An energy condition for the existence of a~rotation

Yu. A. Aminov

Sbornik. Mathematics, Tome 15 (1971) no. 2, pp. 325-334

Résumé

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.