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The anisotropic p-capacity and the anisotropic Minkowski inequality

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Abstract

In this paper, we prove a sharp anisotropic Lp Minkowski inequality involving the total Lp anisotropic mean curvature and the anisotropic p-capacity for any bounded domains with smooth boundary in ℝn. As consequences, we obtain an anisotropic Willmore inequality, a sharp anisotropic Minkowski inequality for outward F-minimising sets and a sharp volumetric anisotropic Minkowski inequality. For the proof, we utilize a nonlinear potential theoretic approach which has been recently developed by Agostiniani et al. (2019).

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11871406). The authors thank Dr. Mattia Fogagnolo for attracting their attention to the recent preprint [17] and explaining the new reformation of the strictly outward minimising hull to them. The authors also thank Professors Virginia Agostiniani, Lorenzo Mazzieri and Deping Ye for their interest.

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  1. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, China

    Chao Xia & Jiabin Yin

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  1. Chao Xia
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  2. Jiabin Yin
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Correspondence to Chao Xia.

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Xia, C., Yin, J. The anisotropic p-capacity and the anisotropic Minkowski inequality. Sci. China Math. 65, 559–582 (2022). https://doi.org/10.1007/s11425-021-1884-1

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