A Density Result in GSBDp with Applications to the Approximation of Brittle Fracture Energies

  • Antonin Chambolle
  • Vito Crismale


We prove that any function in \({GSBD^p(\Omega)}\), with \({\Omega}\) a n-dimensional open bounded set with finite perimeter, is approximated by functions \({u_k\in SBV(\Omega; \mathbb{R}^{n})\cap L^\infty(\Omega; \mathbb{R}^{n})}\) whose jump is a finite union of C1 hypersurfaces. The approximation takes place in the sense of Griffith-type energies \({\int_\Omega W(e(u)) {\rm dx} +\mathcal{H}^{n-1}(J_u)}\), e(u) and Ju being the approximate symmetric gradient and the jump set of u, and W a nonnegative function with p-growth, p > 1. The difference between uk and u is small in Lp outside a sequence of sets \({E_k\subset \Omega}\) whose measure tends to 0 and if \({|u|^r \in L^1(\Omega)}\) with \({r\in (0,p]}\), then \({|u_k-u|^r \to 0}\) in \({L^1(\Omega)}\). Moreover, an approximation property for the (truncation of the) amplitude of the jump holds. We apply the density result to deduce \({\Gamma}\)-convergence approximation à la Ambrosio-Tortorelli for Griffith-type energies with either Dirichlet boundary condition or a mild fidelity term, such that minimisers are a priori not even in \({L^1(\Omega; \mathbb{R}^{n})}\).


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V. Crismale has been supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, and is currently funded by the Marie Skłodowska-Curie Standard European Fellowship No.793018. The authors wish to thank the anonymous referees for their valuable comments.

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The authors declare no competing financial interest.

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.CMAP, École Polytechnique, CNRSPalaiseau CedexFrance

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