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Advanced Nonlinear Studies

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Volume 17, Issue 2 (May 2017)

Issues

Local Elliptic Regularity for the Dirichlet Fractional Laplacian

Umberto Biccari
  • DeustoTech, University of Deusto, 48007 Bilbao, Basque Country; and Facultad de Ingeniería, Universidad de Deusto, Avda Universidades 24, 48007 Bilbao, Basque Country, Spain
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/ Mahamadi Warma
  • Department of Mathematics, College of Natural Sciences, University of Puerto Rico (Rio Piedras Campus), PO Box 70377, San Juan, PR 00936-8377, USA
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/ Enrique Zuazua
  • Corresponding author
  • DeustoTech, University of Deusto, 48007 Bilbao, Basque Country; and Facultad de Ingeniería, Universidad de Deusto, Avda Universidades 24, 48007 Bilbao, Basque Country; and Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain
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Published Online: 2017-04-21 | DOI: https://doi.org/10.1515/ans-2017-0014

Abstract

We prove the Wloc2s,p local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian on an arbitrary bounded open set of N. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different methods. First working directly in the variational formulation of the elliptic problem and then employing the heat kernel representation of solutions.

Keywords: Fractional Laplacian; Dirichlet Boundary Condition; Weak Solutions; Local Regularity

MSC 2010: 35B65; 35R11; 35S05

Dedicated to Ireneo Peral on the occasion of his 70th birthday: Gracias Ireneo por tantos años de amistad y ejemplo

References

  • [1]

    Adams D. R. and Hedberg L. I., Function Spaces and Potential Theory, Grundlehren Math. Wiss. 314, Springer, Berlin, 1996. Google Scholar

  • [2]

    Bakunin O. G., Turbulence and Diffusion: Scaling Versus Equations, Springer, Berlin, 2008. Google Scholar

  • [3]

    Bernard C., Regularity of solutions to the fractional Laplace equation, preprint 2014, http://math.uchicago.edu/~may/REU2014/REUPapers/Bernard.pdf.

  • [4]

    Biccari U., Internal control for non-local Schrödinger and wave equations involving the fractional Laplace operator, preprint 2017, https://arxiv.org/abs/1411.7800v2.

  • [5]

    Bologna M., Tsallis C. and Grigolini P., Anomalous diffusion associated with non-linear fractional derivative Fokker–Planck-like equation: Exact time-dependent solutions, Phys. Rev. E 62 (2000), 2213–2218. Google Scholar

  • [6]

    Cozzi M., Interior regularity of solutions of non-local equations in Sobolev and Nikol’skii spaces, Ann. Mat. Pura Appl. (2) 196 (2017), 555–578. Google Scholar

  • [7]

    Di Nezza E., Palatucci G. and Valdinoci E., Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573. Google Scholar

  • [8]

    Dipierro S., Palatucci G. and Valdinoci E., Dislocation dynamics in crystals: A macroscopic theory in a fractional Laplace setting, Comm. Math. Phys. 333 (2015), no. 2, 1061–1105. Google Scholar

  • [9]

    Duoandikoetxea J. and Zuazua E., Moments, masses de Dirac et décomposition de fonctions, C. R. Acad. Sci. Paris Sér. 1 315 (1992), no. 6, 693–698. Google Scholar

  • [10]

    Engel K.-J. and Nagel R., One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math. 194, Springer, New York, 2000. Google Scholar

  • [11]

    Fiscella A., Servadei R. and Valdinoci E., Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math. 40 (2015), 235–253. Google Scholar

  • [12]

    Folland G. B., Real Analysis: Modern Techniques and Their Applications, John Wiley & Sons, New York, 2013. Google Scholar

  • [13]

    Gal C. G. and Warma M., Nonlocal transmission problems with fractional diffusion and boundary conditions on non-smooth interfaces, Comm. Partial Differential Equations (2017), 10.1080/03605302.2017.1295060. Google Scholar

  • [14]

    Gilboa G. and Osher S., Nonlocal operators with applications to image processing, Multiscale Model. Simul. 7 (2008), no. 3, 1005–1028. Google Scholar

  • [15]

    Grisvard P., Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math. 24, Pitman, Boston, 1985. Google Scholar

