Geodesic
Estimates for torsional rigidity of convex domain via new geometric characteristics
Ufa mathematical journal, Tome 16 (2024) no. 3, pp. 21-39 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We introduce new geometric characteristics of a convex domain with finite boundary length and provide an algorithm for calculating them. A series of isoperimetric inequalities between new functionals and known integral characteristics of the domain are proved. Some of the inequalities have a wide class of extremal domains. We consider applications of new characteristics to the problem on estimating the torsional rigidity of a convex domain.
Keywords: function of distance to the boundary, torsional rigidity, isoperimetric inequality, extremal domain.
Mots-clés : convex domain 
@article{UFA_2024_16_3_a1,
     author = {L. I. Gafiyatullina and R. G. Salakhudinov},
     title = {Estimates for torsional rigidity of convex domain via new geometric characteristics},
     journal = {Ufa mathematical journal},
     pages = {21--39},
     year = {2024},
     volume = {16},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/UFA_2024_16_3_a1/}
}
TY  - JOUR
AU  - L. I. Gafiyatullina
AU  - R. G. Salakhudinov
TI  - Estimates for torsional rigidity of convex domain via new geometric characteristics
JO  - Ufa mathematical journal
PY  - 2024
SP  - 21
EP  - 39
VL  - 16
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/UFA_2024_16_3_a1/
LA  - en
ID  - UFA_2024_16_3_a1
ER  - 
%0 Journal Article
%A L. I. Gafiyatullina
%A R. G. Salakhudinov
%T Estimates for torsional rigidity of convex domain via new geometric characteristics
%J Ufa mathematical journal
%D 2024
%P 21-39
%V 16
%N 3
%U http://geodesic.mathdoc.fr/item/UFA_2024_16_3_a1/
%G en
%F UFA_2024_16_3_a1
L. I. Gafiyatullina; R. G. Salakhudinov. Estimates for torsional rigidity of convex domain via new geometric characteristics. Ufa mathematical journal, Tome 16 (2024) no. 3, pp. 21-39. http://geodesic.mathdoc.fr/item/UFA_2024_16_3_a1/

[1] G. Pólya, G. Szegö, Isoperimetric inequalities in mathematical physics, Princeton University Press, Princeton, 1951 | MR | MR | Zbl

[2] S.P. Timoshenko, History of strength of materials. With a brief account of the history of theory of elasticity and theory of structures, McGraw–Hill Publishing Company, Ltd., London, 1954 | MR | Zbl

[3] L.E. Payne, “Some isoperimetric inequalities in the torsional problem for multiply connected regions”, Stud. Math. Anal. related Topics, Essays in Honor of G. Polya, Standford University Press, California, 1962, 270–280 | MR

[4] E. Makai, “On the principal frequency of a membrane and the torsional rigidity of a beam”, Stud. Math. Anal. related Topics, Essays in Honor of G. Polya, Standford University Press, California, 1962, 227–231 | MR

[5] F.G. Avkhadiev, “Solution of the generalized Saint Venant problem”, Sb. Math., 189:12 (1998), 1739–1748 | DOI | DOI | MR | Zbl

[6] L.I. Gafiyatullina, R.G. Salakhudinov, “A generalization of the Polia — Szego and Makai inequalities for torsional rigidity”, Russ. Math., 65:11 (2021), 76–80 | DOI | MR | Zbl

[7] R.G. Salakhudinov, L.I. Gafiyatullina, “Two-Sided Estimate for the Torsional Rigidity of Convex Domain Generalizing the Polya-Szego and Makai Inequalities”, Lobachevskii J. Math., 43:10 (2022), 3020–3032 | DOI | MR | Zbl

[8] E. Makai, “A proof of Saint-Venant's theorem on torsional rigidity”, Acta Math. Acad. Sci. Hung., 17 (1966), 419–422 | DOI | MR | Zbl

[9] R.G. Salakhudinov, “Isoperimetric properties of Euclidean boundary moments of a simply connected domain”, Russ. Math., 57:8 (2013), 57–69 | DOI | MR | Zbl

[10] R.G. Salahudinov, “Refined inequalities for euclidean moments of a domain with respect to its boundary”, SIAM J. Math. Anal., 44:4 (2012), 2949–2961 | DOI | MR

[11] R.G. Salahudinov, “Some properties of functionals on level sets”, Ufa Math. J., 11:2 (2019), 114–124 | DOI | MR | Zbl

[12] R.G. Salahudinov, “Torsional rigidity and euclidian moments of a convex domain”, Q. J. Math., 67:4 (2016), 669–681 | MR

[13] V.M. Tikhomirov, Stories about maxima and minima, American Mathematical Society, Providence, RI; Mathematical Association of America, Washington, DC, 1990 | MR | Zbl