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Partial regularity for the crack set minimizing the two-dimensional griffith energy. (English) Zbl 1501.49024

The paper deals with regularity analysis of cracked elastic bodies into two spatial dimensions. The model problem is governed by the Griffith theory of brittle fracture. The basic idea consists in extending the regularity theory for minimizers of the classical Mumford-Shah functional to the Griffith energy. However, this extension is not straightforward at all. Actually, in the context of Griffith theory, many fundamental ingredients are not available, including Korn inequality, Euler-Lagrange equation, coarea formula, monotonicity formula for the elastic energy and appropriated extension techniques. Therefore, a decay estimate on the flatness and a decay of the renormalized energy are derived by using Airy function, which restrict the analysis to the two-dimensional setting. However, these estimates allow for deriving a \(\mathcal{C}^{1,\alpha}\)-regularity result for minimizers of the Griffith functional, with \(\alpha\in(0,1)\). In particular, a partial \(\mathcal{C}^{1,\alpha}\)-regularity of the isolated connected components of the crack is obtained, which represents the main contribution of the paper. It is important to point out that only minimizers of the global functional are considered, but the obtained results seem to be also applied to almost minimizers. Although restricted to the two-dimensional case, the obtained results are general and useful for practical applications.

MSC:

49Q20 Variational problems in a geometric measure-theoretic setting
49Q05 Minimal surfaces and optimization
74R10 Brittle fracture
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