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On the Regularity of Optimal Transports Between Degenerate Densities

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Abstract

We study the most common image and informal description of the optimal transport problem for quadratic cost, also known as the second boundary value problem for the Monge–Ampère equation—what is the most efficient way to fill a hole with a given pile of sand?—by proving regularity results for optimal transports between degenerate densities. In particular, our work contains an analysis of the setting in which holes and sandpiles are represented by absolutely continuous measures concentrated on bounded convex domains whose densities behave like nonnegative powers of the distance functions to the boundaries of these domains.

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Authors and Affiliations

  1. Department of Mathematics, Columbia University, 2990 Broadway, New York, NY, 10027, USA

    Yash Jhaveri & Ovidiu Savin

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  1. Yash Jhaveri
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  2. Ovidiu Savin
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Correspondence to Yash Jhaveri.

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Communicated by A. Figalli.

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Yash Jhaveri was supporting in part by NSF Grant DMS-1954363. Ovidiu Savin was supported in part by NSF Grants DMS-1800645 and DMS-2055617.

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Jhaveri, Y., Savin, O. On the Regularity of Optimal Transports Between Degenerate Densities. Arch Rational Mech Anal 245, 819–861 (2022). https://doi.org/10.1007/s00205-022-01796-y

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