Chinese Annals of Mathematics, Series B

, Volume 38, Issue 2, pp 661–686 | Cite as

Symmetrization for fractional elliptic and parabolic equations and an isoperimetric application

  • Yannick Sire
  • Juan Luis Vázquez
  • Bruno Volzone


This paper develops further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors. This theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations. The authors extend the theory for the so-called restricted fractional Laplacian defined on a bounded domain Ω of ℝ N with zero Dirichlet conditions outside of Ω. As an application, an original proof of the corresponding fractional Faber-Krahn inequality is derived. A more classical variational proof of the inequality is also provided.


Symmetrization Fractional Laplacian Nonlocal elliptic and parabolic equations Faber-Krahn inequality 

2000 MR Subject Classification

35B45 35R11 35J61 35K55 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Applebaum, D., Lévy processes and stochastic calculus, Cambridge Studies in Advanced Mathematics, 2nd edition, 116, Cambridge University Press, Cambridge, 2009.CrossRefGoogle Scholar
  2. [2]
    Bandle, C., Isoperimetric inequalities and applications, Monographs and Studies in Mathematics, 7, Pitman (Advanced Publishing Program), Boston, London, 1980.Google Scholar
  3. [3]
    Bandle, C., On symmetrizations in parabolic equations, J. Analyse Math., 30, 1976, 98–112.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    Banuelos, R., Latala, R. and Méndez-Hernández, P. J., A Brascamp-Lieb-Luttinger-type inequality and applications to symmetric stable processes, Proc. Amer. Math. Society, 129(10), 2001, 2997–3008.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    Barbu, V., Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1975.Google Scholar
  6. [6]
    Bénilan, Ph., Equations d’évolution dans un espace de Banach quelconque et applications (in French), Ph. D. Thesis, Univ. Orsay, 1972.MATHGoogle Scholar
  7. [7]
    Bertoin, J., Lévy processes, Cambridge Tracts in Mathematics, 121, Cambridge University Press, Cambridge, 1996.Google Scholar
  8. [8]
    Bénilan, P., Brezis, H. and Crandall, M. G., A semilinear equation in L1(RN), Ann. Scuola Norm. Sup. Pisa Cl. Sci., 2(4), 1975, 523–555.MathSciNetMATHGoogle Scholar
  9. [9]
    Betsakos, D., Symmetrization, symmetric stable processes, and Riesz capacities (electronic), Trans. Amer. Math. Soc., 356, 2004, 735–755.MathSciNetMATHGoogle Scholar
  10. [10]
    Bonforte, M., Sire, Y. and Vázquez, J. L., Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst. A, 35(12), 2015, 5725–5767.MATHGoogle Scholar
  11. [11]
    Bonforte, M. and Vázquez, J. L., Quantitative local and global a priori estimates for fractional nonlinear diffusion equations, Advances in Math., 250, 2014, 242–284.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    Bonforte, M. and Vázquez, J. L., A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains, Arch. Ration. Mech. Anal., 218(1), 2015, 317–362.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    Brasco, L., Lindgren, E. and Parini, E., The fractional Cheeger problem, Interfaces Free Bound., 16(3), 419–458.Google Scholar
  14. [14]
    Brothers, J. and Ziemer, W., Minimal rearrangements of Sobolev functions, J. Reine Angew. Math., 384, 1988, 153–179.MathSciNetMATHGoogle Scholar
  15. [15]
    Bucur, D. and Freitas, P., A new proof of the Faber-Krahn inequality and the symmetry of optimal domains for higher eigenvalues, 2012, preprint.Google Scholar
  16. [16]
    Cabré, X. and Sire, Y., Nonlinear equations for fractional Laplacians, I: Regularity, maximum principles, and Hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31(1), 2014, 23–53.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    Cabré, X. and Tan, J. G., Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224(5), 2010, 2052–2093MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    Caffarelli, L. and Silvestre, L., An extension problem related to the fractional Laplacian, Comm. Part. Diff. Eq., 32(7–9), 2007, 1245–1260.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    Caffarelli, L. and Stinga, P., Fractional elliptic equations, Caccioppoli estimates and regularity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33(3), 2016, 767–807.CrossRefMATHGoogle Scholar
  20. [20]
