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Optimal regularity for the Signorini problem

Calculus of Variations and Partial Differential Equations volume 36, Article number: 533 (2009) Cite this article

Abstract

We prove under general assumptions that solutions of the thin obstacle or Signorini problem in any space dimension achieve the optimal regularity C 1,1/2. This improves the known optimal regularity results by allowing the thin obstacle to be defined in an arbitrary C 1,β hypersurface, β > 1/2, additionally, our proof covers any linear elliptic operator in divergence form with smooth coefficients. The main ingredients of the proof are a version of Almgren’s monotonicity formula and the optimal regularity of global solutions.

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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), no.Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35(34), 49–66, 226

  2. Athanasopoulos I., Caffarelli L.A., Salsa S.: The structure of the free boundary for lower dimensional obstacle problems. Am. J. Math. 130(2), 485–498 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Caffarelli L.A.: Further regularity for the Signorini problem. Comm. Partial Differ. Equ. 4(9), 1067–1075 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  4. Caffarelli L., Silvestre L.: An extension problem related to the fractional Laplacian. Comm. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Caffarelli L.A., Salsa S., Silvestre L.: Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent. Math. 171(2), 425–461 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Duvaut, G., Lions, J.-L.: Les inéquations en mécanique et en physique, Dunod, Paris, Travaux et Recherches Mathématiques, No. 21 (1972)

  7. Frehse J.: On Signorini’s problem and variational problems with thin obstacles. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4(2), 343–362 (1977)

    MATH  MathSciNet  Google Scholar 

  8. Friedman A.: Variational Principles and Free-Boundary Problems Pure and Applied Mathematics. A Wiley-Interscience Publication, Wiley, New York (1982)

    Google Scholar 

  9. Richardson, D.: Variational problems with thin obstacles. Ph.D. Thesis, University of British Columbia, Vancouver, BC (1978)

  10. Silvestre L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60(1), 67–112 (2007)

    Article  MATH  MathSciNet  Google Scholar 

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

    Nestor Guillen

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  1. Nestor Guillen
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Correspondence to Nestor Guillen.

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Guillen, N. Optimal regularity for the Signorini problem. Calc. Var. 36, 533 (2009). https://doi.org/10.1007/s00526-009-0242-5

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  • DOI: https://doi.org/10.1007/s00526-009-0242-5

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