Universal inequalities for the eigenvalues of a power of the Laplace operator

Said Ilias (1) , Ola Makhoul (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

In this paper, we obtain a new abstract formula relating eigenvalues of a self-adjoint operator to two families of symmetric and skew-symmetric operators and their commutators. This formula generalizes earlier ones obtained by Harrell, Stubbe, Hook, Ashbaugh, Hermi, Levitin and Parnovski. We also show how one can use this abstract formulation both for giving dierent and simpler proofs for all the known results obtained for the eigenvalues of a power of the Laplace operator (i.e. the Dirichlet Laplacian, the clamped plate problem for the bilaplacian and more generally for the polyharmonic problem on a bounded Euclidean domain) and to obtain new ones. In a last paragraph, we derive new bounds for eigenvalues of any power of the Kohn Laplacian on the Heisenberg group.

Keywords: eigenvalues; Laplacian; polyharmonic operator; biharmonic operator; clamped plate; Payne-Polya-Weinberger inequality; Hile-Protter inequality; Yang inequality; universal inequalities; commutators; Kohn Laplacian; Heisenberg group

Domains: Spectral Theory [math.SP]; Differential Geometry [math.DG]

Said Ilias, Ola Makhoul. Universal inequalities for the eigenvalues of a power of the Laplace operator. 2009. ⟨hal-00450662⟩