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This article is cited in 4 scientific papers (total in 4 papers)
On the Constancy of the Extremal Function in the Embedding Theorem of Fractional Order
N. S. Ustinov Saint Petersburg State University, St. Petersburg, Russia
Abstract:
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.
Received: 12.08.2020 Revised: 12.08.2020 Accepted: 23.08.2020
Citation:
N. S. Ustinov, “On the Constancy of the Extremal Function in the Embedding Theorem of Fractional Order”, Funktsional. Anal. i Prilozhen., 54:4 (2020), 85–97; Funct. Anal. Appl., 54:4 (2020), 295–305
Linking options:
https://www.mathnet.ru/eng/faa3828https://doi.org/10.4213/faa3828 https://www.mathnet.ru/eng/faa/v54/i4/p85
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Abstract page: | 251 | Full-text PDF : | 35 | References: | 79 | First page: | 6 |
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