    Chavel, I., Eigenvalues in Riemannian Geometry, Series in Pure and Applied Mathematics, 115, Academic Press, Orlando, 1984.Google Scholar
  21. [21]
    Chong, K. M., Some extensions of a theorem of Hardy, Littlewood and Pólya and their applications, Canad. J. Math., 26, 1974, 1321–1340.MATHGoogle Scholar
  22. [22]
    Brändle, C., Colorado, E. and de Pablo, A., A concave-convex elliptic problem involving the fractional laplacian, Proceedings of the Royal Society of Edinburgh, 143A, 2013, 39–71.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    Crandall, M. G., Nonlinear semigroups and evolution governed by accretive operators, Proceedings of Symposium in Pure Math., Part I, F. Browder (ed.), A. M. S., Providence, RI, 1986, 305–338.Google Scholar
  24. [24]
    Crandall, M. G. and Liggett, T. M., Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math., 93, 1971, 265–298.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    Di Blasio, G. and Volzone, B., Comparison and regularity results for the fractional Laplacian via symmetrization methods, J. Differential Equations, 253(9), 2012, 2593–2615.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    De Pablo, A., Quirós, F., Rodríguez, A. and Vázquez, J. L., A fractional porous medium equation, Adv. Math., 226(2), 2011, 1378–1409.MathSciNetCrossRefMATHGoogle Scholar
  27. [27]
    De Pablo, A., Quirós, F., Rodríguez, A. and Vázquez, J. L., A general fractional porous medium equation, Comm. Pure Appl. Math., 65(9), 2012, 1242–1284.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    De Pablo, A., Quirós, F., Rodríguez, A. and Vázquez, J. L., Classical solutions for a logarithmic fractional diffusion equation, preprint.Google Scholar
  29. [29]
    De Pablo, A., Quirós, F., Rodriguez, A. and Vázquez, J. L., Classical solutions and higher regularity for nonlinear fractional diffusion equations, JEMS, to appear.Google Scholar
  30. [30]
    Ferone, A. and Volpicelli, R., Minimal rearrangements of Sobolev functions: A new proof, Ann. Inst. H. Poincaré Anal. Non Linéaire, 20(2), 2003, 333–339.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    Faber, C., Beweiss, dass unter allen homogenen Membrane von gleicher Fläche und gleicher Spannung die kreisförmige die tiefsten Grundton gibt, Sitzungsber. Bayer. Akad. Wiss., Math. Phys. Munich., 1923, 169–172.Google Scholar
  32. [32]
    Frank, R. L. and Seiringer, R., Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., 255, 2008, 3407–3430.MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    Friedman, A., On the regularity of the solutions of nonlinear elliptic and parabolic systems of partial differential equations, J. Math. Mech., 7, 1958, 43–59.MathSciNetMATHGoogle Scholar
  34. [34]
    Hardy, G. H., Littlewood, J. E. and Pólya, G., Some simple inequalities satisfied by convex functions, Messenger Math., 58, 1929, 145–152, Inequalities, 2nd edition, Cambridge University Press, Cambridge, 1952.Google Scholar
  35. [35]
    Hashimoto, Y., A remark on the analyticity of the solutions for non-linear elliptic partial differential equations, Tokyo J. Math., 29(2), 2006, 271–281.MathSciNetCrossRefMATHGoogle Scholar
  36. [36]
    Kawohl, B., Rearrangements and convexity of level sets in PDE, Lecture Notes in Mathematics, 1150, Springer-Verlag, Berlin, 1985.CrossRefMATHGoogle Scholar
  37. [37]
    Kesavan, S., Symmetrization and applications, Series in Analysis, Vol. 3, World Scientific Publishing, Hackensack, NJ,2006.CrossRefMATHGoogle Scholar
  38. [38]
    Krahn, E., Uber eine von Rayleigh formulierte Minmaleigenschaft des Kreises, Math. Ann., 94, 1925, 97–100.MathSciNetCrossRefMATHGoogle Scholar
  39. [39]
    Landkof, N. S., Foundations of modern potential theory, Die Grundlehren der Mathematischen Wissenschaften, Band 180, Springer-Verlag, New York, Heidelberg, 1972.Google Scholar
  40. [40]
    Lions, J.-L. and Magenes, E., Non-homogeneous boundary value problems and applications. Vol. I, GMW 181, Springer-Verlag, New York, Heidelberg, 1972.CrossRefMATHGoogle Scholar
  41. [41]
    Luttinger, J. M., Generalized isoperimetric inequalities, Proc. Nat. Acad. Sci. U.S.A., 70, 1973, 1005–1006.MathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    Maz’ja, V. G., Weak solutions of the Dirichlet and Neumann problems (in Russian), Trudy Moskov. Mat. Obsuc., 20, 1969, 137–172.Google Scholar
  43. [43]
