Curl bounds grad on SO(3)

Let $F^{\mathrm{p}}\in \mathrm{GL}\left(3\right)$ be the plastic deformation from the multiplicative decomposition in elasto-plasticity. We show that the geometric dislocation density tensor of Gurtin in the form $\mathrm{Curl}\left[F^{\mathrm{p}}\right]·{\left(F^{\mathrm{p}}\right)}^T$ applied to rotations controls the gradient in the sense that pointwise $\forall R\in C^1({ℝ}^3,\mathrm{SO}\left(3\right)):\parallel \mathrm{Curl}\left[R\right]·R^T{\parallel }_{{𝕄}^{3\times 3}}^2\ge \frac{1}{2}{\parallel \mathrm{D}R\parallel }_{{ℝ}^{27}}^2$. This result complements rigidity results [Friesecke, James and Müller, Comme Pure Appl. Math. 55 (2002) 1461-1506; John, Comme Pure Appl. Math. 14 (1961) 391-413; Reshetnyak, Siberian Math. J. 8 (1967) 631-653)] as well as an associated linearized theorem saying that $\forall A\in C^1({ℝ}^3,\mathrm{𝔰𝔬}\left(3\right)){:\parallel \mathrm{Curl}\left[A\right]\parallel }_{{𝕄}^{3\times 3}}^2\ge \frac{1}{2}{\parallel \mathrm{D}A\parallel }_{{ℝ}^{27}}^2={\parallel \nabla \mathrm{axl}\left[A\right]\parallel }_{{ℝ}^9}^2$.

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