A sharp upper bound for the torsional rigidity of rods by means of web functions

Archive for Rational Mechanics and Analysis

Volumn 164, Issue 3, 2002, Pages 189-211

  Crasta, Graziano ^a   Fragalà, Ilaria ^b   Gazzola, Filippo ^c  

^a UNIVERSITY OF MODENA AND REGGIO EMILIA   (Italy)

^b Dlpartimento Di Matematica Del Politecnico   (Italy)

^c UNIVERSITY OF EASTERN PIEDMONT   (Italy)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0036026059     PISSN: 00039527     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002050200205     Document Type: Article

References (34)

1. C. BANDLE, Isoperimetric inequalities and applications, Pitman, 1980
2. P. BAUMAN & D. PHILLIPS, A non convex variational problem related to change of phase, Appl. Math. Optim. 21, 1990, 113-138
3. W. BLASCHKE, Eine Frage über konvexe Körper, Jahresber. Deutsch. Math.-Verein. 25, 1916, 121-125
4. G. BOL, Isoperimetrische ungleichungen für bereiche auf flächen, Jahresber. Deutsch. Math.-Verein. 51, 1941, 219-257
5. T. BONNESEN & W. FENCHEL, Theory of convex bodies, BSC Associates, Moscow, Idaho, 1987
6. G. BUTTAZZO, Private communication on the occasion of the FGI2000 Conference on Optimization, Montpellier, 2000
7. A. CELLINA, Minimizing a functional depending on ∇u and on u, Ann. Inst. H. Poincaré Anal. Non Linéaire 14, 1997, 339-352
8. G. CRASTA, Variational problems for a class of functionals on convex domains, J. Differential Equations 178, 2002, 609-629
9. G. CRASTA & F. GAZZOLA, Some estimates of the minimizing properties of web functions, To appear in Calc. Var. Partial Differential Equations
10. G. CRASTA & F. GAZZOLA, Web functions: survey of results and perspectives, To appear in Rend. Ist. Mat. Univ. Trieste
11. F. FIALA, Le problème des isopérimètres sur les surfaces ouvertes à courbure positive, Comment. Math. Helv. 13, 1940/41, 293-346
12. F. GAZZOLA, Existence of minima for nonconvex functionals in spaces of functions depending on the distance from the boundary, Arch. Rational Mech. Anal. 150, 1999, 57-76
13. J. GOODMAN & R.V. KOHN & L. REYNA, Numerical study of a relaxed variational problem from optimal design, Computer Methods Appl. Mech. Engrg. 57, 1986, 107-127
14. P. HARTMAN, Geodesic parallel coordinates in the large, Amen. J. Math. 86, 1964, 705-727
15. B. KAWOHL, On the location of maxima of the gradient for solutions to quasilinear elliptic problems and a problem raised by Saint Venant, J. Elasticity 17, 1987, 195-206.
16. B. KAWOHL & J. STARA & G. WITTUM, Analysis and numerical studies of a problem of shape design, Arch. Rational Mech. Anal. 114, 1991, 349-363
17. L. D. LANDAU & E. M. LIFSHITZ, Theory of Elasticity, Course of Theoretical Physics 7, Pergamon Press, London, 1959
18. E. MAKAI, On the principal frequency of a convex membrane and related problems, Czechoslovak Math. J. 9, 1959, 66-70
19. F. MURAT & L. TARTAR, Calcul des variations et homogénéisation, Les Méthodes de l'Homogénéisation, Eds. D. BERGMAN et al., Collection de la Direction des Etudes et Recherche de l'Électricité de France, 57, 1985, 319-369
20. I. NIVEN, Maxima and minima without calculus, The Dolciani Math. Exp., The Mathematical Association of America, 1981
21. R. OSSERMAN, The isoperimetric inequality, Bull. Amen. Math. Soc. 84, 1978, 1182-1238
22. L.E. PAYNE, Some isoperimetric inequalities in the torsion problem for multiply connected regions. In Studies in Mathematical Analysis and Related Topics, University of California Press, Standford, 1962
23. L. E. PAYNE & H. F. WEINBERGER, Some isoperimetric inequalities for membrane frequencies and torsional rigidity, J. Math. Anal. Appl. 2, 1961, 210-216
24. G. PÓLYA, Torsional rigidity, principal frequency, electrostatic capacity and symmetrization, Quart. Appl. Math. 6, 1948, 267-277
25. G. PÓLYA, Two more inequalities between physical and geometrical quantities, J. Indian Math. Soc. 24, 1960, 413-419
26. G. PÓLYA & G. SZEGÖ, Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, 1951
27. G. PÓLYA & A. WEINSTEIN, On the torsional rigidity of multiply connected crosssections, Ann. of Math. 52, 1950, 154-163
28. F. RIESZ, Sur une inégalité intégrale, J. London Math. Soc. 5, 1930, 162-168
29. B. SAINT-VENANT, Mémoire sur la torsion des prismes, Mémoires présentées par divers savants à l'Académie des Sciences 14, 1856, 233-560
30. L.A. SANTALÓ, Sobre los sistemas completos de desigualdades entre tres elementos de una figura convexa plana, Math. Notae 17, 1961, 82-104
31. R. SCHNEIDER, Convex bodies: the Brunn-Minkowski theory, Cambridge Univ. Press, 1993
32. J. SERRIN, A symmetry problem in potential theory, Arch. Rational Mech. Anal. 43, 1971, 304-318
33. J. STEINER, Über parallele Flächen, Monatsber. preuß. Acad. Wiss., Berlin, 114-118. Ges. Werke, vol. 2, Reimer, Berlin, 1882, 171-176
34. H.F. WEINBERGER, Remark on the preceding paper of Serrin, Arch. Rational Mech. Anal. 43, 1971, 319-320