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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     author = {N. S. Ustinov},
     title = {On the {Constancy} of the {Extremal} {Function} in the {Embedding} {Theorem} of {Fractional} {Order}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {85--97},
     publisher = {mathdoc},
     volume = {54},
     number = {4},
     year = {2020},
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     url = {http://geodesic.mathdoc.fr/item/FAA_2020_54_4_a6/}
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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/