Annals of Functional Analysis

Density properties for fractional Sobolev spaces with variable exponents

Azeddine Baalal and Mohamed Berghout

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Abstract

In this article we show some density properties of smooth and compactly supported functions in fractional Sobolev spaces with variable exponents. The additional difficulty in this nonlocal setting is caused by the fact that the variable exponent Lebesgue spaces are not translation-invariant.

Article information

Source
Ann. Funct. Anal., Volume 10, Number 3 (2019), 308-324.

Dates
Received: 12 August 2018
Accepted: 5 November 2018
First available in Project Euclid: 6 August 2019

Permanent link to this document
https://projecteuclid.org/euclid.afa/1565078417

Digital Object Identifier
doi:10.1215/20088752-2018-0031

Mathematical Reviews number (MathSciNet)
MR3989177

Zentralblatt MATH identifier
07089119

Subjects
Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems
Secondary: 35A15: Variational methods

Keywords
fractional Sobolev spaces with variable exponents density properties

Citation

Baalal, Azeddine; Berghout, Mohamed. Density properties for fractional Sobolev spaces with variable exponents. Ann. Funct. Anal. 10 (2019), no. 3, 308--324. doi:10.1215/20088752-2018-0031. https://projecteuclid.org/euclid.afa/1565078417


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References

  • [1] A. Baalal and M. Berghout, Traces and fractional Sobolev extension domains with variable exponent, Int. J. Math. Anal. (N.S.) 12 (2018), no. 2, 85–98, available online at http://www.m-hikari.com/ijma/ijma-2018/ijma-1-4-2018/p/berghoutIJMA1-4-2018.pdf.
  • [2] A. Bahrouni and V. D. Rădulescu, On a new fractional Sobolev space and applications to nonlocal variational problems with variable exponent, Discrete Contin. Dyn. Syst. Ser. S 11 (2018), no. 3, 379–389.
  • [3] H. Brezis, Analyse fonctionnelle: Théorie et applications, Masson, Paris, 1983.
  • [4] L. Del Pezzo and J. D. Rossi, Traces for fractional Sobolev spaces with variable exponents, Adv. Oper. Theory 2 (2017), no. 4, 435–446.
  • [5] L. Diening, Maximal function on generalized Lebesgue spaces $L^{p(\cdot)}$, Math. Inequal. Appl. 7 (2004), no. 2, 245–253.
  • [6] L. Diening, P. Harjulehto, P. Hästö, and M. Růžička, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Math. 2017, Springer, Heidelberg, 2011.
  • [7] S. Dipierro and E. Valdinoci, A density property for fractional weighted Sobolev spaces, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26 (2015), no. 4, 397–422.
  • [8] D. E. Edmunds and J. Rákosník, Density of smooth functions in $W^{k,p(x)}(\Omega)$, Proc. Roy. Soc. London Ser. A 437 (1992), no. 1899, 229–236.
  • [9] X. Fan, S. Wang, and D. Zhao, Density of $C^{\infty }(\Omega)$ in $W^{1,p(x)}(\Omega)$ with discontinuous exponent $p(x)$, Math. Nachr. 279 (2006), no. 1–2, 142–149.
  • [10] X. Fan and D. Zhao, On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$, J. Math. Anal. Appl. 263 (2001), no. 2, 424–446.
  • [11] A. Fiscella, R. Servadei, and E. Valdinoci, Density properties for fractional Sobolev spaces, Ann. Acad. Sci. Fenn. Math. 40 (2015), no. 1, 235–253.
  • [12] G. B. Folland, Real Analysis: Modern Techniques and Their Applications, Pure Appl. Math., Wiley, New York, 1984.
  • [13] P. Harjulehto, P. Hästö, Ú. V. Lê, and M. Nuortio, Overview of differential equations with non-standard growth, Nonlinear Anal. 72 (2010), no. 12, 4551–4574.
  • [14] P. A. Hästö, “Counter-examples of regularity in variable exponent Sobolev spaces” in The $p$-harmonic Equation and Recent Advances in Analysis (Manhattan, 2003), Contemp. Math. 370, Amer. Math. Soc., Providence, 2005, 133–143.
  • [15] P. A. Hästö, On the density of continuous functions in variable exponent Sobolev space, Rev. Mat. Iberoam. 23 (2007), no. 1, 213–234.
  • [16] U. Kaufmann, J. D. Rossi, and R. Vidal, Fractional Sobolev spaces with variable exponents and fractional $p(x)$-Laplacians, Electron. J. Qual. Theory Differ. Equ. 2017, no. 76.
  • [17] V. Kokilashvili, A. Meskhi, H. Rafeiro, and S. Samko, Integral Operators in Non-Standard Function Spaces, Vol. 1: Variable Exponent Lebesgue and Amalgam Spaces, Oper. Theory Adv. Appl. 248, Birkhaüser/Springer, Cham, 2016.
  • [18] V. Kokilashvili, A. Meskhi, H. Rafeiro, and S. Samko, Integral Operators in Non-Standard Function Spaces, Vol. 2: Variable Exponent Hölder, Morrey-Campanato and Grand Spaces, Oper. Theory Adv. Appl. 249, Birkhaüser/Springer, Cham, 2016.
  • [19] H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen, Tokyo, 1950.
  • [20] W. Orlicz, Über konjugierte Exponentenfolgen, Studia Math. 3 (1931), 200–211.
  • [21] S. Samko, Density of $C^{\infty }_{0}(\mathbb{R}^{n})$ in the generalized Sobolev spaces $W^{m,p(x)}(\mathbb{R}^{n})$ (in Russian), Dokl. Ross. Akad. Nauk 369 (1999), no. 4, 451–454; English translation in Dokl. Math. 60 (1999), no. 3, 382–385.
  • [22] A. Scapellato, Homogeneous Herz spaces with variable exponents and regularity results, Electron. J. Qual. Theory Differ. Equ. 2018, no. 82.
  • [23] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory (in Russian), Izv. Ross. Akad. Nauk Ser. Mat. 50 (1986), no. 4, 675–710; English translation in Izv. Math. 29 (1987), no. 1, 33–66.
  • [24] V. V. Zhikov, On Lavrentiev’s phenomenon, Russ. J. Math. Phys. 3 (1995), no. 2, 249–269.
  • [25] V. V. Zhikov, Meyer-type estimates for solving the nonlinear Stokes system (in Russian), Differ. Uravn. 33 (1997), no. 1, 107–114; English translation in Differ. Equ. 33 (1997), no. 1, 108–115.
  • [26] V. V. Zhikov, On some variational problems, Russ. J. Math. Phys. 5 (1997), no. 1, 105–116.
  • [27] V. V. Zhikov, On the density of smooth functions in Sobolev-Orlicz spaces (in Russian), Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 310 (2004), 67–81; English translation in J. Math. Sci. (N.Y.) 132 (2006), no. 3, 285–294.