Geodesic
An energy condition for the existence of a rotation
Sbornik. Mathematics, Tome 15 (1971) no. 2, pp. 325-334 
Yu. A. Aminov. An energy condition for the existence of a rotation. Sbornik. Mathematics, Tome 15 (1971) no. 2, pp. 325-334. http://geodesic.mathdoc.fr/item/SM_1971_15_2_a8/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper the following assertion is proved. Let the regular vector field $\mathbf u=(u^1,u^2,u^3)$ be defined in a cube in the space $E^3$. If the sum of the principal minors of the matrix $\|\partial u^i/\partial x_j\|$ is majorized by the quantity $c^2\bigl(|1|+|\mathbf u|^2\bigr)^2$ and, moreover, $|\operatorname{rot}\mathbf u|\leqslant\mu$, then the length $a$ of the side of the square is bounded above: $a\leqslant a_0(\mu,c)$. As an application there is an interpretation of the results in terms of the mechanics of elastic media. Thus, it is established that if a deformable body contains a sufficiently large cube and if a large part of the energy does not involve the spatial divergence, then there exists a nonzero rotational force field. Bibliography: 8 titles.

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[4] Yu. A. Aminov, “$n$-mernye analogi integralnoi formuly S. N. Bernshteina”, Matem. sb., 75(117) (1968), 385–399 | MR

[5] Yu. A. Aminov, “Nekotorye globalnye voprosy geometrii vektornogo polya”, Ukr. geometr. sb., 1970, no. 8, 3–15 | MR | Zbl

[6] Yu. A. Aminov, “Istochniki krivizny vektornogo polya”, Matem. sb., 80(122) (1969), 210–224 | MR

[7] A. Lyav, Matematicheskaya teoriya uprugosti, ONTI, Moskva–Leningrad, 1935

[8] Dzh. Sing, Obschaya teoriya otnositelnosti, IL, Moskva, 1963