Calculus of Variations and Geometric Measure Theory
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A. Arroyo-Rabasa

Characterization of generalized Young measures generated by $\mathcal A$-free measures

created by arroyorabasa on 19 Aug 2019

[BibTeX]

preprint

Inserted: 19 aug 2019

Year: 2019

ArXiv: 1908.03186 PDF

Abstract:

We organize a robust analytical framework in terms of the space $\mathrm{BV}^{\mathcal A}(\mathbb R^d)$ of functions with bounded $\mathcal A$-variation, where $\mathcal A$ is a partial differential operator satisfying Murat's constant rank property. This perspective enable us to carry constructions available for gradients into the $\mathcal A$-free framework (introduced by Dacorogna and Fonseca & M\"uller). In particular, this allows the gluing and localization of $\mathcal A$-free measures without modifying the underlying $\mathcal A$-free constraint. We combine these advances with delicate geometric constructions to give a full characterization of the class of generalized Young measures generated by sequences of $\mathcal A$-free measures (where $\mathcal A$ is an operator of arbitrary order). The main characterization result is stated in terms of a well-known separation property involving the class of $\mathcal A$-quasiconvex integrands. We give a second characterization in terms of the tangent Young measures being $\mathcal A$-free Young measures. Lastly, we show that the inclusion \[ \mathrm L^1(\Omega) \cap \ker \mathcal A \hookrightarrow \mathcal M(\Omega) \cap \ker \mathcal A \] is dense with respect to the area-strict convergence of measures.

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