Some estimations of positive and negative eigenvalues for an inhomogeneous membrane

Abstract

In this paper we give some simple estimates of positive and negative eigenvalues for the inhomogeneous membrane boundary problem $$-\mathrm{\Delta }u=\lambda mu,u{|}_{\partial D=0}$$if $D\subset R^2$ is a simple connected domain and $m$ is a real bounded function (possibly changing the sign).