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Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang

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Volume 10, Issue 2


Lower semicontinuity of a class of integral functionals on the space of functions of bounded deformation

Gianni Dal MasoORCID iD: http://orcid.org/0000-0002-1010-4968 / Gianluca Orlando / Rodica Toader
Published Online: 2016-04-07 | DOI: https://doi.org/10.1515/acv-2015-0036


We study the lower semicontinuity of some free discontinuity functionals with linear growth defined on the space of functions with bounded deformation. The volume term is convex and depends only on the Euclidean norm of the symmetrized gradient. We introduce a suitable class of surface terms, which make the functional lower semicontinuous with respect to L1 convergence.

Keywords: Free discontinuity problems; lower semicontinuity; functions of bounded deformation

MSC 2010: 49J45; 26B30


  • [1]

    Ambrosio L., Variational problems in SBV and image segmentation, Acta Appl. Math. 17 (1989), 1–40. Google Scholar

  • [2]

    Ambrosio L., Coscia A. and Dal Maso G., Fine properties of functions with bounded deformation, Arch. Ration. Mech. Anal. 139 (1997), 201–238. Google Scholar

  • [3]

    Ambrosio L., Fusco N. and Pallara D., Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., Clarendon Press, Oxford, 2000. Google Scholar

  • [4]

    Barroso A. C., Fonseca I. and Toader R., A relaxation theorem in the space of functions of bounded deformation, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 29 (2000), 19–49. Google Scholar

  • [5]

    Bellettini G., Coscia A. and Dal Maso G., Compactness and lower semicontinuity properties in SBD(Ω), Math. Z. 228 (1998), 337–351. Google Scholar

  • [6]

    Bouchitté G., Braides A. and Buttazzo G., Relaxation results for some free discontinuity problems, J. Reine Angew. Math. 458 (1995), 1–18. Google Scholar

  • [7]

    Bouchitté G. and Buttazzo G., New lower semicontinuity results for nonconvex functionals defined on measures, Nonlinear Anal. 15 (1990), 679–692. Google Scholar

  • [8]

    Bouchitté G., Fonseca I., Leoni G. and Mascarenhas L., A global method for relaxation in W1,p and in SBVp, Arch. Ration. Mech. Anal. 165 (2002), 187–242. Google Scholar

  • [9]

    Bouchitté G., Fonseca I. and Mascarenhas L., A global method for relaxation, Arch. Ration. Mech. Anal. 145 (1998), 51–98. Google Scholar

  • [10]

    Bourdin B., Francfort G. A. and Marigo J.-J., The Variational Approach to Fracture, Springer, New York, 2008. Google Scholar

  • [11]

    Braides A., Γ-Convergence for Beginners, Oxford Lecture Ser. Math. Appl. 22, Oxford University Press, Oxford, 2002. Google Scholar

  • [12]

    Braides A. and Chiadò Piat V., Integral representation results for functionals defined on SBV(Ω;m), J. Math. Pures Appl. (9) 75 (1996), 595–626. Google Scholar

  • [13]

    Braides A. and De Cicco V., New lower semicontinuity and relaxation results for functionals defined on BV(Ω), Adv. Math. Sci. Appl. 6 (1996), 1–30. Google Scholar

  • [14]

    Braides A., Defranceschi A. and Vitali E., Relaxation of elastic energies with free discontinuities and constraint on the strain, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), 275–317. Google Scholar

  • [15]

    Dal Maso G., An Introduction to Γ-Convergence, Progr. Nonlinear Differential Equations Appl. 8, Birkhäuser, Boston, 1993. Google Scholar

  • [16]

    De Giorgi E., Free discontinuity problems in calculus of variations, Frontiers in Pure and Applied Mathematics, North-Holland, Amsterdam (1991), 55–62. Google Scholar

  • [17]

    Gargiulo G. and Zappale E., A lower semicontinuity result in SBD for surface integral functionals of fracture mechanics, Asymptot. Anal. 72 (2011), 231–249. Google Scholar

  • [18]

    Horn R. A. and Johnson C. R., Matrix Analysis, 2nd ed., Cambridge University Press, Cambridge, 2013. Google Scholar

  • [19]

    Kuczma M., An Introduction to the Theory of Functional Equations and Inequalities, 2nd ed., Birkhäuser, Basel, 2009. Google Scholar

  • [20]

    Morel J.-M. and Solimini S., Variational Methods in Image Segmentation, Progr. Nonlinear Differential Equations Appl. 14, Birkhäuser, Boston, 1995. Google Scholar

  • [21]

    Temam R., Problèmes mathématiques en plasticité, Méthodes Math. Inform. 12, Gauthier-Villars, Montrouge, 1983. Google Scholar

  • [22]

    Virga E. G., Variational Theories for Liquid Crystals, Appl. Math. and Math. Comput. 8, Chapman & Hall, London, 1994. Google Scholar

About the article

Received: 2015-09-28

Revised: 2016-02-06

Accepted: 2016-03-05

Published Online: 2016-04-07

Published in Print: 2017-04-01

This material is based on work supported by the Italian Ministry of Education, University, and Research under the project “Calculus of Variations” (PRIN 2010-11) and by the European Research Council under Grant No. 290888 “Quasistatic and Dynamic Evolution Problems in Plasticity and Fracture”. 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).

Citation Information: Advances in Calculus of Variations, Volume 10, Issue 2, Pages 183–207, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2015-0036.

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