  • [16]

    Grubb G., Fractional Laplacians on domains, a development of Hörmander’s theory of μ-transmission pseudodifferential operators, Adv. Math. 268 (2015), 478–528. Google Scholar

  • [17]

    Jonsson A. and Wallin H., Function Spaces on Subsets of N, Math. Rep. 2, Harwood Academic Publishers, Reading, 1984. Google Scholar

  • [18]

    Kato T., Strong Lp solutions of the Navier–Stokes equation in m, with applications to weak solutions, Math. Z. 187 (1984), no. 4, 471–480. Google Scholar

  • [19]

    Leonori T., Peral I., Primo A. and Soria F., Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations, Discrete Contin. Dyn. Syst. 35 (2015), 6031–6068. Google Scholar

  • [20]

    Levendorski S., Pricing of the American put under Lévy processes, Int. J. Theor. Appl. Finance 7 (2004), no. 3, 303–335. Google Scholar

  • [21]

    Meerschaert M. M., Fractional calculus, anomalous diffusion, and probability, Fractional Dynamics, World Scientific, Hackensack (2012), 265–284. Google Scholar

  • [22]

    Moser J., A new proof of de Giorgi’s theorem concerning the regularity problem for elliptic differential equations, Comm. Pure Appl. Math. 13 (1960), 457–468. Google Scholar

  • [23]

    Pham H., Optimal stopping, free boundary, and American option in a jump-diffusion model, Appl. Math. Optim. 35 (1997), no. 2, 145–164. Google Scholar

  • [24]

    Ros-Oton X. and Serra J., The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl. 101 (2014), no. 9, 275–302. Google Scholar

  • [25]

    Ros-Oton X. and Serra J., The extremal solution for the fractional Laplacian, Calc. Var. Partial Differential Equations 50 (2014), 723–750. Google Scholar

  • [26]

    Ros-Oton X. and Serra J., The Pohozaev identity for the fractional Laplacian, Arch. Ration. Mech. Anal. 213 (2014), 587–628. Google Scholar

  • [27]

    Ros-Oton X. and Serra J., Boundary regularity for fully nonlinear integro-differential equations, Duke Math. J. 165 (2016), 2079–2154. Google Scholar

  • [28]

    Servadei R. and Valdinoci E., On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), 831–855. Google Scholar

  • [29]

    Stein E., Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970. Google Scholar

  • [30]

    Stinga P. R., Fractional powers of second order partial differential operators: Extension problem and regularity theory, Ph.D. Dissertation, Universidad Autónoma de Madrid, Spain, 2010. Google Scholar

  • [31]

    Tartar L., An Introduction to Sobolev Spaces and Interpolation Spaces, Springer, Berlin, 2007. Google Scholar

  • [32]

    Vázquez J. L., Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations (Oslo 2010), Abel Symp. 7, Springer, Heidelberg (2012), 271–298. Google Scholar

  • [33]

    Zhu T. and Harris J. M., Modeling acoustic wave propagation in heterogeneous attenuating media using decoupled fractional Laplacians, Geophysics 79 (2014), no. 3, T105–T116. Google Scholar

About the article


Received: 2017-02-06

Revised: 2017-04-13

Accepted: 2017-04-19

Published Online: 2017-04-21

Published in Print: 2017-05-01


Funding Source: U.S. Air Force

Award identifier / Grant number: FA9550-15-1-0027

Award identifier / Grant number: FA9550-14-1-0214

Funding Source: Ministerio de Economía y Competitividad

Award identifier / Grant number: MTM2014-52347

Funding Source: Agence Nationale de la Recherche

Award identifier / Grant number: ICON

All authors were supported by the Air Force Office of Scientific Research through award no. FA9550-15-1-0027. Umberto Biccari and Enrique Zuazua were supported by Ministerio de Economía y Competitividad (Spain) through grant MTM2014-52347 and by the European Research Council Executive Agency through Advanced Grant DYCON (Dynamic Control). Enrique Zuazua was supported by EOARD-AFOSR through award no. FA9550-14-1-0214 and by Agence Nationale de la Recherche (France) through ICON.


Citation Information: Advanced Nonlinear Studies, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2017-0014.

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