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A Lp Brunn-Minkowski Theory for Logarithmic Capacity

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Abstract

In the present paper, we first introduce the concepts of the Lp n-dimensional Logarithmic-Capacity measure and Lp mixed n-dimensional Logarithmic-Capacity and then prove some geometric properties of Lp n-dimensional Logarithmic-Capacity measure and a Lp Minkowksi inequality for the n-dimensional Logarithmic-Capacity for any fixed n ≥ 2 and p ≥ 1. As an application, we establish a Hadamard variational formula for the n-dimensional Logarithmic-Capacity under p-sum for any fixed n ≥ 2 and p ≥ 1, which extends some results of Borell (Ann. Sci. Ecole Norm. Sup. 17(3), 451–467 1984), Jerison (Adv. Math. 122(2), 262–279 1996), Colesanti-Cuoghi (Potential Anal. 22(3), 289–304 2005), Akman-Lewis-Saari-Vogel (in press) and Xiao (2013, 2018). With the Hadamard variational formula and Minkowksi inequality mentioned above, we prove the existence and uniqueness of the solution for the Lp Minkowski problem for the Logarithmic-Capacity.

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Acknowledgements

The author would like to thank heartfeltly to anonymous referee for the invaluable comments which are helpful to improve the quality of this paper.

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  1. LCSM(Ministry of Education), College of Mathematics & Statistics, Hunan Normal University, Changsha, 410081, Hunan, People’s Republic of China

    Zhengmao Chen

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The work was supported by National Natural Science Foundation of China (Grant: No.11671128). The author would like to thank Professor Qiuyi Dai for the constant encouragement and professional guidance.

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Chen, Z. A Lp Brunn-Minkowski Theory for Logarithmic Capacity. Potential Anal 54, 273–298 (2021). https://doi.org/10.1007/s11118-020-09826-8

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