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We establish some generalizations of the classical inequalities by Polya–Szego and Makai about torsional rigidity of convex domains. The main idea of the proof is in using an exact isoperimetric inequality for Euclidean moments of domains. This inequality has a wide class of extremal regions and is of independent interest. [ABSTRACT FROM AUTHOR]

Let u( x,G) be the stress function of a multiply connected plane domain G. We construct new domain functionals depending on this stress function which are isoperimetrically monotone with respect to the free parameter. A particular case of the proved result is the Payne inequality for the torsional rigidity of G. [ABSTRACT FROM AUTHOR]

We consider integral functionals of a simply connected domain which depend on the distance to the domain boundary. We prove an isoperimetric inequality generalizing theorems derived by the Schwarz symmetrization method. For L-norms of the distance function we prove an analog of the Payne inequality for the torsional rigidity of the domain. In compare with the Payne inequality we find new extremal domains different from a disk. [ABSTRACT FROM AUTHOR]

We consider the Saint-Venant functional P for the torsional rigidity in an arbitrary plane or spatial domain. The main result of this paper is the sharp estimate P ≤ (4/ n) m, where n is the dimension of the space, and m is the harmonic mean of moments of inertia of the domain about the coordinate planes. The extremal domains are ellipsoids of a special kind. Thus, we obtain a generalization of the isoperimetric inequality proved by E. Nicolay for the torsional rigidity of simply connected plane domains. [ABSTRACT FROM AUTHOR]

Let G be a simply connected domain and let u( x,G) be its warping function. We prove that L p-norms of functions u and u−1 are monotone with respect to the parameter p. This monotony also gives isoperimetric inequalities for norms that correspond to different values of the parameter p. The main result of this paper is a generalization of classical isoperimetric inequalities of St.Venant-Pólya and the Payne inequalities. [ABSTRACT FROM AUTHOR]