Advertisement

Inventiones mathematicae

, Volume 171, Issue 2, pp 425–461 | Cite as

Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

Article

Abstract

We use a characterization of the fractional Laplacian as a Dirichlet to Neumann operator for an appropriate differential equation to study its obstacle problem. We write an equivalent characterization as a thin obstacle problem. In this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary.

Keywords

Free Boundary Harnack Inequality Obstacle Problem Weighted Sobolev Space Monotonicity Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Athanasopoulos, I., Caffarelli, L.A.: Optimal regularity of lower dimensional obstacle problems. Zap. Nauchn. Semin. POMI 310 (Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35 [34]), 49–66, 226 (2004)Google Scholar
  2. 2.
    Athanasopoulos, I., Caffarelli, L.A., Salsa, S.: The structure of the free boundary for lower dimesional obstacle problems. To appear in Am. J. Math. (2007)Google Scholar
  3. 3.
    Bouchaud, J.P., Georges, A.: Anomalous diffusion in disordered media: Statistical mechanics, models and physical applications. Phys. Rep. 195(4–5), 127–293 (1990)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Caffarelli, L.A.: The obstacle problem revisited. J. Fourier Anal. Appl. 4(4–5), 383–402 (1998)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Caffarelli, L.A., Cabré, X.: Fully Nonlinear Elliptic Equations. Colloq. Publ., Am. Math. Soc., vol. 43. Am. Math. Soc., Providence, RI (1995)MATHGoogle Scholar
  6. 6.
    Caffarelli, L., Salsa, S.: A Geometric Approach to Free Boundary Problems. Grad. Stud. Math., vol. 68. Am. Math. Soc., Providence, RI (2005)MATHGoogle Scholar
  7. 7.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equations 32(8), 1245–1260 (2007)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Caffarelli, L., Vasseur, A.: Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation. PreprintGoogle Scholar
  9. 9.
    Constantin, P.: Euler equations, Navier-Stokes equations and turbulence. In: Mathematical Foundation of Turbulent Viscous Flows. Lect. Notes Math., vol. 1871, pp. 1–43. Springer, Berlin (2006)CrossRefGoogle Scholar
  10. 10.
    Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman Hall/CRC Financ. Math. Ser. Chapman & Hall/CRC, Boca Raton, FL (2004)Google Scholar
  11. 11.
    Fabes, E.B., Kenig, C.E., Jerison, D.: Boundary behavior of solutions to degenerate elliptic equations. In: Conference on Harmonic Analysis in Honor of Antoni Zygmund, vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., pp. 577–589. Wadsworth, Belmont, CA (1983)Google Scholar
  12. 12.
    Fabes, E.B., Kenig, C.E., Serapioni, R.P.: The local regularity of solutions of degenerate elliptic equations. Commun. Partial Differ. Equations 7(1), 77–116 (1982)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Garofalo, N.: Unique continuation for a class of elliptic operators which degenerate on a manifold of arbitrary codimension. J. Differ. Equations 104(1), 117–146 (1993)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kilpeläinen, T.: Smooth approximation in weighted Sobolev spaces. Commentat. Math. Univ. Carol. 38(1), 29–35 (1997)MATHGoogle Scholar
  15. 15.
    Milakis, E., Silvestre, L.E.: Regularity for fully nonlinear elliptic equations with Neumann boundary data. Commun. Partial Differ. Equations 31(7–9), 1227–1252 (2006)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Silvestre, L.: The 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 2007

Authors and Affiliations

  • Luis A. Caffarelli
    • 1
  • Sandro Salsa
    • 2
  • Luis Silvestre
    • 3
  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA
  2. 2.Department of MathematicsPolytechnic Institute of MilanMilanItaly
  3. 3.Courant Institute of Mathematical SciencesNew YorkUSA

Personalised recommendations