Abstract
This paper presents three new families of fractional Sobolev spaces and their accompanying theory in one dimension. The new construction and theory are based on a newly developed notion of weak fractional derivatives, which are natural generalizations of the well-established integer order Sobolev spaces and theory. In particular, two new families of one-sided fractional Sobolev spaces are introduced and analyzed, and they reveal more insights about another family of so-called symmetric fractional Sobolev spaces. Many key theorems/properties, such as density/approximation theorem, extension theorems, one-sided trace theorem, and various embedding theorems and Sobolev inequalities in those Sobolev spaces are established. Moreover, a few relationships with existing fractional Sobolev spaces are also uncovered. The results of this paper lay down a solid theoretical foundation for systematically developing a fractional calculus of variations theory and a fractional PDE theory as well as their numerical solutions in subsequent works.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of data and materials
Not applicable.
Code Availability Statement
Not applicable.
References
Adams, R.A.: Sobolev Spaces. Academic Press, New York (1975)
Besov, O.V., Il’in, V.P., Nikol’skii, S.M.: Integral Representations of Functions and Embedding Theorems. Izdat (1975)
Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin (2011)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. PDEs 32, 1245–1260 (2007)
Du, Q.: Nonlocal Modeling, Analysis, and Computation. SIAM, Philadelphia (2019)
Ervin, V., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods PDEs 22, 558–576 (2006)
Evans, L.C.: Partial Differential Equations. AMS, New York (2010)
Feng, X., Sutton, M.: A New Theory of Fractional Differential Calculus and Fractional Sobolev Spaces: One-Dimensional Case. Research notes. arXiv:2004.10833
Feng, X., Sutton, M.: A new theory of fractional differential calculus. Anal. Appl. 19, 715–750 (2021)
Feng, X., Sutton, M.: On a new class of fractional calculus of variations and related fractional differential equations. Adv. Differ. Equ. 35, 299–338 (2022)
Fujishiro, K., Yamamoto, M.: Approximate controllability for fractional diffusion equations by interior control. Appl. Anal. 94, 1793–1801 (2014)
Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1998)
Gorenflo, R., Kilbas, A., Mainardi, F., Rogosin, S.: Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin (2014)
Guo, B., Pu, X., Huang, F.: Fractional Partial Differential Equations and their Numerical Solutions. World Scientific Publishing Co., Singapore (2015)
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific Press, Singapore (2000)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006)
Lions, J.L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications. Springer, Berlin (1972)
Mainardi, F.: Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics. Springer, Berlin (1997)
Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. Imperial College Press, London (2010)
Malinowska, A.B., Odzijewicz, T., Torres, D.F.M.: Advanced Methods in the Fractional Calculus of Variations. Springer, Berlin (2015)
Meerschaert, M., Sikorskii, A.: Stochastic Models for Fractional Calculus. de Gruyter, Berlin (2012)
Meyers, N.G., Serrin, J.: H = W. Proc. Natl. Acad. Sci. 51, 1055–1056 (1964)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2021)
Podlubny, I.: Fractional Differential Equations. Academic Press, Cambridge (1999)
Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382, 426–447 (2011)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. CRC Press (1993)
Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Taheri, A.: Function Spaces and Partial Differential Equations, vol. 2. Oxford University Press, Oxford (2015)
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. Birkhäuser Verlag, Basel (1983)
Triebel, H.: Theory of Function Spaces. North-Holland Publishing Co., Amsterdam (1978)
Funding
The authors are partially supported by the NSF grants DMS-1620168 and DMS-2012414.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
Not applicable.
Additional information
Communicated by Tom ter Elst.
Rights and permissions
About this article
Cite this article
Feng, X., Sutton, M. New families of fractional Sobolev spaces. Banach J. Math. Anal. 16, 46 (2022). https://doi.org/10.1007/s43037-022-00198-2
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s43037-022-00198-2
Keywords
- Weak fractional derivatives
- Fundamental theorem of weak fractional calculus
- One-sided and symmetric fractional Sobolev spaces
- Density theorem
- Extension theorems
- One-sided trace theorem
- Embedding theorems