Inequalities between Dirichlet and Neumann eigenvalues in large dimensions

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 $\Omega$.