分数阶Malmheden定理

IF 1.4 4区工程技术 Q3 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Mathematics in Engineering Pub Date : 2022-03-14 DOI:10.3934/mine.2023024

S. Dipierro, G. Giacomin, E. Valdinoci

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引用次数: 2

摘要

我们提供了Schwarz和Malmheden关于调和函数的经典结果的分数对应物。由此得到$ s $-调和函数作为加权经典调和函数的线性叠加的表示公式，并对分数阶的哈纳克不等式作了新的证明。这个证明也得到了分数阶哈纳克不等式在球中的最优常数。

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The fractional Malmheden theorem

We provide a fractional counterpart of the classical results by Schwarz and Malmheden on harmonic functions. From that we obtain a representation formula for $ s $-harmonic functions as a linear superposition of weighted classical harmonic functions which also entails a new proof of the fractional Harnack inequality. This proof also leads to optimal constants for the fractional Harnack inequality in the ball.

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