Inequalities between Dirichlet and Neumann eigenvalues in large dimensions

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