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\(C^{1,\alpha }\) estimates for the fully nonlinear Signorini problem

Abstract

We study the regularity of solutions to the fully nonlinear thin obstacle problem. We establish local \(C^{1,\alpha }\) estimates on each side of the smooth obstacle, for some small \(\alpha > 0\). Our results extend those of Milakis–Silvestre [9] in two ways: first, we do not assume solutions nor operators to be symmetric, and second, our estimates are local, in the sense that do not rely on the boundary data. As a consequence, we prove \(C^{1,\alpha }\) regularity even when the problem is posed in general Lipschitz domains.

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References

  1. Athanasopoulos, I., Caffarelli, L.A.: Optimal regularity of lower dimensional obstacle problems. Zap. Nauchn. Sem. S. Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310 (2004), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35(34), 49–66, 226 [translation in J. Math. Sci. (N. Y.) 132(3), 274–284]

  2. Caffarelli, L.: Further regularity for the Signorini problem. Comm. Partial Differ. Equ. 4, 1067–1075 (1979)

    MathSciNet  Article  MATH  Google Scholar 

  3. Caffarelli, L.: Interior a priori estimates for solutions of fully nonlinear equations. Ann. Math. 130, 189–213 (1989)

    MathSciNet  Article  MATH  Google Scholar 

  4. Caffarelli, L., Cabré, X.: Fully nonlinear elliptic equations. AMS Colloquium Publications 43 (1995)

  5. Garofalo, N., Garcia, M.S.V.: New monotonicity formulas and the optimal regularity in the Signorini problem with variable coefficients. Adv. Math. 262, 682–750 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  6. Guillen, N.: Optimal regularity for the Signorini problem. Calc. Var. 36, 533–546 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  7. Koch, H., Rüland, A., Shi, W: The variable coefficient thin obstacle problem: Carleman inequalities, arXiv preprint

  8. Milakis, E., Silvestre, L.: Regularity for fully nonlinear elliptic equations with Neumann boundary data. Comm. Partial Diff. Equ. 31, 1227–1252 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  9. Milakis, E., Silvestre, L.: Regularity for the nonlinear Signorini problem. Adv. Math. 217, 1301–1312 (2008)

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgments

The author would like to thank Alessio Figalli and Xavier Ros-Oton for their guidance and useful discussions on the topics of this paper.

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Correspondence to Xavier Fernández-Real.

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Communicated by L. Caffarelli.

The author is supported by a fellowship from “Obra Social la Caixa”.

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Fernández-Real, X. \(C^{1,\alpha }\) estimates for the fully nonlinear Signorini problem. Calc. Var. 55, 94 (2016). https://doi.org/10.1007/s00526-016-1034-3

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  • DOI: https://doi.org/10.1007/s00526-016-1034-3

Mathematics Subject Classification

  • 35J60
  • 35B65