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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
Article

Abstract

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.

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References

  1. [1]
    R. A. Adams, Sobolev Spaces, Academics Press, New York, 1975.zbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.CrossRefzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    W. Beckner, Pitt’s inequality and the uncertainty principle, Proc. Amer. Math. Soc. 123 (1995), 1897–1905.MathSciNetzbMATHGoogle 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), 1–37MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    C. Bjorland, L. Caffarelli, and A. Figalli, Non-local gradient dependent operators, Adv. Math. 230 (2012), 1859–1894.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    L. Caffarelli and L. Silvestre, Regularity theory for fully nonlinear integro-differential equations, Comm. Pure Appl. Math. 62 (2009), 597–638.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    L. Caffarelli and L. Silvestre, Regularity results for nonlocal equations by approximation, Arch. Rational Mech. Anal. 200 (2011), 59–88.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.CrossRefzbMATHGoogle 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.MathSciNetzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    K. Ito, Lectures on Stochastic Processes, Springer-Verlag, Berlin, 1984.zbMATHGoogle Scholar
  37. [37]
    N. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York-Heidelberg, 1972.CrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. of Math. (2) 118 (1983), 349–374.MathSciNetCrossRefzbMATHGoogle Scholar
  40. [40]
    L. Ma and D. Chen, Radial symmetry and monotonicity for an integral equation, J. Math. Anal. Appl. 342 (2008), 943–949.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    O. Savin and E. Valdinoci, Elliptic PDEs with fibered nonlinearities, J. Geom. Anal. 19 (2009), 420–432l.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  44. [44]
    J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43 (1971), 304–318.MathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    R. Servadei and E. Valdinocci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl. 389 (2012), 887–898.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetzbMATHGoogle Scholar
  47. [47]
    A. Signorini, Questioni di elasticitá non linearizzata e semilinearizzata, Rendiconti di Matematica e delle sue applicazioni 18 (1959), 95–139.zbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.zbMATHGoogle Scholar
  51. [51]
    E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993.zbMATHGoogle Scholar
  52. [52]
    J. Toland, The Peierls-Nabarro and Benjamin-Ono equations, J. Funct. Anal. 145 (1997), 136–150.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetzbMATHGoogle 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

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