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 Mat. Sb. (N.S.), 1971, Volume 86(128), Number 2(10), Pages 325–334 (Mi msb3298)

An energy condition for the existence of a rotation

Yu. A. Aminov

Abstract: 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 $c^2(|1|+|\mathbf u|^2)^2$ 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.
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English version:
Mathematics of the USSR-Sbornik, 1971, 15:2, 325–334

Bibliographic databases:

UDC: 516.8
MSC: Primary 53A05, 53A45, 53C99; Secondary 57D25

Citation: Yu. A. Aminov, “An energy condition for the existence of a rotation”, Mat. Sb. (N.S.), 86(128):2(10) (1971), 325–334; Math. USSR-Sb., 15:2 (1971), 325–334

Citation in format AMSBIB
\Bibitem{Ami71} \by Yu.~A.~Aminov \paper An energy condition for the existence of a~rotation \jour Mat. Sb. (N.S.) \yr 1971 \vol 86(128) \issue 2(10) \pages 325--334 \mathnet{http://mi.mathnet.ru/msb3298} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=291998} \zmath{https://zbmath.org/?q=an:0221.52005} \transl \jour Math. USSR-Sb. \yr 1971 \vol 15 \issue 2 \pages 325--334 \crossref{https://doi.org/10.1070/SM1971v015n02ABEH001548}