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A note on interior \(W^{2,1+ \varepsilon }\)estimates for the Monge–Ampère equation

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Abstract

By a variant of the techniques introduced by the first two authors in De Philippis and Figalli (Invent Math 2012) to prove that second derivatives of solutions to the Monge–Ampère equation are locally in \(L\log L\), we obtain interior \(W^{2,1+\varepsilon }\) estimates.

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Notes

  1. Although this will not be used here, we point out for completeness that (10) is equivalent to the so-called Condition \((\mu _\infty )\), first introduced by Caffarelli and Gutierrez in [4]. Indeed, using (10) with \(E=S{\setminus }F\) one sees that \(|F|/|S| \ll 1\) implies \(\mu (F)/\mu (S) \le 1-\gamma /2\), and then an iteration and covering argument in the spirit of [4, Theorem 6] shows that (10) is actually equivalent to Condition \((\mu _\infty )\).

References

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  2. Ambrosio, L., Colombo, M., De Philippis, G., Figalli, A.: A global existence result for the semigeostrophic equations in three dimensional convex domains (2012) (Preprint)

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  7. Stein, T.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton (1993)

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Authors and Affiliations

  1. Scuola Normale Superiore di Pisa, 56126 , Pisa, Italy

    G. De Philippis

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

    A. Figalli

  3. Department of Mathematics, Columbia University, New York, NY, 10027, USA

    O. Savin

Authors
  1. G. De Philippis
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  2. A. Figalli
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  3. O. Savin
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Corresponding author

Correspondence to A. Figalli.

Additional information

A. Figalli was partially supported by NSF grant 0969962. O. Savin was partially supported by NSF grant 0701037.

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De Philippis, G., Figalli, A. & Savin, O. A note on interior \(W^{2,1+ \varepsilon }\)estimates for the Monge–Ampère equation. Math. Ann. 357, 11–22 (2013). https://doi.org/10.1007/s00208-012-0895-9

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  • DOI: https://doi.org/10.1007/s00208-012-0895-9

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