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Interior Sobolev regularity for fully nonlinear parabolic equations

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Abstract

In the present paper, we establish sharp Sobolev estimates for solutions of fully nonlinear parabolic equations, under minimal, asymptotic, assumptions on the governing operator. In particular, we prove that solutions are in \(W^{2,1;p}_{ loc }\). Our argument unfolds by importing improved regularity from a limiting configuration. In this concrete case, we recur to the recession function associated with F. This machinery allows us to impose conditions solely on the original operator at the infinity of \(\mathcal {S}(d)\). From a heuristic viewpoint, integral regularity would be set by the behavior of F at the ends of that space. Moreover, we explore a number of consequences of our findings, and develop some related results; these include a parabolic version of Escauriaza’s exponent, a universal modulus of continuity for the solutions and estimates in p-BMO spaces.

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Acknowledgements

For valuable comments and suggestions on the material in this paper, the authors are grateful to B. Sirakov, A. Świȩch, E. Teixeira and an anonymous referee. R. Castillo is funded by CAPES-Brazil; E. Pimentel was partially supported by FAPESP (Grant # 2015/13011-6) and PUC-Rio baseline funds.

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Authors and Affiliations

  1. Department of Mathematics, Universidade Federal de Pernambuco, Recife, PE, 50670-901, Brazil

    Ricardo Castillo

  2. Department of Mathematics, Pontifical Catholic University of Rio de Janeiro – PUC-Rio, Gávea, Rio de Janeiro, RJ, 22451-900, Brazil

    Edgard A. Pimentel

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  1. Ricardo Castillo
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  2. Edgard A. Pimentel
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Correspondence to Edgard A. Pimentel.

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Communicated by L. Caffarelli.

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Castillo, R., Pimentel, E.A. Interior Sobolev regularity for fully nonlinear parabolic equations. Calc. Var. 56, 127 (2017). https://doi.org/10.1007/s00526-017-1227-4

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