Potential Analysis

, Volume 42, Issue 2, pp 499–547 | Cite as

The Fractional Relative Capacity and the Fractional Laplacian with Neumann and Robin Boundary Conditions on Open Sets

  • Mahamadi WarmaEmail author


Let \(\Omega \subset \mathbb {R}^{N}\) be an arbitrary open set with boundary Ω. Let \(p\in [1,\infty )\) and s∈(0,1). In the first part of the article we give some useful properties of the fractional order Sobolev spaces. We define a relative (s,p)-capacity on \(\overline {\Omega }\) with the fractional order Sobolev spaces, give its properties and its connection with the classical Bessel (s,p)-capacity and the Hausdorff measure. We also use the relative capacity to characterize completely the zero trace fractional order Sobolev spaces. In the second part of the article, we clarify the Neumann and Robin boundary conditions associated with the fractional Laplace operator on open subsets of \(\mathbb {R}^{N}\). Contrary to the classical Laplace operator, it turns out that Dirichlet, Neumann and Robin boundary conditions may coincide for the fractional Laplacian on bounded domains. In the last part of the article we consider some nonlocal elliptic problems associated with the fractional Laplacian with Neumann and Robin type boundary conditions. We show some existence and regularity results of weak solutions on non smooth domains.


Fractional order Sobolev spaces Arbitrary domains Capacity Fractional Laplacian Neumann and Robin boundary conditions Elliptic boundary value problems Existence and regularity of weak solutions Semigroup Ultracontractivity 

Mathematics Subject Classifications (2010)

