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Optimal Regularity for the Convex Envelope and Semiconvex Functions Related to Supersolutions of Fully Nonlinear Elliptic Equations

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Abstract

In this paper we prove optimal regularity for the convex envelope of supersolutions to general fully nonlinear elliptic equations with unbounded coefficients. More precisely, we deal with coefficients and right hand sides (RHS) in Lq with \({q \geq n}\). This extends the result of Caffarelli on the \({C_{loc}^{1,1}}\) regularity of the convex envelope of supersolutions of fully nonlinear elliptic equations with bounded RHS. Moreover, we also provide a regularity result with estimates for \({\omega}\)-semiconvex functions that are supersolutions to the same type of equations with unbounded RHS (i.e, RHS in \({L^{q}, q \geq n}\)). By a completely different method, our results here extend the recent regularity results obtained by Braga et al. (Adv Math 334:184–242, 2018) for \({q > n}\), as far as fully nonlinear PDEs are concerned. These results include, in particular, the apriori estimate obtained by Caffarelli et al. (Commun Pure Appl Math 38(2):209–252, 1985) on the modulus of continuity of the gradient of \({\omega}\)-semiconvex supersolutions (for linear equations and bounded RHS) that have a Hölder modulus of semiconvexity.

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Acknowledgements

The work of A. Figalli is supported by the ERC Grant “Regularity and Stability in Partial Differential Equations (RSPDE)”. The work of D. Moreira is supported by CNPq grant “Universal-2014" -447536/2014-1. The authors would like to thank Lihe Wang for sharing nice ideas contained the appendix of this paper. The authors also thank the anonymous referees for their useful comments on a preliminary version of this paper.

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Authors and Affiliations

  1. Departamento de Matemática, Universidade Federal do Ceará, Campus do Pici - Bloco 914, Fortaleza, Ceará, CEP 60455-760, Brazil

    J. Ederson M. Braga & Diego Moreira

  2. Department of Mathematics, ETH Zürich, HG G 63.2, Rämistrasse 101, 8092, Zürich, Switzerland

    Alessio Figalli

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  1. J. Ederson M. Braga
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  2. Alessio Figalli
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  3. Diego Moreira
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Correspondence to Diego Moreira.

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Communicated by H.-T. Yau

Dedicated to Luis Caffarelli on his 70th Birthday.

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Braga, J.E.M., Figalli, A. & Moreira, D. Optimal Regularity for the Convex Envelope and Semiconvex Functions Related to Supersolutions of Fully Nonlinear Elliptic Equations. Commun. Math. Phys. 367, 1–32 (2019). https://doi.org/10.1007/s00220-019-03370-2

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