Geodesic
Isoperimetric inequality for torsional rigidity in multidimensional domains
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2012), pp. 45-49

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We consider the Saint Venant functional $P$ for the torsional rigidity in arbitrary plane and space domains. Our main result is the following sharp estimate: $P\leq(4/n)m$, where $n$ is the dimension of domains and $m$ is the harmonic mean of inertial moments of a domain with respect to coordinate planes. Extremal domains are some ellipsoids. Hence, we obtain a generalization of the isoperimetric inequality, proved by E. Nicolay for the torsional rigidity of simply connected planar domains.
Keywords: isoperimetric inequality, torsional rigidity
Mots-clés : inertial moments. 
@article{IVM_2012_7_a4,
     author = {F. G. Avkhadiev},
     title = {Isoperimetric inequality for torsional rigidity in multidimensional domains},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {45--49},
     publisher = {mathdoc},
     number = {7},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2012_7_a4/}
}
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F. G. Avkhadiev. Isoperimetric inequality for torsional rigidity in multidimensional domains. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2012), pp. 45-49. http://geodesic.mathdoc.fr/item/IVM_2012_7_a4/