On the geometry of star domains and the spectra of Hodge-Laplace operators

TL;DR

This paper investigates Poincaré--Friedrichs--Weber constants for Sobolev forms on convex and star-shaped domains, establishing their monotonicity and providing new estimates and proofs related to domain geometry.

Contribution

It generalizes existing results by showing the nonincreasing nature of these constants in convex domains and offers improved estimates for star-shaped domains.

Findings

Poincaré--Friedrichs--Weber constants are nonincreasing with form degree on convex domains.

The Poincaré constant bounds the Poincaré--Friedrichs--Weber constants.

New Lipschitz estimates for gauge and expansion functions of convex and star domains.

Abstract

We study Poincar\'e--Friedrichs--Weber constants for Sobolev differential forms on bounded convex domains and on domains star-shaped with respect to a ball. Generalizing work by Guerini and Savo, our main result shows that the Poincar\'e--Friedrichs--Weber constants in the Sobolev de~Rham complexes on bounded convex domains are nonincreasing in the degree of the differential forms. In particular, the Poincar\'e constant is an upper bound for the Poincar\'e--Friedrichs--Weber constants. We also obtain estimates for the Poincar\'e--Friedrichs--Weber constants on domains star-shaped with respect to a ball. As preparatory work, which may be of independent interest, we study the gauge function and the expansion function of bounded convex sets and star domains, providing new proofs of Lipschitz estimates by Vre\'{c}ica and Toranzos for the expansion function and improving a Lipschitz estimate for the gauge function due to Beer.