Isoperimetric inequality for some eigenvalues of an inhomogeneous, free membrane

Szegos inequality concerning the second eigenvalue of a homogeneous, free membrane is extended to the case of an inhomogeneous free membrane. With the help of a variational principle and the conformal mapping technic upper bounds are constructed for the sum VjU2 + V j ^ > where ^ and ^ denote the second and third eigenvalue. These bounds only depend on the total mass of the domain and on a simple expression involving the mass distribution and its logarithm. (*) v 'This work was supported by the NSF Grant GU-2056 ISOPERIMETRIC INEQUALITY FOR SOME EIGENVALUES OF AN INHOMOGENEOUS, FREE MEMBRANE Catherine Bandle 1. This paper is concerned with the eigenvalue problem 2 a2 (A) Acp+ppcp = O in G [A = —2 + —j J IS where G is a simply connected domain in the z-plane (z=x+iy), T is its boundary and n denotes the outer normal. p(z) > O is the mass distribution. We assume that there exists a countable set of eigenvalues 0 = /i, < fi2 <. ̂ 3 <. Szego [5] proved for membranes with constant p the isoperimetric inequality