Advertisement

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
Article

Abstract

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.

Keywords

Perforated domains Nonlocal equations Neumann problem Dirichlet problem 

Mathematics Subject Classification (2010)

45A05 45M05 49J40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

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).

References

  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 http://mate.dm.uba.ar/~jrossi/PD_final_version.pdf
  21. 21.
    Rauch, J., Taylor, M.: Potential and scattering theory on wildly perturbed domains. J. Funct. Anal. 18, 27–59 (1975)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© 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

Personalised recommendations