On some convergence results for fractional periodic Sobolev spaces

• Autorzy: Ambrosio Vincenzo

Identyfikatory

DOI

10.7494/OpMath.2020.40.1.5

• Języki publikacji: EN

Abstrakty

EN

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 Γ-convergence result in the spirit of Ponce. The main theorems are obtained by using the nice structure of Fourier series.

Słowa kluczowe

EN

fractional periodic Sobolev spaces Fourier series; Γ-convergence

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2020
• Tom: Vol. 40, no. 1
• Strony: 5--20
• Opis fizyczny: Bibliogr. 20 poz.

Twórcy

autor

Ambrosio Vincenzo

• Dipartimento di Ingegneria Industriale e Scienze Matematiche Università Politecnica delle Marche Via Brecce Bianche, 12 60131 Ancona, Italy

Bibliografia

• [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 [formula], 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, Γ-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 Γ-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 Hµ (Rn), 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 Γ-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.

Uwagi

PL

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).