Some isoperimetric inequalities for membrane frequencies and torsional rigidity

Some Isoperimetric Inequalities and Their Applications

We denote by Sn the Euclidean sphere of R" , equipped with the geodesic distance Q and a rotation invariant probability pn. For a (measurable) subset A of S,„ consider the set Au of points of Sn within geodesic distance u of A. The isoperimetric inequality for the sphere, discovered by P. Levy, is of fundamental importance. It states that p„(Au) > p„(Cu), where C is a cap (intersection of the s...

Equivalence of Some Affine Isoperimetric Inequalities

Some estimates for the torsional rigidity of composite rods

A well-known problem in elasticity consists in placing two linearly elastic materials (of different shear moduli) in a given plane domain Ω, so as to maximize the torsional rigidity of the resulting rod; moreover, the proportion of these materials is prescribed. Such a problem may not have a classical solution as the optimal design may contain homogenization regions, where the two materials are...

Some inequalities related to isoperimetric inequalities with partial free boundary

The main purpose of this paper is to prove a sharp Sobolev inequality in an exterior of a convex bounded domain. There are two ingredients in the proof: One is the observation of some new isoperimetric inequalities with partial free boundary, and the other is an integral inequality (due to Duff [9]) for any nonnegative function under Schwarz equimeasurable rearrangement. These ingredients also ...

An Isoperimetric Inequality for the Torsional Rigidity of Imperfectly Bonded Fiber Reinforced Cylinders

An isoperimetric inequality for the torsional rigidity of imperfectly bonded, fiber reinforced cylinders is found. The fiber cross sections can be simply or multiply connected. The imperfect bonding conditions are given in terms of a flexible interface model. Mathematics Subject Classifications (1991): 35J20, 73C02, 73K20.