Archive for Rational Mechanics and Analysis

, Volume 202, Issue 1, pp 63–113 | Cite as

Lower Semicontinuity for Integral Functionals in the Space of Functions of Bounded Deformation Via Rigidity and Young Measures

  • Filip Rindler


We establish a general weak* lower semicontinuity result in the space BD(Ω) of functions of bounded deformation for functionals of the form
$$ \begin{array}{ll} \,\mathcal{F}(u) := &\int_\Omega f (x, \mathcal{E} u) \;{\rm d} x + \int_\Omega f^\infty \left( x, \frac{{\rm d} E^s u}{{\rm d} |{E^s u}|} \right) \;{\rm d} |{E^s u}| \\ &+ \int_{\partial \Omega} f^\infty \left( x, u|_{\partial \Omega} \odot n_\Omega \right) \;{\rm d} \mathcal{H}^{d-1}, \qquad u \in {\rm BD}(\Omega). \end{array} $$
The main novelty is that we allow for non-vanishing Cantor-parts in the symmetrized derivative Eu. The proof is accomplished via Jensen-type inequalities for generalized Young measures and a construction of good blow-ups, which is based on local rigidity arguments for some differential inclusions involving symmetrized gradients, and an iteration of the blow-up construction. This strategy allows us to establish the lower semicontinuity result without an Alberti-type theorem in BD(Ω), which is not available at present. We also include existence and relaxation results for variational problems in BD(Ω), as well as a complete discussion of some differential inclusions for the symmetrized gradient in two dimensions.


Lower Semicontinuity Differential Inclusion Young Measure Symmetric Tensor Product Tangent Measure 


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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Mathematical InstituteUniversity of OxfordOxfordUK

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