A geometrical approach to the sharp Hardy inequality in Sobolev-Slobodecki\uı spaces

2024/07/11 by Bianchi, Francesca, Stefani, Giorgio, Zagati, Anna Chiara
#46E35 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Primary 49Q20. Secondary 35R11

paper · doi:10.48550/arxiv.2407.08373

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 p=1 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 1-dimensional case, even for non-convex sets, some of which optimal in the case p=1.

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