Journal d'Analyse Mathématique

, Volume 135, Issue 1, pp 37–57 | Cite as

On the moving plane method for nonlocal problems in bounded domains

  • Begoña Barrios
  • Luigi Montoro
  • Berardino Sciunzi


We consider a nonlocal problem involving the fractional Laplacian and the Hardy potential in bounded smooth domains. Exploiting the moving plane method and some weak and strong comparison principles, we deduce symmetry and monotonicity properties of positive solutions under zero Dirichlet boundary conditions.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. A. Adams, Sobolev Spaces, Academics Press, New York, 1975.MATHGoogle Scholar
  2. [2]
    G. Alberti and G. Bellettini, A nonlocal anisotropic model for phase transitions. I. The optimal profile problem, Math. Ann. 310 (1998), 527–560.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    A. D. Alexandrov, A characteristic property of the spheres, Ann. Mat. Pura Appl. 58 (1962), 303–354.MathSciNetCrossRefGoogle Scholar
  4. [4]
    B. Barrios, E. Colorado, R. Servadei, and F. Soria, A critical fractional equation with concaveconvex nonlinearities, Ann. Henri Poincaré 3 (2015), 875–900.CrossRefMATHGoogle Scholar
  5. [5]
    B. Barrios, A. Figalli, and E. Valdinoci, Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (5) 13 (2014), 609–639.MathSciNetMATHGoogle Scholar
  6. [6]
    B. Barrios, M. Medina, and I. Peral, Some remarks on the solvability of non local elliptic problems with the Hardy potential, Commun. Contemp. Math. 16 (2014), no. 4.Google Scholar
  7. [7]
    B. Barrios, I. Peral, F. Soria, and E. Valdinoci, A Widder’s type theorem for the heat equation with nonlocal diffusion, Arch. Rational Mech. Anal. 213 (2014), 629–650.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    R. Bass and M. Kassmann, Hölder continuity of harmonic functions with respect to operators of variable order, Comm. Partial Differential Equations 30 (2005), 1249–1259.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    W. Beckner, Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123 (1995), 1897–1905.MathSciNetMATHGoogle Scholar
  10. [10]
    H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bolletin Soc. Brasil. de Mat Nova Ser, 22 (1991), 1–37MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    C. Bjorland, L. Caffarelli, and A. Figalli, Non-local gradient dependent operators, Adv. Math. 230 (2012), 1859–1894.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song, and Z. Vondracek, Potential Analysis of Stable Processes and its Extensions, Springer-Verlag, Berlin, 2009.CrossRefGoogle Scholar
  13. [13]
    H. Brezis and X. Cabré, Some simple nonlinear PDE’s without solutions, Boll. Unione Mat. Ital. 1-B (1998), 223–262.MathSciNetMATHGoogle Scholar
  14. [14]
    X. Cabré and J. Sola-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math. 58 (2005), 1678–1732.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    X. Cabré and Y. Sire, Nonlinear equations for fractional laplacians II: existence, uniqueness, and qualitative properties of solutions, Ann. Inst. H. Poincare (C) Non Linear Analysis, online 2013.Google Scholar
  16. [16]
    L. Caffarelli, Further regularity for the Signorini problem, Comm. Partial Differential Equations, 4 (1979), 1067–1075.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    L. Caffarelli and A. Figalli, Regularity of solutions to the parabolic fractional obstacle problem, J. Reine Angew. Math. 680 (2013), 191–233.MathSciNetMATHGoogle Scholar
  18. [18]
    L. Caffarelli, J. M. Roquejoffre, and Y. Sire, Variational problems in free boundaries for the fractional Laplacian, J. Eur. Math. Soc. (JEMS) 12 (2010), 1151–1179.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math. 62 (2009), 597–638.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Rational Mech. Anal. 200 (2011), 59–88.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    L. Caffarelli and L. Silvestre, The Evans-Krylov theorem for non local fully non linear equations, Ann. of Math. (2) 174 (2011), 1163–1187.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasigeostrophic equation, Ann. of Math. (2) 171 (2010), 1903–1930.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    W. Chen, C. Li, and B. Ou, Qualitative properties of solutions for an integral equation, Discrete Contin. Dyn. Syst. 12 (2005), 347–354.MathSciNetMATHGoogle Scholar
  24. [24]
