Improved inequalities between Dirichlet and Neumann eigenvalues of the biharmonic operator

Abstract

We prove that the ( k + d ) (k+d) -th Neumann eigenvalue of the biharmonic operator on a bounded connected d d -dimensional ( d ≥ 2 ) (d\ge 2) Lipschitz domain is not larger than its k k -th Dirichlet eigenvalue for all k ∈ N k\in \mathbb {N} . For a special class of domains with symmetries we obtain a stronger inequality. Namely, for this class of domains, we prove that the ( k + d + 1 ) (k+d+1) -th Neumann eigenvalue of the biharmonic operator does not exceed its k k -th Dirichlet eigenvalue for all k ∈ N k\in \mathbb {N} . In particular, in two dimensions, this special class consists of domains having an axis of symmetry.