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.

中文翻译：

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关于分数阶嵌入定理中极值函数的恒常性

摘要

我们考虑分数嵌入定理\(\mathcal{H}^s(\Omega) \hookrightarrow L_q(\Omega)\) 中极小值的恒常性问题，用于有界 Lipschitz 域\(\Omega\)，取决于关于域大小。对于域族\(\varepsilon \Omega\)，我们证明，对于小膨胀系数\(\varepsilon\)，唯一的极小值是常数，而对于大\(\varepsilon\)，常数函数不是甚至是局部最小化器。我们还讨论了如果一个常数函数是一个局部函数，它是否是一个全局极小值。