    W. Chen, C. Li, and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006), 330–343.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    E. Di Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    S. Dipierro, L. Montoro, I. Peral, and B. Sciunzi, Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential, Calc. Var. Partial Differential Equations 55 (2016), Art. 99.Google Scholar
  27. [27]
    S. Dipierro, G. Palatucci, and E. Valdinoci, Existence and symmetry results for a Schrödinger type problem involving the fractional Laplacian, Matematiche (Catania) 68 (2013), 201–216.MathSciNetMATHGoogle Scholar
  28. [28]
    R. L. Frank, E. H. Lieb, and R. Seiringer, Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators, J. Amer.Math. Soc. 21 (2008), 925–950.MathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    P. Felmer and Y. Wang, Radial symmetry of positive solutions to equations involving the fractional Laplacian, Commun. Contemp. Math. 16 (2014), no. 1.Google Scholar
  30. [30]
    A. Figalli, S. Dipierro, and E. Valdinoci, Strongly nonlocal dislocation dynamics in crystals, Commun. Partial Differential Equations 39 (2014), 2351–2387.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    B. Gidas, W. M. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209–243.MathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    I. Herbst, Spectral theory of the operator (p2 +m2)1/2-Ze2/r, Commun. Math. Phys. 53 (1977), 285–294.CrossRefGoogle Scholar
  33. [33]
    R. Husseini and M. Kassmann, Jump processes, L-harmonic functions, continuity estimates and the Feller property, Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009), 1099–1115.CrossRefMATHGoogle Scholar
  34. [34]
    S. Jarohs and T. Weth, Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations, Discrete Contin. Dyn. Syst. 34 (2014), 2581–2615.MathSciNetMATHGoogle Scholar
  35. [35]
    S. Jarohs and T. Weth, Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Ann. Mat. Pura Appl. (4) 195 (2016), 273–291.MathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    K. Ito, Lectures on Stochastic Processes, Springer-Verlag, Berlin, 1984.MATHGoogle Scholar
  37. [37]
    N. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York-Heidelberg, 1972.CrossRefMATHGoogle Scholar
  38. [38]
    T. Leonori, I. Peral, A. Primo, and F. Soria, Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst. 35 (2015), 6031–6068.MathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), 349–374.MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl. 342 (2008), 943–949.MathSciNetCrossRefMATHGoogle Scholar
  41. [41]
    X. Ros-Oton and J. Serra, The Dirichlet problem for the fractional laplacian: regularity up to the boundary, J. Math. Pures Appl. (9) 101 (2014), 275–302.MathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    O. Savin and E. Valdinoci, Elliptic PDEs with fibered nonlinearities, J. Geom. Anal. 19 (2009), 420–432l.MathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    J. Serra, Cs+a regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels, Calc. Var. Partial Differential Equations 54 (2015), 3571–3601.MathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318.MathSciNetCrossRefMATHGoogle Scholar
  45. [45]
    R. Servadei and E. Valdinocci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), 887–898.MathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst. 33 (2013), 2105–2137.MathSciNetMATHGoogle Scholar
  47. [47]
    A. Signorini, Questioni di elasticitá non linearizzata e semilinearizzata, Rendiconti di Matematica e delle sue applicazioni 18 (1959), 95–139.MATHGoogle Scholar
  48. [48]
    L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), 67–112.MathSciNetCrossRefMATHGoogle Scholar
  49. [49]
    Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: a geometric inequality and a symmetry result, J. Funct. Anal. 256 (2009), 1842–1864.MathSciNetCrossRefMATHGoogle Scholar
  50. [50]
    E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.MATHGoogle Scholar
  51. [51]
    E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.MATHGoogle Scholar
  52. [52]
    J. Toland, The Peierls-Nabarro and Benjamin-Ono equations, J. Funct. Anal. 145 (1997), 136–150.MathSciNetCrossRefMATHGoogle Scholar
  53. [53]
    E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA, 49 (2009), 33–44.MathSciNetMATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2018

Authors and Affiliations

  • Begoña Barrios
    • 1
  • Luigi Montoro
    • 2
  • Berardino Sciunzi
    • 2
  1. 1.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain
  2. 2.Dipartimento di Matematica e InformaticaUniversità della CalabriaArcavacata di Rende, CosenzaItaly

Personalised recommendations