31A15 35J25 35B65 46E39 34B10 


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  1. 1.
    Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften, vol. 314. Springer-Verlag, Berlin (1996)Google Scholar
  2. 2.
    Adams, R.A.: Sobolev Spaces. Pure and Applied Mathematics, vol. 65. Academic Press, New York (1975)Google Scholar
  3. 3.
    Adler, R. J., Feldman, R.E., Taqqu, M.S. (eds.): A Practical Guide to Heavy Tails. Statistical Techniques and Applications. Papers from the workshop held in Santa Barbara, CA, December 1995. Birkhäuser (1998)Google Scholar
  4. 4.
    Alt, H.W.: Linear Functional Analysis. An application Oriented Introduction. Springer-Verlag, Berlin (1996)Google Scholar
  5. 5.
    Applebaum, D.: Lévy Processes and Stochastic Calculus. Cambridge Studies in Advanced Mathematics, vol. 93 (2004)Google Scholar
  6. 6.
    Arendt, W., Warma, M.: The Laplacian with Robin boundary conditions on arbitrary domains. Potential Anal. 19, 341–363 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Arendt, W., Warma, M.: Dirichlet and Neumann boundary conditions: What is in between. J. Evol. Equ. 3, 119–135 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Biegert, M.: Elliptic Problems on Varying Domains. Dissertation, Logos Verlag, Berlin (2005)zbMATHGoogle Scholar
  9. 9.
    Biegert, M., Warma, M.: The heat equation with nonlinear generalized Robin boundary conditions. J. Differ. Equat. 247, 1949–1979 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Biegert, M., Warma, M.: Some Quasi-linear elliptic equations with inhomogeneous generalized Robin boundary conditions on “bad” domains. Adv. Differ. Equa. 15, 893–924 (2010)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Bjorland, C., Caffarelli, L., Figalli, A.: Nonlocal tug-of-war and the infinity fractional Laplacian. Comm. Pure Appl. Math. 65, 337–380 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Bliedtner, J., Hansen, W.: Potential Theory. An Analytic and Probabilistic Approach to Balayage. Universitext. Springer-Verlag, Berlin (1986)Google Scholar
  13. 13.
    Bogdan, K., Burdzy, K., Chen, Z.Q.: Censored stable processes. Probab. Theory Relat. Field 127, 89–152 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Bogdan, K., Byczkowski, T.: Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains. Studia Math. 133, 53–92 (1999)zbMATHMathSciNetGoogle Scholar
  15. 15.
    Caetano, A.: Approximation by functions of compact support in Besov-Triebel-Lizorkin spaces on irregular domains. Studia Math. 142, 47–63 (2000)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Caffarelli, L., Roquejoffre, J-M., Sire, Y.: Variational problems for free boundaries for the fractional Laplacian. J. Eur. Math. Soc 12, 1151–1179 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Caffarelli, L., Salsa, S., Silvestre, L.: Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian. Invent. Math. 171, 425–461 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Partial Differ. Equa. 32, 1245–1260 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Chen, Z-Q., Kumagai, T.: Heat kernel estimates for stable-like processes on d-sets. Stoch. Process Appl. 108, 27–62 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Chill, R., Warma, M.: Dirichlet and Neumann boundary conditions for the p-Laplace operator: What is in between. Proc. Roy. Soc. Edinburgh Sect. A 142, 975–1002 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Choquet, G.: Theory of capacities. Ann. Inst. Fourier 5, 131–295 (1954)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 23.
    Daners, D.: Robin boundary value problems on arbitrary domains. Trans. Amer. Math. Soc. 352, 4207–4236 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 24.
    Daners, D., Drábek, P.: A priori estimates for a class of quasi-linear elliptic equations. Trans. Amer. Math. Soc. 361, 6475–6500 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 25.
    Danielli, D., Garofalo, N., Nhieu, D-M.: Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Carathéodory spaces. Mem. Amer. Math. Soc., 182 (2006)Google Scholar
  25. 26.
    Davies, E. B.: Heat Kernels and Spectral Theory. Cambridge University Press, Cambridge (1989)CrossRefzbMATHGoogle Scholar
  26. 27.
    D’Elia, M., Gunzburger, M.: The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator. Comput. Math. Appl. 66, 1245–1260 (2013)CrossRefMathSciNetGoogle Scholar
  27. 28.
    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, 493–540 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 29.
    Du, Q., Gunzburger, M., Lehoucq, R.B., Zhou, K.: Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54, 667–696 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 30.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math 136, 521–573 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 31.
    Doob, J.L.: Classical Potential Theory and its Probabilistic Counterpart. Classics in Mathematics. Springer-Berlin (1984)Google Scholar
  31. 32.
    Farkas, W., Jacob, N.: Sobolev spaces on non-smooth domains and Dirichlet forms related to subordinate reflecting diffusions. Math. Nachr. 224, 75–104 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 33.
    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Second revised and extended edition. De Gruyter Studies in Mathematics, vol. 19. Berlin (2011)Google Scholar
  33. 34.
    Fukushima, M., Uemura, T.: On Sobolev and capacitary inequalities for contractive Besov spaces over d-sets. Potential Anal. 18, 59–77 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 35.
    Guan, Q.Y.: Integration by parts formula for regional fractional Laplacian. Comm. Math. Phys. 266, 289–329 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 36.
    Guan, Q.Y., Ma, Z.M.: Reflected symmetric α-stable processes and regional fractional Laplacian. Probab. Theory Relat. Fields 134, 649–694 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 37.
    Guan, Q.Y., Ma, Z.M.: Boundary problems for fractional Laplacians. Stoch. Dyn. 5, 385–424 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 38.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin (1983)CrossRefzbMATHGoogle Scholar
  38. 39.
    Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, vol. 24. Pitman, Boston (1985)Google Scholar
  39. 40.
    Gunzburger, M., Lehoucq, R.B.: A nonlocal vector calculus with application to nonlocal boundary value problems. Multiscale Model Simul. 8, 1581–1598 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 41.
    Hoh, W., Jacob, J.: On the Dirichlet problem for pseudodifferential operators generating Feller semigroups. J. Funct. Anal. 137, 19–48 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 42.
    Jacob, N., Knopova, V.: Fractional derivatives and fractional powers as tools in understanding Wentzell boundary value problems for pseudo-differential operators generating Markov processes. Fract. Calc. Appl. Anal. 8, 91–112 (2005)zbMATHMathSciNetGoogle Scholar
  42. 43.
    Jones, P.W.: Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 147, 71–88 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  43. 44.
    Jonsson, A., Wallin, H.: Function spaces on subsets of \(\mathbb R^{N}\). Math. Rep, 2 (1984)Google Scholar
  44. 45.
    Lions, J.-L.: Non-homogeneous Boundary Value Problems and Applications, vol. I. Springer-Verlag, New York-Heidelberg (1972)CrossRefzbMATHGoogle Scholar
  45. 46.
    Lions, J.-L., Magenes, E: Non-homogeneous Boundary Value Problems and Applications, vol. II. Springer-Verlag, New York-Heidelberg (1972)CrossRefzbMATHGoogle Scholar
  46. 47.
    Lions, J.-L., Magenes, E: Non-homogeneous Boundary Value Problems and Applications, vol. III. Springer-Verlag, New York-Heidelberg (1973)CrossRefzbMATHGoogle Scholar
  47. 48.
    Maz’ya, V.G.: Sobolev Spaces. Springer-Verlag, Berlin (1985)Google Scholar
  48. 49.
    Maz’ya, V.G, Poborchi, SV: Differentiable Functions on Bad Domains. World Scientific Publishing (1997)Google Scholar
  49. 50.
    Murthy, M.K.V., Stampacchia, G.: Boundary value problems for some degenerate-elliptic operators. Ann. Mat. Pura Appl. 80, 1–122 (1968)CrossRefzbMATHMathSciNetGoogle Scholar
  50. 51.
    Schertzer, D., Larcheveque, M., Duan, J., Yanovsky, V.V., Lovejoy, S.: Fractional Fokker-Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Lévy stable noises. J. Math. Phys. 42, 200–212 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  51. 52.
    Ros-Oton, X., Serra, J.: Fractional Laplacian: Pohozaev identity and nonexistence results. C. R. Math. Acad. Sci. Paris 350, 505–508 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  52. 53.
    Ros-Oton, X., Serra, J.: The Pohozaev identity for the fractional Laplacian. Arch. Rational Mech. Anal 213, 587–628 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  53. 54.
    Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. 101(9), 275–302 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  54. 55.
    Ros-Oton, X., Serra, J.: The extremal solution for the fractional Laplacian. Calc. Var. Partial Differ. Equ. 50, 723–750 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  55. 56.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series. Princeton University Press (1970)Google Scholar
  56. 57.
    Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, 2nd edn. Johann Ambrosius Barth (1995)Google Scholar
  57. 58.
    Velez-Santiago, A., Warma, M.: A class of quasi-linear parabolic and elliptic equations with nonlocal Robin boundary conditions. J. Math. Anal. Appl. 372, 120–139 (2010)CrossRefMathSciNetGoogle Scholar
  58. 59.
    Warma, M.: The p-Laplace operator with the nonlocal Robin boundary conditions on arbitrary open sets. Ann. Mat. Pura Appl. 193(4), 203–235 (2014)CrossRefzbMATHMathSciNetGoogle Scholar

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© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Faculty of Natural Sciences, Department of MathematicsUniversity of Puerto RicoSan JuanUSA

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