On the Constancy of the Extremal Function in the Embedding Theorem of Fractional Order

We consider the problem of the constancy of the minimizer 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 $\varepsilon$, the 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.