Abstract
We prove a compactness result with respect to \(\Gamma \)-convergence for a class of integral functionals which are expressed as a sum of a local and a non-local term. The main feature is that, under our hypotheses, the local part of the \(\Gamma \)-limit depends on the interaction between the local and non-local terms of the converging subsequence. The result is applied to concentration and homogenization problems.
This is a preview of subscription content, log in via an institution to check access.
We’re sorry, something doesn't seem to be working properly.
Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.
References
Alicandro, R., Ansini, N., Braides, A., Piatnitski, A., Tribuzio, A.: A Variational Theory of Convolution-type Functionals. Springer Briefs on PDEs and Data Science. Springer, Berlin (2023)
Bellido, J.C., Mora-Corral, C.: Lower semicontinuity and relaxation via Young measures for nonlocal variational problems and applications to peridynamics. SIAM J. Math. Anal. 50, 779–809 (2018)
Bellido, J.C., Mora-Corral, C., Pedregal, P.: Hyperelasticity as a \(\Gamma \)-limit of peridynamics when the horizon goes to zero. Calc. Var. Part. Differ. Equ. 54, 1643–1670 (2015)
Bourgain, J., Brezis, H., Mironescu, P.: Another look at Sobolev spaces. In: Optimal Control and Partial Differential Equations, pp. 439–455. IOS Press (2001)
Braides, A., Defranceschi, A.: Homogenization of Multiple Integrals. Oxford University Press, Oxford (1998)
Brezis, H., Nguyen, H.-M.: Non-local functionals related to the total variation and connections with image processing. Ann. PDE 4, 9 (2018)
Braides, A., Dal Maso, G.: Continuity of some non-local functionals with respect to a convergence of the underlying measures. J. Math. Pures Appl. 170, 136–149 (2023)
Braides, A., Piatnitski, A.: Homogenization of random convolution energies. J. Lond. Math. Soc. 104, 295–319 (2021)
Dal Maso, G.: An Introduction to \(\Gamma \)-Convergence. Birkhäuser, Basel (1994)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)
Du, Q., Mengesha, T., Tian, X.: \(L^p\) compactness criteria with an application to variational convergence of some nonlocal energy functionals. ArXiv preprint arXiv:1801.08000
Emmrich, E., Lehoucq, R.B., Puhst, D.: Peridynamics: a nonlocal continuum theory. In: Meshfree Methods for Partial Differential Equations VI, Lect. Notes Comput. Sci. Eng., vol. 89, pp. 45–65. Springer, Heidelberg (2013)
Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7, 1005–1028 (2008)
Kreisbeck, C., Zappale, E.: Loss of double-integral character during relaxation. SIAM J. Math. Anal. 53, 351–385 (2021)
Mengesha, T., Du, Q.: On the variational limit of a class of nonlocal functionals related to peridynamics. Nonlinearity 28, 3999–4035 (2015)
Mora-Corral, C., Tellini, A.: Relaxation of a scalar nonlocal variational problem with a double-well potential. Calc. Var. Part. Differ. Equ. 59, 67 (2020)
Muñoz, J.: Characterisation of the weak lower semicontinuity for a type of nonlocal integral functional: the n-dimensional scalar case. J. Math. Anal. Appl. 360, 495–502 (2009)
Pedregal, P.: Weak lower semicontinuity and relaxation for a class of non-local functionals. Rev. Mat. Complut. 29, 485–495 (2016)
Ponce, A.C.: A new approach to Sobolev spaces and connections to \(\Gamma \)-convergence. Calc. Var. Part. Differ. Equ. 19, 229–255 (2004)
Shi, Z., Osher, S., Zhu, W.: Weighted nonlocal Laplacian on interpolation from sparse data. J. Sci. Comput. 73, 1164–1177 (2017)
Silling, S.A.: Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48, 175–209 (2000)
Slepčev, D., Thorpe, M.: Analysis of p-Laplacian regularization in semisupervised learning. SIAM J. Math. Anal. 51, 2085–2120 (2019)
Acknowledgements
This paper is based on work supported by the National Research Project (PRIN 2017BTM7SN) “Variational Methods for Stationary and Evolution Problems with Singularities and Interfaces”, funded by the Italian Ministry of University and Research. The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests. Manuscript has no associated data.
Additional information
Communicated by J. Kristensen.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Braides, A., Maso, G.D. Compactness for a class of integral functionals with interacting local and non-local terms. Calc. Var. 62, 148 (2023). https://doi.org/10.1007/s00526-023-02491-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-023-02491-w