Abstract
In this paper, we consider parametric generalizations of a St. Venant type functional, defined on domains of the Euclidean space of dimension \(n\geq 2\) and connected with the torsional rigidity of a domain as well as with integrals of powers of the St. Venant stress function over simply connected plane domains. We give several estimates for the generalized functionals. In particular, for bounded convex domains we obtain an essential improvement of two known results proved by R. Bañuelos, M. van den Berg, and T. Carrol (see J. London Math. Soc. 66 (2), 499–512 (2002)) and by R.G. Salahudinov (see Russian Math. (Iz. VUZ) 50 (3), 39–46 (2006)). In addition, we examine these functionals over non convex domains in two cases when a domain has uniformly perfect boundary or it is close to convex domains in a certain sense. For such a domain we prove several new estimates using power boundary moments of domains.
