Journal of Elliptic and Parabolic Equations

, Volume 4, Issue 1, pp 223–269 | Cite as

On a fractional (sp)-Dirichlet-to-Neumann operator on bounded Lipschitz domains

  • Mahamadi Warma


Let \(p\in (1,\infty )\) and \(\Omega \subset \mathbb {R}^{N}\) a bounded open set with Lipschitz continuous boundary \(\partial \Omega \). We define a fractional p-Dirichlet-to-Neumann operator associated with the regional fractional p-Laplace operator \((-\Delta )_{p,\Omega }^{s}\), \(0<s<1\), and prove that it generates a strongly continuous semigroup on \(L^{2}(\partial \Omega )\) which is order preserving and non-expansive on \(L^\infty (\partial \Omega )\). We show the convergence as time goes to \(\infty \) of all the trajectories of the semigroup. Some results of existence, regularity and fine a priori estimates of solutions to elliptic and parabolic problems associated with the fractional p-Dirichlet-to-Neumann operator are also obtained.


Fractional p-Laplacian Fractional p-Dirichlet-to-Neumann operator Nonlinear submarkovian semigroup Ultracontractivity of (nonlinear) semigroups Nonlocal quasi-linear elliptic problems Regularity of weak solutions 

Mathematics Subject Classification

35R11 35B65 35K65 47H20 35B40 



We thank the referee for her/his careful reading of the first version of the manuscript and for her/his useful comments that have helped to improve the paper. The work of the author is partially supported by the Air Force Office of Scientific Research under the Award No: FA9550-15-1-0027.


