In 1989 Almgren and Lieb proved a rearrangement inequality for the Sobolev spaces of fractional order Ws; p. The case p ¼ 2 of their result implies the nonlocal isoperimetric inequality PsEÞ jEj N2s N b PsB1Þ jB1j N_2s N ; 0 < s < 1=2; where Ps indicates the fractional s-perimeter, and B1 is the unit ball in RN. In this note we explicitly compute the best constant, and show that for any 0 < s < 1=2, one has PsB1Þ jB1j N2s N ¼ NpN 2 þsG12sÞ sGN 2 þ 12s NG1sÞGNþ22s 2:.
On the best constant in the nonlocal isoperimetric inequality of Almgren and Lieb
Garofalo N.
2020
Abstract
In 1989 Almgren and Lieb proved a rearrangement inequality for the Sobolev spaces of fractional order Ws; p. The case p ¼ 2 of their result implies the nonlocal isoperimetric inequality PsEÞ jEj N2s N b PsB1Þ jB1j N_2s N ; 0 < s < 1=2; where Ps indicates the fractional s-perimeter, and B1 is the unit ball in RN. In this note we explicitly compute the best constant, and show that for any 0 < s < 1=2, one has PsB1Þ jB1j N2s N ¼ NpN 2 þsG12sÞ sGN 2 þ 12s NG1sÞGNþ22s 2:.File in questo prodotto:
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