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Applicable Analysis
An International Journal
Volume 102, 2023 - Issue 1
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Research Article

Embedding and extension results in fractional Musielak–Sobolev spaces

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Pages 195-219 | Received 18 Jan 2021, Accepted 21 Jun 2021, Published online: 03 Jul 2021

References

  • Landkof NS. Foundations of modern potential theory. Translated from Russian, Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences]. Vol. 180, Berlin: Springer- Verlag; 1973.
  • Molica Bisci G, Radulescu V, Servadei R. Variational methods for nonlocal fractional problems. Encyclopedia of mathematics and its applications. Vol. 162, Cambridge University Press; ISBN: 9781107111943. Foreword by J. Mawhin. p. 1–400.
  • Stein EM. Singular integrals and differentiability properties of functions. Princeton (NJ): Princeton University Press; 1970. (Princeton Mathematical Series 30).
  • Alberti G, Bouchitté G, Seppecher P. Phase transition with the line-tension effect. Arch Rational Mech Anal. 1998;144(1):1–46.
  • Biler P, Karch G, Monneau R. Nonlinear diffusion of dislocation density and self-similar solutions. Comm Math Phys. 2010;294(1):145–168.
  • Biler P, Karch G, Woyczynski WA. Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws. Ann Inst H Poincaré Anal Non Linéaire. 2001;18(5):613–637.
  • Cont R, Tankov P. Financial modelling with jump processes. Boca Raton (FL): Chapman and Hall/CRC; 2004. (Chapman and Hall/CRC Financial Mathematics Series).
  • Metzler R, Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys Rep. 2000;339:1–77.
  • Savin O, Valdinoci E. Elliptic PDEs with bered nonlinearities. J Geom Anal. 2009;19(2):420–432.
  • Di. Nezza E, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math. 2012;136(5):521–573. MR 2944369.
  • Calderón AP. Lebesgue spaces of differentiable functions and distributions. Proceedings of Symposia in Pure Mathematics; Vol. 4, American Mathematical Society; 1961. p. 33–49.
  • Diening L, Harjulehto P, Hasto P, et al. Lebesgue and Sobolev Spaces with Variable Exponents. Heidelberg: Springer-Verlag; 2011. (Lecture Notes in Mathematics; 2017).
  • Jones Peter W. Quasiconformal mappings and extendability of functions in Sobolev spaces. Acta Math. 1981;147:71–88.
  • Sobolev SL. Nekotorye Primeneniya Funkcional nogo Analiza v MatematiOceskoj Fizike. Leningrad: Izdat. Leningrad. Gos. Univ.; 1950.
  • Sobolev SL. Applications of functional analysis in mathematical physics. Translated from the Russian by Browder FE, editor. Translations of mathematical monographs. Vol. 7, Providence (RI): American Mathematical Society; 1963.
  • Demengel F, Demengel G. Functional spaces for the theory of elliptic partial differential equations. Springer; 2007. DOI:10.1007/978-1-4471-2807-6.
  • Gagliardo E. Characterizations of the traces on the border related to some classes of functions in n variables. Rend Sem Mat Univ Padova. 1957;27:284–305.
  • Zhou Y. Fractional Sobolev extension and imbedding. Trans Amer Math Soc. 2015;367(2):959–979. DOI:10.1090/s0002-9947-2014-06088-1.
  • Azroul E, Benkirane A, Shimi M, et al. On a class of nonlocal problems in new fractional Musielak-Sobolev spaces. Appl Anal. 2020; DOI:10.1080/00036811.2020.1789601.
  • Leoni G. A first course in Sobolev spaces. Providence, RI: American Mathematical Society; 2009. (Graduate Studies in Mathematics 105).
  • Rajagopal KR, Ruzicka M. Mathematical modeling of electrorheological materials. Contin Mech Thermodyn. 2001;13:59–78.
  • Ruzicka M. Electrorheological fluids: modeling and mathematical theory. Berlin: Springer; 2000. (Lecture Notes in Mathematics).
  • Perona P, Malik J. Scale-space and edge detection using anisotropic diffusion. IEEE Trans Pattern Anal Machine Intell. 1990;12:629–639.
  • Azroul E, Benkirane A, Srati M. Existence of solutions for a nonlocal type problem in fractional Orlicz Sobolev spaces. Adv Oper Theor. 2020; DOI:10.1007/s43036-020-00042-0.
