · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1210.3111

32 pages, 2 figures, accepted for publication in JFA

arxiv created 2013/03/19 · arxiv updated 2013/03/21

Abstract

In the optimal partial transport problem, one is asked to transport a fraction 0<m ≤ min\||f||_L¹, ||g||_L¹\ of the mass of f=f χ_Ω onto g=gχ_Λ while minimizing a transportation cost. If f and g are bounded away from zero and infinity on strictly convex domains Ω and Λ, respectively, and if the cost is quadratic, then away from ∂(Ω∩ Λ) the free boundaries of the active regions are shown to be C_loc^1,α hypersurfaces up to a possible singular set. This improves and generalizes a result of Caffarelli and McCann \citeCM and solves a problem discussed by Figalli \cite[Remark 4.15]Fi. Moreover, a method is developed to estimate the Hausdorff dimension of the singular set: assuming Ω and Λ to be uniformly convex domains with C^1,1 boundaries, we prove that the singular set is H^n-2 σ-finite in the general case and H^n-2 finite if Ω and Λ are separated by a hyperplane.

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• Unidimensional semi-discrete partial optimal transport

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