Density estimates for a nonlocal variational model via the Sobolev inequality

Ovidiu Savin; Enrico Valdinoci

doi:10.1137/110831040

Density estimates for a nonlocal variational model via the Sobolev inequality

Ovidiu Savin, Enrico Valdinoci

Research output: Contribution to journal › Article › peer-review

27 Citations (Scopus)

Abstract

We consider the minimizers of the energy ∥u∥ _H ^s ²(Ω) +∫ΩW(u) dx, with s ∈ (0, 1/2), where ∥u∥ _Hs(Ω) denotes the total contribution from Ω in the H ^s norm of u, and W is a double-well potential. By using a fractional Sobolev inequality, we give a new proof of the fact that the sublevel sets of a minimizer u in a large ball BR occupy a volume comparable with the volume of B _R.

Original language	English
Pages (from-to)	2675-2687
Number of pages	13
Journal	SIAM Journal on Mathematical Analysis
Volume	43
Issue number	6
DOIs	https://doi.org/10.1137/110831040
Publication status	Published - 1 Dec 2011
Externally published	Yes

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abstract = "We consider the minimizers of the energy ∥u∥ H s 2(Ω) +∫ΩW(u) dx, with s ∈ (0, 1/2), where ∥u∥ Hs(Ω) denotes the total contribution from Ω in the H s norm of u, and W is a double-well potential. By using a fractional Sobolev inequality, we give a new proof of the fact that the sublevel sets of a minimizer u in a large ball BR occupy a volume comparable with the volume of B R.",

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AU - Valdinoci, Enrico

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AB - We consider the minimizers of the energy ∥u∥ H s 2(Ω) +∫ΩW(u) dx, with s ∈ (0, 1/2), where ∥u∥ Hs(Ω) denotes the total contribution from Ω in the H s norm of u, and W is a double-well potential. By using a fractional Sobolev inequality, we give a new proof of the fact that the sublevel sets of a minimizer u in a large ball BR occupy a volume comparable with the volume of B R.

KW - Fractional Laplacian

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KW - Phase segregation models

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Density estimates for a nonlocal variational model via the Sobolev inequality

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