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Existence of strong solutions to the Dirichlet problem for the Griffith energy

  • Antonin ChambolleEmail author
  • Vito Crismale
Article
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Abstract

In this paper we continue the study of the Griffith brittle fracture energy minimisation under Dirichlet boundary conditions, suggested by Francfort and Marigo (J Mech Phys Solids 46:1319–1342, 1998). In a recent paper (Chambolle and Crismale in J Eur Math Soc (JEMS), 2018) we proved the existence of weak minimisers of the problem. Now we show that these minimisers are indeed strong solutions, namely their jump set is closed and they are smooth away from the jump set and continuous up to the Dirichlet boundary. This is obtained by extending up to the boundary the recent regularity results of Conti et al. (Ann Inst H Poincaré Anal Non Linéaire 36:455–474, 2019) and Chambolle et al. (J Math Pures Appl, 2019.  https://doi.org/10.1016/j.matpur.2019.02.001).

Mathematics Subject Classification

49Q20 49N60 35R35 26A45 74R10 

Notes

Acknowledgements

V. C. has been supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, is currently funded by the Marie Skłodowska-Curie Standard European Fellowship No. 793018, and acknowledge the financial support from the Laboratory Ypatia of Mathematical Sciences LYSM and the Labo CMAP.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.CMAP, CNRSÉcole PolytechniquePalaiseau CedexFrance

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