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 


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.SISSATriesteItaly

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