Opuscula Math. 40, no. 1 (2020), 5-20
https://doi.org/10.7494/OpMath.2020.40.1.5

 
Opuscula Mathematica

On some convergence results for fractional periodic Sobolev spaces

Vincenzo Ambrosio

Abstract. In this note we extend the well-known limiting formulas due to Bourgain-Brezis-Mironescu and Maz'ya-Shaposhnikova, to the setting of fractional Sobolev spaces on the torus. We also give a \(\Gamma\)-convergence result in the spirit of Ponce. The main theorems are obtained by using the nice structure of Fourier series.

Keywords: fractional periodic Sobolev spaces, Fourier series, \(\Gamma\)-convergence.

Mathematics Subject Classification: 42B05, 46E35, 49J45.

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  1. M.S. Agranovich, Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains, Springer Monographs in Mathematics, Springer, Cham, 2015.
  2. V. Ambrosio, Periodic solutions for a pseudo-relativistic Schrödinger equation, Nonlinear Anal. 120 (2015), 262-284.
  3. V. Ambrosio, Periodic solutions for the non-local operator \((-\Delta+m^{2})^{s}-m^{2s}\) with \(m\geq 0\), Topol. Methods Nonlinear Anal. 49 (2017) 1, 75-104.
  4. A. Bényi, T. Oh, The Sobolev inequality on the torus revisited, Publ. Math. Debrecen 83 (2013) 3, 359-374.
  5. J. Bourgain, H. Brezis, P. Mironescu, Another look at Sobolev spaces, [in:] J.L. Menaldi et al. (eds.), Optimal control and partial differential equations, pp. 439-455 (A volume in honour of A. Benssoussan's 60th birthday), IOS Press, 2001.
  6. A. Braides, \(\Gamma\)-convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications, vol. 22, Oxford University Press, Oxford, 2002.
  7. G. Dal Maso, An Introduction to \(\Gamma\)-convergence, Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston, Inc., Boston, MA, 1993.
  8. E. De Giorgi, T. Franzoni, Su un tipo di convergenza variazionale, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat. (8) 58 (1975) 6, 842-850.
  9. E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521-573.
  10. V.I. Kolyada, A.K. Lerner, On limiting embeddings of Besov spaces, Studia Math. 171 (2005) 1, 1-13.
  11. J.-L. Lions, E. Magenes, Non-homogeneous Boundary Value Problems and Applications, vol. I, Translated from the French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 181. Springer-Verlag, New York-Heidelberg, 1972.
  12. W. Masja, J. Nagel, Über äquivalente Normierung der anisotropen Funktionalräume \(H^{\mu}(\mathbb{R}^{n})\), Beiträge Anal. 12 (1978), 7-17.
  13. V. Maz'ya, T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal. 195 (2002) 2, 230-238.
  14. M. Milman, Notes on limits of Sobolev spaces and the continuity of interpolation scales, Trans. Amer. Math. Soc. 357 (2005) 9, 3425-3442.
  15. G. Molica Bisci, V. Rădulescu, R. Servadei, Variational methods for nonlocal fractional problems, with a foreword by Jean Mawhin. Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016.
  16. A.C. Ponce, A new approach to Sobolev spaces and connections to \(\Gamma\)-convergence, Calc. Var. Partial Differential Equations 19 (2004) 3, 229-255.
  17. E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, vol. 32, Princeton University Press, Princeton, N.J., 1971.
  18. R. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Functional Analysis 52 (1983) 1, 48-79.
  19. H. Triebel, Theory of Function Spaces, Monographs in Mathematics, 78. Birkhäuser Verlag, Basel, 1983.
  20. H. Triebel, Theory of Function Spaces. II, Monographs in Mathematics, 84. Birkhäuser Verlag, Basel, 1992.
  • Vincenzo Ambrosio
  • ORCID iD https://orcid.org/0000-0003-3439-1428
  • Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 12, 60131 Ancona, Italy
  • Communicated by Patrizia Pucci.
  • Received: 2019-12-15.
  • Revised: 2020-01-23.
  • Accepted: 2020-01-23.
  • Published online: 2020-02-17.
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Cite this article as:
Vincenzo Ambrosio, On some convergence results for fractional periodic Sobolev spaces, Opuscula Math. 40, no. 1 (2020), 5-20, https://doi.org/10.7494/OpMath.2020.40.1.5

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