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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.
Bibliography: 8 titles.

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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
Received: 27.07.1970

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}


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