Inequalities between Dirichlet and Neumann eigenvalues of the polyharmonic operators

Abstract

We prove that μ k + m m > λ k m \mu _{k+m}^m >\lambda _k^m , where μ k m \mu _k^m ( λ k m \lambda _k^m ) are the eigenvalues of ( − Δ ) m (-\Delta )^m on Ω ⊂ R d \Omega \subset \mathbb R^d , d ≥ 2 d\geq 2 , with Neumann (Dirichlet) boundary conditions.