Some inequalities involving eigenvalues of the Neumann Laplacian

This paper is concerned with the eigenvalues of the Neumann Laplacian on various classes of domains of given measure: simply‐connected Lipschitz planar domains, n‐sided planar polygons and smooth N‐dimensional domains. In each case, we consider some quantities involving low eigenvalues of the Neumann Laplacian for which we obtain new inequalities. Moreover, we sharpen a universal bound derived by M. Ashbaugh and R. Benguria for sum of reciprocal of Neumann eigenvalues. Our investigations make use of some properties of conformal mappings, Bessel functions, symmetric domains or some isoperimetric inequalities for moments of inertia. Copyright © 2013 John Wiley & Sons, Ltd.