Potential Analysis

, Volume 48, Issue 3, pp 361–373 | Cite as

An Obstacle Problem for Nonlocal Equations in Perforated Domains

  • Marcone C. Pereira
  • Julio D. Rossi


In this paper we analyze the behavior of solutions to a nonlocal equation of the form Ju (x) − u (x) = f (x) in a perforated domain Ω ∖ A 𝜖 with u = 0 in \(A^{\epsilon } \cup {\Omega }^{c}\) and an obstacle constraint, uψ in Ω ∖ A 𝜖 . We show that, assuming that the characteristic function of the domain Ω ∖ A 𝜖 verifies \(\chi _{\epsilon } \rightharpoonup \mathcal {X}\) weakly in \(L^{\infty }({\Omega })\), there exists a weak limit of the solutions u 𝜖 and we find the limit problem that is satisfied in the limit. When \(\mathcal {X} \not \equiv 1\) in this limit problem an extra term appears in the equation as well as a modification of the obstacle constraint inside the domain.


Perforated domains Nonlocal equations Neumann problem Dirichlet problem 

Mathematics Subject Classification (2010)

45A05 45M05 49J40 


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The first author (MCP) is partially supported by CNPq 302960/2014-7 and 471210/2013-7, FAPESP 2015/17702-3 (Brazil) and the second author (JDR) by MINCYT grant MTM2016-68210 (Spain).


  1. 1.
    Andreu-Vaillo, F., Mazón, J.M., Rossi, J.D., Toledo, J.: Nonlocal Diffusion Problems. Mathematical Surveys and Monographs, vol. 165. AMS (2010)Google Scholar
  2. 2.
    Barles, G., Chasseigne, E., Imbert, C.: On the Dirichlet problem for second-order elliptic integro-differential equations. Indiana Univ. Math. J. 57(1), 213–246 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Caffarelli, L.A., Mellet, A.: Random homogenization of an obstacle problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(2), 375–395 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Caffarelli, L.A., Mellet, A.: Random homogenization of fractional obstacle problems. Netw. Heterog. Media 3(3), 523–554 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Calvo-Jurado, C., Casado-Díaz, J., Luna-Laynez, M.: Homogenization of nonlinear Dirichlet problems in random perforated domains. Nonlinear Anal. 133, 250–274 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cioranescu, D., Murat, F.: A strange term coming from nowhere. Progress Nonl. Diff. Eq. Their Appl. 31, 45–93 (1997)zbMATHGoogle Scholar
  7. 7.
    Cioranescu, D., Damlamian, A., Donato, P., Griso, G., Zaki, R.: The periodic unfolding method in domains with holes. SIAM J. Math. Anal. 44(2), 718–760 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cioranescu, D., Donato, P.: An Introduction to Homogenization. Oxford Lecture Series in Mathematics and its Applications, vol. 17. Oxford University Press (1999)Google Scholar
  9. 9.
    Cioranescu, D., Saint Jean Paulin, J.: Homogenization in open sets with holes. J. Math. Anal. Appl. 71(2), 590–607 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chasseigne, E., Felmer, P., Rossi, J.D., Topp, E.: Fractional decay bounds for nonlocal zero order heat equations. Bull. Lond. Math. Soc. 46(5), 943–952 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cortazar, C., Elgueta, M., Rossi, J.D.: Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions. Israel J. Math. 170(1), 53–60 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cortazar, C., Elgueta, M., Rossi, J.D., Wolanski, N.: How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems. Arch. Ration. Mech. Anal. 187(1), 137–156 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Courant, R., Hilbert, D.: Methods of Mathematical Physics, vol. I. Interscience, New York (1953)zbMATHGoogle Scholar
  14. 14.
    Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: A nonlocal vector calculus, nonlocal volume-constrained problems, and nonlocal balance laws. Math. Models Methods Appl. Sci. 23(3), 493–540 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Felmer, P., Topp, E.: Uniform equicontinuity for a family of zero order operators approaching the fractional Laplacian. Comm. Partial Differential Equations 40(9), 1591–1618 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Friedman, A.: Variational Principles and Free-boundary Problems. Wiley (1982)Google Scholar
  17. 17.
    García Melián, J., Rossi, J.D.: On the principal eigenvalue of some nonlocal diffusion problems. J. Differential Equations. 246(1), 21–38 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lehoucq, R.B., Silling, S.A.: Force flux and the peridynamic stress tensor. J. Mech. Phys. Solids 56(4), 1566–1577 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Necas, J.: Les Méthodes Directes En Théorie Des Équations Elliptiques. Masson, Paris (1967)zbMATHGoogle Scholar
  20. 20.
    Pereira, M.C., Rossi J.D.: Nonlocal problems in perforated domains. Preprint. Available at
  21. 21.
    Rauch, J., Taylor, M.: Potential and scattering theory on wildly perturbed domains. J. Funct. Anal. 18, 27–59 (1975)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Dpto. de Matemática Aplicada, IMEUniversidade de São PauloSão PauloBrazil
  2. 2.Dpto. de Matemáticas, FCEyNUniversidad de Buenos AiresBuenos AiresArgentina

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