Skip to main content
SpringerLink
Log in

Reifenberg flatness of free boundaries in obstacle problems with VMO ingredients

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We study the obstacle problem with an elliptic operator in divergence form. We develop all of the basic theory of existence, uniqueness, optimal regularity, and nondegeneracy of the solutions. These results, in turn, allow us to begin the study of the regularity of the free boundary in the case where the coefficients are in VMO.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Blank, I.: Sharp results for the regularity and stability of the free boundary in the obstacle problem. Indiana Univ. Math. J. 50(3), 1077–1112 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Blank, I., Hao, Z.: The mean value theorem and basic properties of the obstacle problem for divergence form elliptic operators (submitted)

  3. Blank, I., Korten, M., Moore, C.: The Hele-Shaw problem as a “Mesa” limit of Stefan problems: Existence, uniqueness, and regularity of the free boundary. Trans. AMS 361(3), 1241–1268 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Blank, I., Teka, K.: The Caffarelli alternative in measure for the nondivergence form elliptic obstacle problem with principal coeffcients in VMO. Commun. Partial Differ. Equ. 39(2), 321–353 (2014)

  5. Caffarelli, L.A.: The regularity of free boundaries in higher dimensions. Acta Math. 139(3–4), 155–184 (1977)

    Article  MathSciNet  Google Scholar 

  6. Caffarelli, L.A.: A remark on the Hausdorff measure of a free boundary, and the convergence of coincidence sets. Boll. Un. Mat. Ital. A (5) 18(1), 109–113 (1981)

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Caffarelli, L.A.: The obstacle problem revisited. J. Fourier Anal. Appl. 4(4–5), 383–402 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Caffarelli, L.A.: The Obstacle Problem. The Fermi Lectures, Accademia Nazionale Dei Lincei Scuola Normale Superiore (1998)

  10. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 2nd edn. Springer, Berlin (1983)

  11. Kenig, C.E., Toro, T.: Harmonic measure on locally flat domains. Duke Math. J. 87(3), 509–551 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Kenig, C.E., Toro, T.: Harmonic measures and Poisson Kernels. Ann. Math. (2) 150(2), 369–454 (1999)

  13. Li, Y.Y., Vogelius, M.: Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients. Arch. Ration. Mech. Anal. 153(2), 91–151 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  14. Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge UniversityPress, Cambridge (1995)

  15. Maugeri, A., Palagachev, D.K., Softova, L.G.: Elliptic and Parabolic Equations with Discontinuous Coefficients. Mathematical Research, vol. 109, Wiley, New York (2000)

  16. Reifenberg, E.R.: Solution of the plateau problem for \(m\)-dimensional surfaces of varying topological type. Acta Math. 104, 1–92 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  17. Weiss, G.S.: A homogeneity improvement approach to the obstacle problem. Invent. Math. 138, 23–50 (1999)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Manhattan, Kansas, USA

    Ivan Blank

  2. Stout, Wisconsin, USA

    Zheng Hao

Authors
  1. Ivan Blank
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zheng Hao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ivan Blank.

Additional information

Communicated by L. Caffarelli.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Blank, I., Hao, Z. Reifenberg flatness of free boundaries in obstacle problems with VMO ingredients. Calc. Var. 53, 943–959 (2015). https://doi.org/10.1007/s00526-014-0772-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-014-0772-3

Mathematics Subject Classification

Search

Navigation