Geodesic
On the Constancy of the Extremal Function in the Embedding Theorem of Fractional Order
Funkcionalʹnyj analiz i ego priloženiâ, Tome 54 (2020) no. 4, pp. 85-97.

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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.
Keywords: fractional Laplace operators, constancy of the minimizer, spectral Dirichlet Laplacian. 
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N. S. Ustinov. On the Constancy of the Extremal Function in the Embedding Theorem of Fractional Order. Funkcionalʹnyj analiz i ego priloženiâ, Tome 54 (2020) no. 4, pp. 85-97. http://geodesic.mathdoc.fr/item/FAA_2020_54_4_a6/

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