Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Partial $W^{2,p}$ regularity for optimal transport maps   We prove that, in the optimal transportation problem with general costs and
positive continuous densities, the potential function is always of class
$W^{2,p}_{loc}$ for any $p \geq 1$ outside of a closed singular set of measure
zero. We also establish global $W^{2,p}$ estimates when the cost is a small
perturbation of the quadratic cost. The latter result is new even when the cost
is exactly the quadratic cost.

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The categories of this paper are: math-ph
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The categories of this paper are: solv-int
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The categories of this paper are: eess
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
16
The categories of this paper are: stat
34
The categories of this paper are: chem-ph
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The categories of this paper are: quant-ph
7
The categories of this paper are: hep-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: alg-geom
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The categories of this paper are: astro-ph
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The categories of this paper are: supr-con
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The categories of this paper are: dg-ga
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The categories of this paper are: funct-an
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The categories of this paper are: cond-mat