A density result for GSBD and its application to the approximation of brittle fracture energies



We present an approximation result for functions \(u:\Omega \rightarrow \mathbb {R}^n\) belonging to the space \(GSBD(\Omega )\cap L^2(\Omega ,{{\mathbb R}}^n)\) with \(e(u)\) square integrable and \(\fancyscript{H}^{n-1}(J_u)\) finite. The approximating functions \(u_k\) are piecewise continuous functions such that \(u_k\rightarrow u\) in \(L^2(\Omega ,\mathbb {R}^n)\), \(e(u_k)\rightarrow e(u)\) in \(L^2(\Omega ,\mathbb {M}^{n{\times }n}_{sym})\), \(\fancyscript{H}^{n-1}(J_{u_k}\triangle J_u)\rightarrow 0\), and \(\int _{J_{u_k}\cup J_u}|u_k^\pm -u^\pm |\wedge 1d{\fancyscript{H}}^{n-1}\rightarrow 0\).  As an application, we provide the extension to the vector-valued case of the \(\Gamma \)-convergence result in \(GSBV(\Omega )\) proved by Ambrosio and Tortorelli (Commun Pure Appl Math 43:999–1036, 1990; Boll. Un. Mat. Ital. B (7) 6:105–123, 1992).

Mathematics Subject Classification (2000)

49Q20 49J45 26B30 74R10 35A35 



This material is based on work supported by the ERC Advanced Grant n. 290888 “QuaDynEvoPro”. The author gratefully acknowledges Prof. Gianni Dal Maso for many interesting discussions.


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.SISSATriesteItaly

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