Free membrane eigenvalues

Abstract

For a simply connected and normalized domain D in the plane it was proven by Polya and Schiffer in 1954 for the fixed membrane eigenvalues $$\sum \limits^{n}_{1} \frac{1}{{\lambda}_j} \geq \sum \limits^{n}_{1} \frac{1}{{\lambda}^{(0)}_j}$$ for any n, where λj(0) are the fixed membrane eigenvalues of the unit disk. The purpose of this paper is to prove an analog inequality for the free membrane eigenvalues.