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<p>We prove that for two-marginal optimal transport with Coulomb cost on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {R}^d\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation>, the optimal map is a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(C^{1,\alpha }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mrow> <mn>1</mn> <mo>,</mo> <mi>α</mi> </mrow> </msup> </math></EquationSource> </InlineEquation> diffeomorphism outside a closed set of Lebesgue measure zero provided the marginals are <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Hölder continuous, bounded, and strictly positive. Excluding a set of measure zero is necessary as optimal maps for the Coulomb cost have long been known to exhibit jump singularities across codimension 1 surfaces (even for smooth marginals on convex domains).</p>

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Partial regularity of optimal transport with Coulomb cost

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年6月26日

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  • Gero Friesecke,
  • Tobias Ried

摘要

We prove that for two-marginal optimal transport with Coulomb cost on \(\mathbb {R}^d\) R d , the optimal map is a \(C^{1,\alpha }\) C 1 , α diffeomorphism outside a closed set of Lebesgue measure zero provided the marginals are \(\alpha \) α -Hölder continuous, bounded, and strictly positive. Excluding a set of measure zero is necessary as optimal maps for the Coulomb cost have long been known to exhibit jump singularities across codimension 1 surfaces (even for smooth marginals on convex domains).