Sharp rigidity estimates for incompatible fields as a consequence of the Bourgain Brezis div-curl result. (Estimées de rigidité pour les champs incompatibles comme conséquence du résultat div-rot de Bourgain et Brezis.)(English. French summary)Zbl 1464.49033

The authors derive a sharp rigidity estimate and a sharp Korn’s inequality for matrix-valued fields. The main results follow:
Given an open, bounded, connected and Lipschitz set $$\Omega \subset \mathbb{R}^n$$, $$n\ge 2$$, there exists $$C>0$$ such that for every $$\beta \in L^1(\Omega;\mathbb{R}^{n\times n})$$ with $$\operatorname{Curl}\beta \in \mathcal{M}(\Omega;\mathbb{R}^{n\times n\times n})$$, there exist a rotation $$R\in \operatorname{SO}(n)$$ such that $\|\beta-R\|_{L^{1^*}}(\Omega)\le C\big(\|\operatorname{dist}(\beta,\operatorname{SO}(n)\| +|\operatorname{Curl}\beta|(\Omega)\big).$ and an antisymmetric matrix $$A$$ such that $\|\beta-A\|_{L^{1^*}}(\Omega)\le C\big(\|\beta+\beta^T\|_{L^{1^*}}(\Omega) +|\operatorname{Curl}\beta|(\Omega)\big)$ where $$\mathcal{M}(\Omega;\mathbb{R}^{n\times n\times n}$$ is the set of Radon measures on $$\Omega$$ with values in $$\mathbb{R}^{n\times n\times n}$$ and $$1^* = n/(n-1)$$ is the Sobolev conjugate exponent.

MSC:

 49Q20 Variational problems in a geometric measure-theoretic setting 74C15 Large-strain, rate-independent theories of plasticity (including nonlinear plasticity) 53C24 Rigidity results
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References:

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