We study the sharp constant in the Hardy inequality for fractional Sobolev spaces defined on open subsets of the Euclidean space. We first list some properties of such a constant, as well as of the associated variational problem. We then restrict the discussion to open convex sets and compute such a sharp constant, by constructing suitable supersolutions by means of the distance function. Such a method of proof works only for \$sp\ge 1\$ or for \$\Omega\$ being a half-space. We exhibit a simple example suggesting that this method can not work for \$sp&lt;1\$ and \$\Omega\$ different from a half-space. The case \$sp&lt;1\$ for a generic convex set is left as an interesting open problem, except in the Hilbertian setting (i.e. for \$p=2\$): in this case we can compute the sharp constant in the whole range \$0&lt;1\$ . This completes a result which was left open in the literature.

### On the sharp Hardy inequality in Sobolev–Slobodeckiĭ spaces

#### Abstract

We study the sharp constant in the Hardy inequality for fractional Sobolev spaces defined on open subsets of the Euclidean space. We first list some properties of such a constant, as well as of the associated variational problem. We then restrict the discussion to open convex sets and compute such a sharp constant, by constructing suitable supersolutions by means of the distance function. Such a method of proof works only for \$sp\ge 1\$ or for \$\Omega\$ being a half-space. We exhibit a simple example suggesting that this method can not work for \$sp<1\$ and \$\Omega\$ different from a half-space. The case \$sp<1\$ for a generic convex set is left as an interesting open problem, except in the Hilbertian setting (i.e. for \$p=2\$): in this case we can compute the sharp constant in the whole range \$0<1\$ . This completes a result which was left open in the literature.
##### Scheda breve Scheda completa Scheda completa (DC)
2023
Bianchi, F.; Brasco, L.; Zagati, A. C.
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in SFERA sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11392/2537011`
##### Attenzione

Attenzione! I dati visualizzati non sono stati sottoposti a validazione da parte dell'ateneo

• ND
• 1
• 2