A geometrical approach to the sharp Hardy inequality in Sobolev-Slobodecki\uı spaces
arXiv:2407.08373 · doi:10.1016/j.na.2025.113948
Abstract
We give a partial negative answer to a question left open in a previous work by Brasco and the first and third-named authors concerning the sharp constant in the fractional Hardy inequality on convex sets. Our approach has a geometrical flavor and equivalently reformulates the sharp constant in the limit case as the Cheeger constant for the fractional perimeter and the Lebesgue measure with a suitable weight. As a by-product, we obtain new lower bounds on the sharp constant in the -dimensional case, even for non-convex sets, some of which optimal in the case .
29 pages. Updates: several improvements in the geometrical approach, minor typos amended