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

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.