Eigenvalue lower bounds through a generalized inradius

2025/09/23 by Frank, Rupert L., Laptev, Ari, Suragan, Durvudkhan
#35P15 #58J50 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2509.18878

Abstract

Lieb has shown a lower bound on the smallest Dirichlet eigenvalue of the Laplace operator in terms of a generalized inradius. We derive similar bounds for Robin eigenvalues, for eigenvalues of the polyharmonic operator and the sub-Laplacian on the Heisenberg group. We propose a method based on Hardy inequalities that is different from Lieb's approach.

Citations

• On Courant and Pleijel theorems for sub-Riemannian Laplacians
• Two consequences of Davies's Hardy inequality
• A geometric characterization of a sharp Hardy inequality
• The Hardy-Rellich Inequality for Polyharmonic Operators

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• Universal inequalities for the eigenvalues of a power of the Laplace operator
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• Estimates for lower bounds of eigenvalues of the poly-Laplacian and quadratic polynomial operator of the Laplacian
• A Generalization of a Levitin and Parnovski Universal Inequality for Eigenvalues