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On the structure of $${\mathcal A}$$-free measures and applications. (English) Zbl 1352.49050
Consider a $$k$$-th order linear constant-coefficients PDE operator $$\mathcal A$$, $$(k\in\mathbb N)$$, i.e. $$\mathcal A=\sum_{|\alpha|\leq k}A_\alpha\partial^\alpha$$, where $$A_\alpha\in\mathbb R^{n\times m}$$ and $$\alpha=(\alpha_1,\dots\alpha_d)$$ is a multi-index. Assume that $$\mu$$ is a finite Radon measure on an open set $$\Omega\subset\mathbb R^d$$ with values in $$\mathbb R^m$$ and it is $$\mathcal A$$-free. Define the wave cone $$\Lambda_{\mathcal A}=\bigcup_{|\xi|=1}\ker \mathcal A^k(\xi)\subset\mathbb R^m$$ with $$\mathcal A^k(\xi)=(2\pi i)^k\sum _{|\alpha |=k}A_\alpha\xi^\alpha$$, where $$\xi^\alpha =\xi_1^{\alpha_1}\dots \xi_d^{\alpha_d}$$. The authors show that for a measure $$\mu$$ solving the initial PDE $$\mathcal A\mu=0$$ , the polar $$d\mu/d|\mu|$$, i.e. the Radon-Nikodým derivative of $$\mu$$ with respect to its total variation measure $$|\mu|$$ lies in the wave cone at almost all singular points.
Theorem 1.1. Let $$\Omega \subset \mathbb R^d$$ be an open set, let $$\mathcal A$$ be a $$k$$-th order linear constant-coefficients differential operator and let $$\mu\in\mathcal M(\Omega;\mathbb R^m)$$ be an $$\mathcal A$$-free Radon measure on $$\Omega$$ with values in $$\mathbb R^m$$. Then, $$d\mu/d|\mu|(x)\in \Lambda_{\mathcal A}$$, for $$|\mu|^s$$-a.e. $$x\in\Omega$$.
By applying Theorem 1.1. to suitably chosen differential operators, one can easily obtain several remarkable consequences. First of all, a simple proof of Alberti’s rank-one theorem and its extensions to functions of bounded deformation are obtained. Then, the authors prove a structure theorem for the singular part of a finite family of normal currents in the spirit of the rank-one theorem. Considering the results of Alberti and Marquese and of Schioppa, it follows that the Rademacher theorem can hold only for absolutely continuous measures and that every top-dimensional Ambrosio-Kirchheim metric current in $$\mathbb R^d$$ is a Federer flat chain.

##### MSC:
 49Q15 Geometric measure and integration theory, integral and normal currents in optimization 35D30 Weak solutions to PDEs 28B05 Vector-valued set functions, measures and integrals 49Q20 Variational problems in a geometric measure-theoretic setting
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