A geometrical isoperimetric inequality and applications to problems of mathematical physics

Abstract

The classical isoperimetric inequality states that among all closed curves of given circumference the circle encloses the largest area. This inequality has been considerably generalized by A. D. Alexandrow. He derived [1] inequalities for the case where the curve lies on an abstract surface, and obtained lower bounds for the length of the curve in terms of the area of the domain and an expression involving the curvature of the surface. In this paper we consider a curve Fo on an abstract surface whose endpoints lie on a curve F 1. With the help of Alexandrow's inequality we construct lower bounds for the length of Fo. These bounds depend on the area of the domain between Fo and F1, the curvature of the surface and the geodesic curvature of F~. By use of the geometrical inequalities we derive a monotony property of the Green's function. The geometrical inequalities lead also to an estimate for the fundamental frequency of an inhomogeneous membrane with partially free boundary. The result extends the Rayleigh-Faber-Krahn inequality [12] and its generalizations obtained by Nehari [11] and the author [2, 3, 6]. At the end we indicate how to generalize the concept of Schwarz symmetrization [12] for functions which do not vanish at the whole boundary. This symmetrization combines in a certain way the ones defined in [2] and [3]. The principal results of this paper have already been announced in [4].