    Mossino, J. and Rakotoson, J.-M., Isoperimetric inequalities in parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13(4), 1986, 51–73.MathSciNetMATHGoogle Scholar
  44. [44]
    Musina, R. and Nazarov, A. I., On fractional Laplacians, Comm. Part. Diff. Eqs., 39(9), 2014, 1780–1790.MathSciNetCrossRefMATHGoogle Scholar
  45. [45]
    Park, Y. J., Fractional Pólya-Szegö inequality, J Chungcheong Math. Soc., 24(2), 2011, 267–271.Google Scholar
  46. [46]
    Petrowskii, I., Sur l’analyticité des solutions des systèmes d’équations différentielles, Mat. Sbornik (N.S.), 5(47), 1939, 3–70.MathSciNetGoogle Scholar
  47. [47]
    Polya, G. and Szegö, C., Isoperimetric inequalities in mathematical physics, Annals of Mathematics Studies, Vol. 27, Princeton University Press, Princeton, N.J., 1951.MATHGoogle Scholar
  48. [48]
    Servadei, R. and Valdinoci, E., Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33(5), 2013, 2105–2137.MathSciNetMATHGoogle Scholar
  49. [49]
    Servadei, R. and Valdinoci, E., On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144(4), 2014, 831–855.MathSciNetCrossRefMATHGoogle Scholar
  50. [50]
    Servadei, R. and Valdinoci, E., Weak and viscosity solutions of the fractional Laplace equation, Publ. Mat., 58(1), 2014, 133–154.MathSciNetCrossRefMATHGoogle Scholar
  51. [51]
    Silvestre, L. E., Hölder estimates for solutions of integro differential equations like the fractional Laplace, Indiana Univ. Math. J., 55(3), 2006, 1155–1174.MathSciNetCrossRefMATHGoogle Scholar
  52. [52]
    Silvestre, L. E., Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60(1), 2007, 67–112.MathSciNetCrossRefMATHGoogle Scholar
  53. [53]
    Stein, E. M., Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.Google Scholar
  54. [54]
    Talenti, G., Elliptic equations and rearrangements, Ann. Scuola Norm. Sup., 3(4), 1976, 697–718.MathSciNetMATHGoogle Scholar
  55. [55]
    Talenti, G., Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces, Annal. Mat. Pura Appl., 4, 120, 1979, 159–184.MathSciNetCrossRefMATHGoogle Scholar
  56. [56]
    Talenti, G., Linear elliptic P.D.E.’s: Level sets, rearrangements and a priori estimates of solutions, Boll. Un. Mat. Ital. B, 4(6), 1985, 917–949.MATHGoogle Scholar
  57. [57]
    Valdinoci, E., From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49, 2009, 33–44.MathSciNetMATHGoogle Scholar
  58. [58]
    Vázquez, J. L., Symétrisation pour ut = (u) et applications, C. R. Acad. Sc. Paris, 295, 1982, 71–74.MATHGoogle Scholar
  59. [59]
    Vázquez, J. L., Symmetrization and mass comparison for degenerate nonlinear parabolic and related elliptic equations, Advances in Nonlinear Studies, 5, 2005, 87–131.MathSciNetCrossRefMATHGoogle Scholar
  60. [60]
    Vázquez, J. L., The porous medium equation. Mathematical Theory, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.Google Scholar
  61. [61]
    Vázquez, J. L., Smoothing and decay estimates for nonlinear diffusion equations: Equations of porous medium type, Oxford Lecture Series in Mathematics and Its Applications, 33, Oxford University Press, Oxford, 2006.CrossRefGoogle Scholar
  62. [62]
    Vázquez, J. L., Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations: The Abel Symposium 2010, Holden, H. and Karlsen, K. H. (eds.), Springer-Verlag, New York, 2012, 271–298.Google Scholar
  63. [63]
    Vázquez, J. L., Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type, J. Eur. Math. Soc., 16, 2014, 769–803.MathSciNetCrossRefMATHGoogle Scholar
  64. [64]
    Vázquez, J. L. and Volzone, B., Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type, J. Math. Pures Appl., 101, 2014, 553–582.MathSciNetCrossRefMATHGoogle Scholar
  65. [65]
    Vázquez, J. L. and Volzone, B., Optimal estimates for fractional fast diffusion equations, J. Math. Pures Appl. (9), 103(2), 2015, 535–556.MathSciNetCrossRefMATHGoogle Scholar
  66. [66]
    Weinberger, H., Symmetrization in uniformly elliptic problems, Studies in Mathematical Analysis, Stanford University Press, California, 1962, 424–428.Google Scholar

Copyright information

© Fudan University and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • Yannick Sire
    • 1
  • Juan Luis Vázquez
    • 2
  • Bruno Volzone
    • 3
  1. 1.Johns Hopkins UniversityBaltimoreUSA
  2. 2.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain
  3. 3.Dipartimento di IngegneriaUniversit degli Studi di Napoli “Parthenope”NapoliItalia

Personalised recommendations