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

TL;DR

This paper investigates how the minimizer in the fractional embedding theorem behaves with domain size, showing that for small domains the minimizer is constant, but for large domains it is not, revealing domain-dependent properties.

Contribution

It demonstrates the domain size dependence of the minimizer's constancy in the fractional embedding theorem, providing new insights into the problem's geometric aspects.

Findings

For small domains, the minimizer is uniquely constant.

For large domains, the constant function is not a local minimizer.

Discussion on whether a local minimizer can be globally optimal.

Abstract

We consider the problem of the minimizer constancy in the fractional embedding theorem $\mathcal{H}^s(\Omega) \hookrightarrow L_q(\Omega)$ for a bounded Lipschitz domain $\Omega,$ depending on the domain size. For the family of domains $\varepsilon \Omega,$ we prove that for small dilation coefficients $\varepsilon$ a unique minimizer is constant, whereas for large $\varepsilon$ 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.