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Global \(C^{1,\alpha }\) Regularity for Monge–Ampère Equation and Convex Envelope

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Abstract

In this paper we establish the global \(C^{1,\alpha }\) regularity for solutions to the Dirichlet problem of the Monge–Ampère equation \(\det D^2 u=f\). By examples we show that our conditions are optimal. Our proof allows the degenerate case \(f\ge 0\), including the special case \(f\equiv 0\). We also prove the global \(C^{1,\alpha }\) regularity for the convex envelope of a given function under optimal conditions.

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Authors and Affiliations

  1. Department of Mathematics, The University of Texas at Austin, Austin, TX, 78712, USA

    Luis A. Caffarelli

  2. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Lan Tang

  3. Centre for Mathematics and Its Applications, Australian National University, Canberra, ACT, 0200, Australia

    Xu-Jia Wang

Authors
  1. Luis A. Caffarelli
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  2. Lan Tang
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  3. Xu-Jia Wang
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Correspondence to Lan Tang.

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Communicated by A. Figalli

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L. Caffarelli is partially supported by NSF Grant 1500871. L. Tang is partially supported by NNSFC 11831109 and Fundamental Research Grant for Central Universities (CCNU19TS032). X.-J. Wang is partially supported by ARC DP170100929.

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Caffarelli, L.A., Tang, L. & Wang, XJ. Global \(C^{1,\alpha }\) Regularity for Monge–Ampère Equation and Convex Envelope. Arch Rational Mech Anal 244, 127–155 (2022). https://doi.org/10.1007/s00205-022-01757-5

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  • DOI: https://doi.org/10.1007/s00205-022-01757-5

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