On eigenvalue inequalities of Schmuckenschlager
Classical Analysis and ODEs
2023-05-10 v1 Functional Analysis
Abstract
About ten years ago, Schmuckenschl\"ager proved that the lowest eigenvalue of Dirichlet Laplacian for the intersection of two balls (i.e., convex, symmetric and compact subsets of with non-empty interior) is less than the sum of the lowest eigenvalue for each. His arguments rely on Kac's formula, the log-concavity of Gaussian measures, the symmetry of balls and Lieb's classical result for the intersection of two domains. In this note we revisit Schmuckenschl\"ager's proof and propose a direct and elementary Lieb-free approach to these inequalities.
Keywords
Cite
@article{arxiv.2305.05264,
title = {On eigenvalue inequalities of Schmuckenschlager},
author = {Yi C. Huang},
journal= {arXiv preprint arXiv:2305.05264},
year = {2023}
}