On the Constancy of the Extremal Function in the Embedding Theorem of Fractional Order
Nikita Ustinov · Functional Analysis and Its Applications · 2020
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.