Optimal regularity for the Signorini problem

Article

Abstract

We prove under general assumptions that solutions of the thin obstacle or Signorini problem in any space dimension achieve the optimal regularity C1,1/2. This improves the known optimal regularity results by allowing the thin obstacle to be defined in an arbitrary C1,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.

Mathematics Subject Classification (2000)

35R35 74G40 

References

  1. 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, 226Google Scholar
  2. 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)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Caffarelli L.A.: Further regularity for the Signorini problem. Comm. Partial Differ. Equ. 4(9), 1067–1075 (1979)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Caffarelli L., Silvestre L.: An extension problem related to the fractional Laplacian. Comm. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)MATHCrossRefMathSciNetGoogle Scholar
  5. 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)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Duvaut, G., Lions, J.-L.: Les inéquations en mécanique et en physique, Dunod, Paris, Travaux et Recherches Mathématiques, No. 21 (1972)Google Scholar
  7. 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)MATHMathSciNetGoogle Scholar
  8. 8.
    Friedman A.: Variational Principles and Free-Boundary Problems Pure and Applied Mathematics. A Wiley-Interscience Publication, Wiley, New York (1982)Google Scholar
  9. 9.
    Richardson, D.: Variational problems with thin obstacles. Ph.D. Thesis, University of British Columbia, Vancouver, BC (1978)Google Scholar
  10. 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)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA

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