Skip to main content
SpringerLink
Log in

Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We construct two new one-parameter families of monotonicity formulas to study the free boundary points in the lower dimensional obstacle problem. The first one is a family of Weiss type formulas geared for points of any given homogeneity and the second one is a family of Monneau type formulas suited for the study of singular points. We show the uniqueness and continuous dependence of the blowups at singular points of given homogeneity. This allows to prove a structural theorem for the singular set.

Our approach works both for zero and smooth non-zero lower dimensional obstacles. The study in the latter case is based on a generalization of Almgren’s frequency formula, first established by Caffarelli, Salsa, and Silvestre.

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. Almgren, F.J. Jr.: Dirichlet’s problem for multiple valued functions and the regularity of mass minimizing integral currents. In: Minimal Submanifolds and Geodesics. Proc. Japan-United States Sem., Tokyo, 1977, pp. 1–6. North-Holland, Amsterdam (1979)

    Google Scholar 

  2. Alt, H.W., Caffarelli, L.A., Friedman, A.: Variational problems with two phases and their free boundaries. Trans. Am. Math. Soc. 282(2), 431–461 (1984)

    Article  MathSciNet  Google Scholar 

  3. Athanasopoulos, I., Caffarelli, L.A.: Optimal regularity of lower dimensional obstacle problems. Zap. Naucn. Semin. St.-Peterb. Otd. Mat. Inst. Steklova (POMI) 310(35), 49–66 (2004). Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. [34], 226 (English, with English and Russian summaries); English transl. J. Math. Sci. (NY) 132(3), 274–284 (2006)

    MATH  Google Scholar 

  4. 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 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  7. Caffarelli, L.A., Rivière, N.M.: Asymptotic behaviour of free boundaries at their singular points. Ann. Math. (2) 106(2), 309–317 (1977)

    Article  MathSciNet  Google Scholar 

  8. Caffarelli, L., 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 

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman and Hall, CRC Financial Mathematics Series. Chapman and Hall, CRC, Boca Raton (2004)

    MATH  Google Scholar 

  11. Duvat, G., Lions, J.L.: Les inequations en mechanique et en physique. Dunod, Paris (1972)

    Google Scholar 

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

    MATH  Google Scholar 

  13. Garofalo, N., Lin, F.-H.: Monotonicity properties of variational integrals A p weights and unique continuation. Indiana Univ. Math. J. 35(2), 245–268 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  14. Garofalo, N., Lin, F.-H.: Unique continuation for elliptic operators: a geometric-variational approach. Commun. Pure Appl. Math. 40(3), 347–366 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  15. Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM Studies in Applied Mathematics, vol. 8. Society for Industrial and Applied Mathematics, Philadelphia (1988)

    MATH  Google Scholar 

  16. Monneau, R.: On the number of singularities for the obstacle problem in two dimensions. J. Geom. Anal. 13(2), 359–389 (2003)

    MathSciNet  Google Scholar 

  17. Monneau, R.: Pointwise estimates for Laplace equation. Applications to the free boundary of the obstacle problem with Dini coefficients. Preprint (2007)

  18. 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 

  19. Ural’tseva, N.N.: On the regularity of solutions of variational inequalities. Usp. Mat. Nauk 42(6), 151–174 (1987) (Russian); English transl., Russ. Math. Surv. 42(6), 191–219 (1987)

    MathSciNet  Google Scholar 

  20. Weiss, G.S.: A homogeneity improvement approach to the obstacle problem. Invent. Math. 138(1), 23–50 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  21. Whitney, H.: Analytic extensions of differentiable functions defined in closed sets. Trans. Am. Math. Soc. 36(1), 63–89 (1934)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Nicola Garofalo & Arshak Petrosyan

Authors
  1. Nicola Garofalo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arshak Petrosyan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Arshak Petrosyan.

Additional information

N. Garofalo is supported in part by NSF grant DMS-0701001.

A. Petrosyan is supported in part by NSF grant DMS-0701015.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Garofalo, N., Petrosyan, A. Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem. Invent. math. 177, 415–461 (2009). https://doi.org/10.1007/s00222-009-0188-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-009-0188-4

Mathematics Subject Classification (2000)

Search

Navigation