On the best constant in the nonlocal isoperimetric inequality of Almgren and Lieb

Nicola Garofalo

doi:10.4171/rlm/900

JournalsrlmVol. 31, No. 2pp. 465–470

On the best constant in the nonlocal isoperimetric inequality of Almgren and Lieb

Nicola Garofalo
Università di Padova, Italy

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Abstract

In 1989 Almgren and Lieb proved a rearrangement inequality for the Sobolev spaces of fractional order $W^{s, p}$ . The case $p = 2$ of their result implies the nonlocal isoperimetric inequality

\frac{P _{s} ( E )}{∣ E ∣ ^{\frac{N - 2 s}{N}}} \geq \frac{P _{s} ( B _{1} )}{∣ B _{1} ∣ ^{\frac{N - 2 s}{N}}}, 0 < s < 1/2,

where $P_{s}$ indicates the fractional $s$ -perimeter, and $B_{1}$ is the unit ball in $R^{N}$ . In this note we explicitly compute the best constant, and show that for any $0 < s < 1/2$ , one has

\frac{P _{s} ( B _{1} )}{∣ B _{1} ∣ ^{\frac{N - 2 s}{N}}} = \frac{N π ^{\frac{N}{2} + s} Γ ( 1 - 2 s )}{s Γ ( \frac{N}{2} + 1 ) ^{\frac{2 s}{N}} Γ ( 1 - s ) Γ ( \frac{N + 2 - 2 s}{2} )} .

Cite this article

Nicola Garofalo, On the best constant in the nonlocal isoperimetric inequality of Almgren and Lieb. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 31 (2020), no. 2, pp. 465–470

DOI 10.4171/RLM/900