Inequalities between Dirichlet and Neumann eigenvalues in large dimensions

Abstract

Let $\Omega$ be a bounded domain in $R^d$. Denote by ${\lambda }_k$ (resp. ${\mu }_k$) the eigenvalues of the Laplace operator in $\Omega$ with Dirichlet (resp. Neumann) boundary conditions. Denote by $\Psi =\Psi (d,k,\Omega )$ the shift of indices in the inequality ${\mu }_{k+\Psi }\le {\lambda }_k$. We are interested to describe the behaviour of $\Psi$ for large $d$. We prove that a) $\Psi (d,1,\Omega )\ge C(e/2{)}^d$ for all domains $\Omega$; and b) $\Psi (d,k,\Omega )\ge C(e/2{)}^d$ for all $k$ and all convex domains $\Omega$.