Upper bounds for higher-order Poincaré constants

Abstract

Here we introduce higher-order Poincaré constants for compact weighted manifolds and estimate them from above in terms of subsets. These estimates imply upper bounds for eigenvalues of the weighted Laplacian and the first nontrivial eigenvalue of the p p -Laplacian. In the case of the closed eigenvalue problem and the Neumann eigenvalue problem these are related to the estimates obtained by Chung-Grigor’yan-Yau and Gozlan-Herry. We also obtain similar upper bounds for Dirichlet eigenvalues and multi-way isoperimetric constants. As an application, for manifolds with boundary of nonnegative dimensional weighted Ricci curvature, we give upper bounds for inscribed radii in terms of dimension and the first Dirichlet Poincaré constant.