Advertisement

Monotonicity formulas for obstacle problems with Lipschitz coefficients

  • M. FocardiEmail author
  • M. S. Gelli
  • E. Spadaro
Article

Abstract

We prove quasi-monotonicity formulas for classical obstacle-type problems with energies being the sum of a quadratic form with Lipschitz coefficients, and a Hölder continuous linear term. With the help of those formulas we are able to carry out the full analysis of the regularity of free-boundary points following the approaches by Caffarelli (J Fourier Anal Appl 4(4–5), 383–402, 1998), Monneau (J Geom Anal 13(2), 359–389, 2003), and Weiss (Invent Math 138(1), 23–50, 1999).

Mathematics Subject Classification

35R35 49N60 

Notes

Acknowledgments

The first two authors have been supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

References

  1. 1.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Athanasopoulos, I., Caffarelli, L.A.: A theorem of real analysis and its application to free boundary problems. Commun. Pure Appl. Math. 38(5), 499–502 (1985)MathSciNetCrossRefGoogle Scholar
  3. 3.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Caffarelli, L.A.: The regularity of free boundaries in higher dimensions. Acta Math. 139(3–4), 155–184 (1977)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Caffarelli, L.A.: Compactness methods in free boundary problems. Commun. Partial Differ. Equ. 5(4), 427–448 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Caffarelli, L.A.: The obstacle problem revisited. Lezioni Fermiane [Fermi Lectures]. Accademia Nazionale dei Lincei, Rome; Scuola Normale Superiore, Pisa, ii+54 pp (1998)Google Scholar
  7. 7.
    Caffarelli, L.A.: The obstacle problem revisited. J. Fourier Anal. Appl. 4(4–5), 383–402 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Caffarelli, L.A., Fabes, E., Mortola, S., Salsa, S.: Boundary behavior of nonnegative solutions of elliptic operators in divergence form. Indiana Univ. Math. J. 30(4), 621–640 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Caffarelli, L.A., Salsa, S.: A geometric approach to free boundary problems. Graduate Studies in Mathematics, vol. 68. American Mathematical Society, Providence, x+270 pp (2005)Google Scholar
  10. 10.
    Caffarelli, L.A., Karp, L., Shahgholian, H.: Regularity of a free boundary with application to the Pompeiu problem. Ann. Math. (2) 151(1), 269–292 (2000)Google Scholar
  11. 11.
    Cerutti, M.C., Ferrari, F., Salsa, S.: Two-phase problems for linear elliptic operators with variable coefficients: Lipschitz free boundaries are \(C^{1,\gamma }\). Arch. Ration. Mech. Anal. 171(3), 329–348 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ferrari, F., Salsa, S.: Regularity of the free boundary in two-phase problems for linear elliptic operators. Adv. Math. 214(1), 288–322 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ferrari, F., Salsa, S.: Regularity of the solutions for parabolic two-phase free boundary problems. Commun. Partial Differ. Equ. 35(6), 1095–1129 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Friedman, A.: Variational Principles and Free Boundary Problems, 2nd edn. Robert E. Krieger Publishing Co., Inc, Malabar, x+710 pp (1988)Google Scholar
  15. 15.
    Giusti, E.: Equazioni ellittiche del secondo ordine, Quaderni Unione Matematica Italiana, vol. 6. Pitagora Editrice, Bologna, v+213 pp (1978)Google Scholar
  16. 16.
    Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Reprint of the 1998 edition. Classics in Mathematics. Springer, Berlin, xiv+517 pp (2001)Google Scholar
  17. 17.
    Jerison, D., Kenig, C.: Boundary behaviour of harmonic functions in nontangentially accessible domains. Adv. Math. 46(1), 80–147 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kinderlehrer, D., Stampacchia, G.: An introduction to variational inequalities and their applications. Pure and Applied Mathematics, vol. 88. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, xiv+313 pp (1980)Google Scholar
  19. 19.
    Kukavica, I.: Quantitative uniqueness for second-order elliptic operators. Duke Math. J. 91(2), 225–240 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Lin, F.: On regularity and singularity of free boundaries in obstacle problems. Chin. Ann. Math. Ser. B 30(5), 645–652 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Matevosyan, N., Petrosyan, A.: Almost monotonicity formulas for elliptic and parabolic operators with variable coefficients. Commun. Pure Appl. Math. 64(2), 271–311 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Monneau, R.: On the number of singularities for the obstacle problem in two dimensions. J. Geom. Anal. 13(2), 359–389 (2003)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Petrosyan, A., Shahgholian, H.: Geometric and energetic criteria for the free boundary regularity in an obstacle-type problem. Am. J. Math. 129(6), 1659–1688 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Petrosyan, A., Shahgholian, H., Uraltseva, N.: Regularity of free boundaries in obstacle-type problems. Graduate Studies in Mathematics, vol. 136. American Mathematical Society, Providence, x+221 pp (2012)Google Scholar
  25. 25.
    Rodrigues, J.F.: Obstacle problems in mathematical physics. North-Holland Mathematics Studies, vol. 134. Notas de Matemtica [Mathematical Notes], vol. 114. North-Holland Publishing Co., Amsterdam, xvi+352 pp (1987)Google Scholar
  26. 26.
    Wang, P.Y.: Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order. I. Lipschitz free boundaries are \(C^{1,\alpha }\). Commun. Pure Appl. Math. 53(7), 799–810 (2000)CrossRefzbMATHGoogle Scholar
  27. 27.
    Wang, P.Y.: Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order. II. Flat free boundaries are Lipschitz. Commun. Partial Differ. Equ. 27(7–8), 1497–1514 (2002)CrossRefzbMATHGoogle Scholar
  28. 28.
    Weiss, G.S.: A homogeneity improvement approach to the obstacle problem. Invent. Math. 138(1), 23–50 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Weiss, G.S.: An obstacle-problem-like equation with two phases: pointwise regularity of the solution and an estimate of the Hausdorff dimension of the free boundary. Interfaces Free Bound. 3(2), 121–128 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Ziemer, W.P.: Weakly differentiable functions. Sobolev spaces and functions of bounded variation. GTM, vol. 120. Springer, New York, xvi+308 pp (1989)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.DiMaI “U. Dini”Università di FirenzeFlorenceItaly
  2. 2.Dipartimento di MatematicaUniversità di PisaPisaItaly
  3. 3.Max-Planck-Institut für Mathematik in den NaturwissenschaftenLeipzigGermany

Personalised recommendations