Inventiones mathematicae

, Volume 215, Issue 1, pp 311–366 | Cite as

On the fine structure of the free boundary for the classical obstacle problem

  • Alessio FigalliEmail author
  • Joaquim Serra


In the classical obstacle problem, the free boundary can be decomposed into “regular” and “singular” points. As shown by Caffarelli in his seminal papers (Caffarelli in Acta Math 139:155–184, 1977; J Fourier Anal Appl 4:383–402, 1998), regular points consist of smooth hypersurfaces, while singular points are contained in a stratified union of \(C^1\) manifolds of varying dimension. In two dimensions, this \(C^1\) result has been improved to \(C^{1,\alpha }\) by Weiss (Invent Math 138:23–50, 1999). In this paper we prove that, for \(n=2\) singular points are locally contained in a \(C^2\) curve. In higher dimension \(n\ge 3\), we show that the same result holds with \(C^{1,1}\) manifolds (or with countably many \(C^2\) manifolds), up to the presence of some “anomalous” points of higher codimension. In addition, we prove that the higher dimensional stratum is always contained in a \(C^{1,\alpha }\) manifold, thus extending to every dimension the result in Weiss (1999). We note that, in terms of density decay estimates for the contact set, our result is optimal. In addition, for \(n\ge 3\) we construct examples of very symmetric solutions exhibiting linear spaces of anomalous points, proving that our bound on their Hausdorff dimension is sharp.



both authors are supported by ERC Grant “Regularity and Stability in Partial Differential Equations (RSPDE)”. The authors thank the anonymous referees for useful comments on a preliminary version of the manuscript.


  1. 1.
    Athanasopoulos, I., Caffarelli, L.: Optimal regularity of lower dimensional obstacle problems. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310 (2004), Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35 [34], 49–66, 226; translation in J. Math. Sci. (N. Y.) 132 (2006), no. 3, 274–284Google Scholar
  2. 2.
    Athanasopoulos, I., Caffarelli, L., Salsa, S.: The structure of the free boundary for lower dimensional obstacle problems. Am. J. Math. 130, 485–498 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Brézis, H., Kinderlehrer, D.: The smoothness of solutions to nonlinear variational inequalities. Indiana Univ. Math. J. 23, 831–844 (1973/74)Google Scholar
  4. 4.
    Caffarelli, L.: The regularity of free boundaries in higher dimensions. Acta Math. 139, 155–184 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Caffarelli, L.: The obstacle problem revisited. J. Fourier Anal. Appl. 4, 383–402 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Caffarelli, L., Cabré, X.: Fully Nonlinear Elliptic Equations. American Mathematical Society Colloquium Publications, vol. 43. American Mathematical Society, Providence, RI (1995)zbMATHGoogle Scholar
  7. 7.
    Caffarelli, L., Rivière, N.: Smoothness and analyticity of free boundries in variational inequalities. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 3, 289–310 (1976)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Caffarelli, L., Rivière, N.: Asymptotic behavior of free boundaries at their singular points. Ann. Math. 106, 309–317 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Colombo, M., Spolaor, L., Velichkov, B.: A logarithmic epiperimetric inequality for the obstacle problem (preprint). arXiv:1708.02045v1
  10. 10.
    Fefferman, C.: Extension of \(C^{m,\omega }\)-smooth functions by linear operators. Rev. Mat. Iberoam. 25(1), 1–48 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Focardi, M., Spadaro, E.: On the measure and structure of the free boundary of the lower dimensional obstacle problem (preprint). arXiv:1703.00678v2
  12. 12.
    Garofalo, N., Petrosyan, A.: Some new monotonicity formulas and the singular set in the lower dimensional obstacle problem. Invent. Math. 177, 415–461 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Reprint of the 1998 edition. Classics in Mathematics. Springer, Berlin (2001)zbMATHGoogle Scholar
  14. 14.
    Kinderlehrer, D., Nirenberg, L.: Regularity in free boundary problems. Ann. Sc. Norm. Sup. Pisa 4, 373–391 (1977)Google Scholar
  15. 15.
    Monneau, R.: A brief overview on the obstacle problem. In: European Congress of Mathematics, Vol. II (Barcelona, 2000), volume 202 of Progr. Math., pp. 303–312. Birkhäuser, Basel (2001)Google Scholar
  16. 16.
    Monneau, R.: On the number of singularities for the obstacle problem in two dimensions. J. Geom. Anal. 13, 359–389 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Petrosyan, A., Shahgholian, H., Usaltseva, N.: Regularity of Free Boundaries in Obstacle-Type Problems. Graduate Studies in Mathematics, vol. 136. American Mathematical Society, Providence, RI (2012)CrossRefGoogle Scholar
  18. 18.
    Sakai, M.: Regularity of a boundary having a Schwarz function. Acta Math. 166(3–4), 263–297 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sakai, M.: Regularity of free boundaries in two dimensions. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 20(3), 323–339 (1993)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Schaeffer, D.: Some examples of singularities in a free boundary. Ann. Scuola Norm. Sup. Pisa 4, 131–144 (1976)Google Scholar
  21. 21.
    Simon, L.: Lectures on geometric measure theory. In: Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3. Australian National University, Centre for Mathematical Analysis, Canberra (1983)Google Scholar
  22. 22.
    Weiss, G.: A homogeneity improvement approach to the obstacle problem. Invent. Math. 138, 23–50 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Yeressian, K.: Obstacle problem with a degenerate force term. Anal. PDE 9(2), 397–437 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Mathematics DepartmentETH ZürichZurichSwitzerland

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