Solution of the generalized Saint Venant problem

F. G. Avkhadiev

Sbornik. Mathematics, Tome 189 (1998) no. 12, pp. 1739-1748

Résumé

A well-known problem in the mathematical theory of elasticity about the torsional rigidity $P(\Omega)$ of a bar whose cross-section is an arbitrary simply connected domain $\Omega$ is considered. It is shown that $P(\Omega)$ is equivalent to the moment of inertia of the domain relative to its boundary. Thus, a new interpretation of the well-known Coulomb's formula is suggested, and on this basis the following problem, which has its origins in works of Cauchy and Saint Venant, is solved: find a geometric parameter equivalent to the torsional rigidity coefficient of elastic bars with simply connected cross-sections. The proof is based on the definition of the torsional rigidity as the norm of a certain embedding operator in a Sobolev space and on the theory of conformal maps. In particular, some conformally invariant inequalities are established.

Bibliographie

[1] Sen-Venan B., Memuar o kruchenii prizm. Memuar ob izgibe prizm, GIFML, M., 1961

[2] Timoshenko S. P., Istoriya nauki o soprotivlenii materialov, GITTL, M., 1957

[3] Polia G., Sege G., Izoperimetricheskie neravenstva v matematicheskoi fizike, GIFML, M., 1962

[4] Goldshtein R. V., Entov V. M., Kachestvennye metody v mekhanike sploshnykh sred, Nauka, M., 1989 | MR

[5] Arutyunyan N. Kh., Abramyan B. L., Kruchenie uprugikh tel, GIFML, M., 1963

[6] Bañuelos R., Carroll T., “Brownian motion and the fundamental frequency of a drum”, Duke Math. J., 75:3 (1994), 575–602 | DOI | MR | Zbl

[7] Burago Yu. D., Zalgaller V. A., Geometricheskie neravenstva, Nauka, L., 1980 | MR | Zbl

[8] Kudryavtsev L. D., Nikolskii S. M., “Prostranstva differentsiruemykh funktsii mnogikh peremennykh i teoremy vlozheniya”, Itogi nauki i tekhniki. Sovr. problemy matem. Fundament. napravleniya, 26, VINITI, M., 1988, 5–157 | MR

[9] Mazya V. G., “Klassy oblastei, mer i emkostei v teorii prostranstv differentsiruemykh funktsii”, Itogi nauki i tekhniki. Sovr. problemy matem. Fundament. napravleniya, 26, VINITI, M., 1988, 159–228 | MR

[10] Fujii N., “Second order necessary conditions in a domain optimization problem”, J. Optim. Theory Appl., 65:2 (1990), 223–244 | DOI | MR | Zbl

[11] Hebey E., Sobolov Spaces on Riemannian Manifolds, Lecture Notes in Math., 1635, 1996 | MR | Zbl

[12] Avkhadiev F. G., “Reshenie zadach B. Sen-Venana i Dzh. Releya ob otsenkakh zhestkosti krucheniya i osnovnoi chastoty”, Teoriya funktsii i ee prilozheniya, Tezisy dokladov shkoly-konferentsii (15–22 iyunya 1995 g.), Kazanskii fond “Matematika”, Kazan, 1995, 3–4

[13] Avkhadiev F. G., “Geometricheskie kharakteristiki oblastei, ekvivalentnye normam nekotorykh operatorov vlozheniya”, Materialy Mezhd. konf. i chebyshevskikh chtenii, T. 1, Izd-vo mekhmata MGU, M., 1996, 12–14

[14] Goluzin G. M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966 | MR

[15] Nevanlinna R., Uniformizatsiya, IL, M., 1955

[16] Khardi G., Littlvud D., Polia G., Neravenstva, IL, M., 1948

[17] Bañuelos R., Carroll T., “An Improvement of the Osserman Constant for the Bass Note of a Drum”, Proc. Sympos. Pure Math., 57 (1995), 3–10 | MR | Zbl

[18] Avkhadiev F. G., Konformnye otobrazheniya i kraevye zadachi, Kazanskii fond “Matematika”, Kazan, 1996 | MR | Zbl

[19] Kra I., Avtomorfnye formy i kleinovy gruppy, Mir, M., 1975 | MR | Zbl