  1. 1.
    Antil, H., Warma, M.: Optimal control of the coefficient for fractional and regional fractional p-Laplace equations: approximation and convergence. arXiv preprintarXiv:1612.08201,2016Google Scholar
  2. 2.
    Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems, Volume 96 of Monographs in Mathematics, 2nd ed. Birkhäuser/Springer Basel AG, Basel (2011)Google Scholar
  3. 3.
    Arendt, W., ter Elst, A.F.M.: The Dirichlet-to-Neumann operator on rough domains. J. Differ. Equ. 251(8), 2100–2124 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arendt, W., ter Elst, A.F.M., Kennedy, J.B., Sauter, M.: The Dirichlet-to-Neumann operator via hidden compactness. J. Funct. Anal. 266(3), 1757–1786 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Arendt, W., Ter Elst, A.F.M., Warma, M.: Fractional powers of sectorial operators via the Drichlet-to-Neumann operator. Commun. Part. Differ. Equ. (2017). Google Scholar
  6. 6.
    Attouch, H., Buttazzo, G., Michaille, G.: Variational Analysis in Sobolev and BV Spaces: Applications to PDEs and Optimization, volume 17 of MOS-SIAM Series on Optimization, 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA; Mathematical Optimization Society, Philadelphia, PA (2014)Google Scholar
  7. 7.
    Behrndt, J., ter Elst, A.F.M.: Dirichlet-to-Neumann maps on bounded Lipschitz domains. J. Differ. Equ. 259(11), 5903–5926 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bogdan, K., Burdzy, K., Chen, Z.-Q.: Censored stable processes. Probab. Theory Relat. Fields 127(1), 89–152 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bonforte, M., Grillo, G.: Ultracontractive bounds for nonlinear evolution equations governed by the subcritical p-Laplacian. In: Trends in partial differential equations of mathematical physics, volume 61 of Progr. Nonlinear Differential Equations Appl., pp. 15–26. Birkhäuser, Basel (2005) Corrected version available electronically from the publisherGoogle Scholar
  10. 10.
    Brander, T., Kar, M., Salo, M.: Enclosure method for the p-Laplace equation. Inverse Probl. 31(4), 045001 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bucur, C., Valdinoci, E.: Nonlocal diffusion and applications, volume 20 of Lecture Notes of the Unione Matematica Italiana. Springer, Cham; Unione Matematica Italiana, Bologna (2016)Google Scholar
  12. 12.
    Caffarelli, L.A., Roquejoffre, J.-M., Sire, Y.: Variational problems for free boundaries for the fractional Laplacian. J. Eur. Math. Soc. 12(5), 1151–1179 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Caffarelli, L.A., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Part. Differ. Equ. 32(7–9), 1245–1260 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cipriani, F., Grillo, G.: Uniform bounds for solutions to quasilinear parabolic equations. J. Differ. Equ. 177(1), 209–234 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cipriani, F., Grillo, G.: Nonlinear Markov semigroups, nonlinear Dirichlet forms and applications to minimal surfaces. J. Reine Angew. Math. 562, 201–235 (2003)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Daners, D.: Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator. Positivity 18(2), 235–256 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Di Castro, A., Kuusi, T., Palatucci, G.: Nonlocal Harnack inequalities. J. Funct. Anal. 267(6), 1807–1836 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Di Castro, A., Kuusi, T., Palatucci, G.: Local behavior of fractional \(p\)-minimizers. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(5), 1279–1299 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Dipierro, S., Ros-Oton, X., Valdinoci, E.: Nonlocal problems with Neumann boundary conditions. Rev. Mat. Iberoam. 33(2), 377–416 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Drábek, P., Milota, J.: Methods of Nonlinear Analysis: Applications to Differential Equations. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser/Springer Basel AG, Basel, 2nd edn, (2013)Google Scholar
  22. 22.
    Gal, C.G., Warma, M.: Bounded solutions for nonlocal boundary value problems on Lipschitz manifolds with boundary. Adv. Nonlinear Stud. 16(3), 529–550 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Gal, C.G., Warma, M.: Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces. Commun. Part. Differ. Equ. 42(4), 579–625 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gal, C.G., Warma, M.: On some degenerate non-local parabolic equation associated with the fractional \(p\)-Laplacian. Dyn. Part. Differ. Equ. 14(1), 47–77 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Ghosh, T., Salo, M., Uhlmann, G.: The Calderón problem for the fractional Schrödinger equation. arXiv:1609.09248
  26. 26.
    Gorenflo, R., Mainardi, F.: Random walk models approximating symmetric space-fractional diffusion processes. In: Problems and Methods in Mathematical Physics (Chemnitz, 1999), Volume 121 of Operotor Theory Advance Application, pp. 120–145. Birkhäuser, Basel (2001)Google Scholar
  27. 27.
    Grisvard, P.: Elliptic problems in nonsmooth domains, volume 69 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA: Reprint of the 1985 original. With a foreword by Susanne C, Brenner (2011)Google Scholar
  28. 28.
    Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math. 97(4), 1061–1083 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Grubb, G.: Fractional Laplacians on domains, a development of Hörmander’s theory of \(\mu \)-transmission pseudodifferential operators. Adv. Math. 268, 478–528 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Guan, Q.-Y.: Integration by parts formula for regional fractional Laplacian. Commun. Math. Phys. 266(2), 289–329 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Guan, Q.-Y., Ma, Z.-M.: Boundary problems for fractional Laplacians. Stoch. Dyn. 5(3), 385–424 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Guan, Q.-Y., Ma, Z.-M.: Reflected symmetric \(\alpha \)-stable processes and regional fractional Laplacian. Probab. Theory Relat. Fields 134(4), 649–694 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Hauer, D.: The \(p\)-Dirichlet-to-Neumann operator with applications to elliptic and parabolic problems. J. Differ. Equ. 259(8), 3615–3655 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Hoh, W., Jacob, N.: On the Dirichlet problem for pseudodifferential operators generating Feller semigroups. J. Funct. Anal. 137(1), 19–48 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Iannizzotto, A., Mosconi, S., Squassina, M.: Global Hölder regularity for the fractional \(p\)-Laplacian. Rev. Mat. Iberoam. 32(4), 1353–1392 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Jonsson, A., Wallin, H.: Function spaces on subsets of R\(^{n}\). Math. Rep. 2(1), 221 (1984)zbMATHGoogle Scholar
  37. 37.
    Kuusi, T., Mingione, G., Sire, Y.: Nonlocal equations with measure data. Commun. Math. Phys. 337(3), 1317–1368 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Minty, G.J.: Monotone (nonlinear) operators in Hilbert space. Duke Math. J. 29, 341–346 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Muralidhar, R., Ramkrishna, D., Nakanishi, H.H., Jacobs, D.: Anomalous diffusion: a dynamic perspective. Phys. A Stat. Mech. Appl. 167, 539–559 (1990)CrossRefGoogle Scholar
  40. 40.
    Murthy, M.K.V., Stampacchia, G.: Boundary value problems for some degenerate-elliptic operators. Ann. Mat. Pura Appl. 4(80), 1–122 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. 101(3), 275–302 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Ros-Oton, X., Serra, J.: The Pohozaev identity for the fractional Laplacian. Arch. Ration. Mech. Anal. 213(2), 587–628 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Servadei, R., Valdinoci, E.: On the spectrum of two different fractional operators. Proc. R. Soc. Edinb. Sect. A 144(4), 831–855 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Sire, Y., Valdinoci, E.: Rigidity results for some boundary quasilinear phase transitions. Commun. Part. Differ. Equ. 34(7–9), 765–784 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Stinga, P.R., Torrea, J.L.: Extension problem and Harnack’s inequality for some fractional operators. Commun. Part. Differ. Equ. 35(11), 2092–2122 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Umarov, S., Gorenflo, R.: On multi-dimensional random walk models approximating symmetric space-fractional diffusion processes. Fract. Calc. Appl. Anal. 8(1), 73–88 (2005)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Valdinoci, E.: From the long jump random walk to the fractional Laplacian. Bol. Soc. Esp. Mat. Apl. SeMA 49, 33–44 (2009)MathSciNetzbMATHGoogle Scholar
  48. 48.
    Vázquez, J.-L.: The Dirichlet problem for the fractional \(p\)-Laplacian evolution equation. J. Differ. Equ. 260(7), 6038–6056 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Warma, M.: A fractional Dirichlet-to-Neumann operator on bounded Lipschitz domains. Commun. Pure Appl. Anal. 14(5), 2043–2067 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Warma, M.: The fractional relative capacity and the fractional Laplacian with Neumann and Robin boundary conditions on open sets. Potential Anal. 42(2), 499–547 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Warma, M.: The fractional Neumann and Robin type boundary conditions for the regional fractional \(p\)-Laplacian. NoDEA Nonlinear Differ. Equ. Appl. 23(1), 1–46 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Warma, M.: Local Lipschitz continuity of the inverse of the fractional \(p\)-Laplacian, Hölder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains. Nonlinear Anal. 135, 129–157 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Orthogonal Publishing and Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics (Rio Piedras Campus), Faculty of Natural SciencesUniversity of Puerto RicoSan JuanUSA

Personalised recommendations