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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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