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Anisotropic Sobolev Capacity with Fractional Order

Published:2016-03-16
Printed: Aug 2017
• Jie Xiao,
Department of Mathematics and Statistics, Memorial University, St. John's, NL A1C 5S7, Canada
• Deping Ye,
Department of Mathematics and Statistics, Memorial University, St. John's, NL A1C 5S7, Canada
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Abstract

In this paper, we introduce the anisotropic Sobolev capacity with fractional order and develop some basic properties for this new object. Applications to the theory of anisotropic fractional Sobolev spaces are provided. In particular, we give geometric characterizations for a nonnegative Radon measure $\mu$ that naturally induces an embedding of the anisotropic fractional Sobolev class $\dot{\Lambda}_{\alpha,K}^{1,1}$ into the $\mu$-based-Lebesgue-space $L^{n/\beta}_\mu$ with $0\lt \beta\le n$. Also, we investigate the anisotropic fractional $\alpha$-perimeter. Such a geometric quantity can be used to approximate the anisotropic Sobolev capacity with fractional order. Estimation on the constant in the related Minkowski inequality, which is asymptotically optimal as $\alpha\rightarrow 0^+$, will be provided.
 Keywords: sharpness, isoperimetric inequality, Minkowski inequality, fractional Sobolev capacity, fractional perimeter
 MSC Classifications: 52A38 - Length, area, volume [See also 26B15, 28A75, 49Q20] 53A15 - Affine differential geometry 53A30 - Conformal differential geometry

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