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Integral Representation for Functionals Defined on SBDp in Dimension Two

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Abstract

We prove an integral representation result for functionals with growth conditions which give coercivity on the space \({SBD^p(\Omega)}\), for \({\Omega\subset\mathbb{R}^{2}}\), which is a bounded open Lipschitz set, and \({p\in(1,\infty)}\). The space SBD p of functions whose distributional strain is the sum of an L p part and a bounded measure supported on a set of finite \({\mathcal{H}^{1}}\)-dimensional measure appears naturally in the study of fracture and damage models. Our result is based on the construction of a local approximation by W 1,p functions. We also obtain a generalization of Korn’s inequality in the SBD p setting.

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Authors and Affiliations

  1. Institut für Angewandte Mathematik, Universität Bonn, 53115, Bonn, Germany

    Sergio Conti & Flaviana Iurlano

  2. DiMaI, Università di Firenze, 50134, Florence, Italy

    Matteo Focardi

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  1. Sergio Conti
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  2. Matteo Focardi
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  3. Flaviana Iurlano
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Correspondence to Matteo Focardi.

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Communicated by G. Dal Maso

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Conti, S., Focardi, M. & Iurlano, F. Integral Representation for Functionals Defined on SBDp in Dimension Two. Arch Rational Mech Anal 223, 1337–1374 (2017). https://doi.org/10.1007/s00205-016-1059-y

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