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
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.
- M.S. Agranovich, Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains, Springer Monographs in Mathematics, Springer, Cham, 2015.
- V. Ambrosio, Periodic solutions for a pseudo-relativistic Schrödinger equation, Nonlinear Anal. 120 (2015), 262-284.
- 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.
- A. Bényi, T. Oh, The Sobolev inequality on the torus revisited, Publ. Math. Debrecen 83 (2013) 3, 359-374.
- 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.
- A. Braides, \(\Gamma\)-convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications, vol. 22, Oxford University Press, Oxford, 2002.
- G. Dal Maso, An Introduction to \(\Gamma\)-convergence, Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston, Inc., Boston, MA, 1993.
- 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.
- E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521-573.
- V.I. Kolyada, A.K. Lerner, On limiting embeddings of Besov spaces, Studia Math. 171 (2005) 1, 1-13.
- 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.
- W. Masja, J. Nagel, Über äquivalente Normierung der anisotropen Funktionalräume \(H^{\mu}(\mathbb{R}^{n})\), Beiträge Anal. 12 (1978), 7-17.
- 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.
- M. Milman, Notes on limits of Sobolev spaces and the continuity of interpolation scales, Trans. Amer. Math. Soc. 357 (2005) 9, 3425-3442.
- 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.
- A.C. Ponce, A new approach to Sobolev spaces and connections to \(\Gamma\)-convergence, Calc. Var. Partial Differential Equations 19 (2004) 3, 229-255.
- E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, vol. 32, Princeton University Press, Princeton, N.J., 1971.
- R. Strichartz, Analysis of the Laplacian on the complete Riemannian manifold, J. Functional Analysis 52 (1983) 1, 48-79.
- H. Triebel, Theory of Function Spaces, Monographs in Mathematics, 78. Birkhäuser Verlag, Basel, 1983.
- H. Triebel, Theory of Function Spaces. II, Monographs in Mathematics, 84. Birkhäuser Verlag, Basel, 1992.
- Vincenzo Ambrosio
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.
