On the constancy of the extremal function in the embedding theorem of fractional order

2020/11/21 by Nikita Ustinov, Ustinov, Nikita
Computer Science · Mathematics · #34A05 #34A08 #35R11 #47J30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations #math.AP #msc:34A05 #msc:34A08 #msc:35R11 #msc:47J30

paper · pdf · doi:10.48550/arxiv.2011.10811

12 pages

arxiv created 2020/11/21 · openalex publication_date 2020/11/21 · arxiv updated 2020/11/24 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

We consider the problem of the minimizer constancy in the fractional embedding theorem H^s(Ω) \hookrightarrow L_q(Ω) for a bounded Lipschitz domain Ω, depending on the domain size. For the family of domains ε Ω, we prove that for small dilation coefficients ε a unique minimizer is constant, whereas for large ε a constant function is not even a local minimizer. We also discuss whether a constant function is a global minimizer if it is a local one.

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