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arXiv · 2407.08373

A geometrical approach to the sharp Hardy inequality in Sobolev-Slobodecki\u{\i} spaces

Francesca Bianchi·Giorgio Stefani·Anna Chiara Zagati
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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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Francesca Bianchi, Giorgio Stefani, Anna Chiara Zagati. 2024-07-11. A geometrical approach to the sharp Hardy inequality in Sobolev-Slobodecki\u{\i} spaces. https://doi.org/10.1016/j.na.2025.113948

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