  • Azroul E, Benkirane A, Srati M. Eigenvalue problem associated with nonhomogeneous integro-differential operators. J Elliptic Parabol Equ. 2021; DOI:10.1007/s41808-020-00092-8.
  • Azroul E, Benkirane A, Srati M. Nonlocal eigenvalue type problem in fractional Orlicz-Sobolev space. Adv Oper Theor. 2020; DOI: 10.1007/s43036-020-00067-5.
  • Azroul E, Benkirane A, Srati M. Mountain pass type solutions for a nonlacal fractional a(.)-Kirchhoff type problems. J Nonlinear Funct Anal. 2021;2021:1–18. Article ID 3.
  • Azroul E, Benkirane A, Srati M, et al. Infinitely many solutions for a nonlocal type problem with sign-changing weight function. Electron J Differ Equ. 2021;2021(16):1–15.
  • Bahrouni S, Ounaies H, Tavares L. Basic results of fractional Orlicz-Sobolev space and applications to non-local problems. Topol Meth Nonlinear Anal. 2020;55(2):681–695. DOI:10.12775/TMNA.2019.111.
  • Boumazourh A, Srati M. Leray-Schauder's solution for a nonlocal problem in a fractional Orlicz-Sobolev space. Moroccan J Pure and Appl Anal (MJPAA). 2020;6:42–52. DOI: 10.2478/mjpaa-2020-0004.
  • Bonder JF, Salort AM. Fractional order Orlicz-Soblev spaces. J Funct Anal. 2019; Available From: https://doi.org/10.1016/j.jfa.2019.04.003.
  • Bonder JF, Llanos MP, Salort AM. A Hölder infinity Laplacian obtained as limit of Orlicz fractional Laplacians. arXiv:1807.01669.
  • Azroul E, Benkirane A, Shimi M. Eigenvalue problems involving the fractional p(x)-Laplacian operator. Adv Oper Theor. 2019;4(2):539–555. DOI: 10.15352/aot.1809-1420.
  • Azroul E, Benkirane A, Shimi M. Existence and multiplicity of solutions for fractional p(x,.)-Kirchhoff type problems in RN. Applicable Anal. 2019; DOI:10.1080/00036811.2019.1673373.
  • Azroul E, Benkirane A, Shimi M. General fractional Sobolev space with variable exponent and applications to nonlocal problems. Adv Oper Theor. 2020; Available From: https://doi.org/10.1007/s43036-020-00062-w.
  • Azroul E, Benkirane A, Shimi M. Existence results for fractional p(x,.)-Laplacian problem via the Nehari manifold approach. Appl Math Optim. 2020; DOI: 10.1007/s00245-020-09686-z:
  • Azroul E, Shimi M. Nonlocal eigenvalue problems with variable exponent. Moroccan J Pure and Appl Anal. 2018;4(1):46–61.
  • Azroul E, Benkirane A, Shimi M, et al. On a class of fractional p(x)-Kirchhoff type problems. J Applicable Anal. 2021;100(2):383–402. Available From: https://doi.org/10.1080/00036811.2019.1603372.
  • Bahrouni A, Rădulescu V. On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent. Discrete Contin Dyn Syst. 2018;11:379–389.
  • Kaufmann U, Rossi JD, Vidal R. Fractional Sobolev spaces with variable exponents and fractional p(x)-Laplacians. Elec J Qual Th Diff Equ. 2017;76:1–10.
  • Musielak J. Orlicz Spaces and Modular Spaces. Berlin,: Springer; 1983. (Lecture Notes in Mathematics; 1034).
  • Bonanno G, Molica Bisci G, Radulescu V. Quasilinear elliptic non-homogeneous Dirichlet problems through Orlicz-Sobolev spaces. Nonlinear Anal. 2012;75:4441–4456.
  • Bonanno G, Molica Bisci G, Radulescu V. Arbitrarily small weak solutions for a nonlinear eigenvalue problem in Orlicz-Sobolev spaces. Monatsh Math. 2011;165(3–4):305–318.
  • Bonanno G, Molica Bisci G, Radulescu V. Existence of three solutions for a non-homogeneous Neumann problem through Orlicz-Sobolev spaces. Nonlinear Anal. 2011;74:4785–4795.
  • Mihäilescu M, Rädulescu V. Neumann problems associated to nonhomogeneous differential operators in Orlicz-Soboliv spaces. Ann Inst Fourier. 2008;58(6):2087–2111.
  • Adams RA. Sobolev Spaces. New York (NY): Academic Press; 1975.
  • Benkirane A, Sidi El Vally M. An existence result for nonlinear elliptic equations in Musielak-Orlicz-Sobolev spaces. Bull Belg Math Soc Simon Stevin. 2013;20(1):57–75. DOI: 10.36045/bbms/1366306714.
  • Ambrosetti A, Rabinowitz P. Dual variational methods in critical point theory and applications. J Funct Anal. 1973;14:349–381.
  • Alberico A, Cianchi A, Pick L, et al. Fractional Orlicz-Sobolev embeddings. 15 Jan 2020. arXiv:2001.05565.

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