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

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

Ps(E)∣E∣N−2sN≥Ps(B1)∣B1∣N−2sN,       0<s<1/2,\frac{P_s(E)}{|E|^{\frac{N-2s}N}} \ge \frac{P_s(B_1)}{|B_1|^{\frac{N-2s}N}},\ \ \ \ \ \ \ 0 < s < 1/2,

where PsP_s indicates the fractional ss-perimeter, and B1B_1 is the unit ball in RN\mathbb R^N. In this note we explicitly compute the best constant, and show that for any 0<s<1/20 < s < 1/2, one has

Ps(B1)∣B1∣N−2sN=NπN2+sΓ(1−2s)sΓ(N2+1)2sNΓ(1−s)Γ(N+2−2s2).\frac{P_s(B_1)}{|B_1|^{\frac{N-2s}N}} = \frac{N \pi^{\frac N2 + s} \Gamma (1-2s)}{s \Gamma (\frac N2+1)^{\frac{2s}N} \Gamma (1-s)\Gamma (\frac{N+2-2s}{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