In the present paper, we first introduce the concepts of the Lpq-capacity measure and Lp mixed q-capacity and then prove some geometric properties of Lpq-capacity measure and a Lp Minkowski inequality for the q-capacity for any fixed p ⩾ 1 and q > n. As an application of the Lp Minkowski inequality mentioned above, we establish a Hadamard variational formula for the q-capacity under p-sum for any fixed p ⩾ 1 and q > n, which extends results of Akman et al. (Adv. Calc. Var. (in press)). With the Hadamard variational formula, variational method and Lp Minkowski inequality mentioned above, we prove the existence and uniqueness of the solution for the Lp Minkowski problem for the q-capacity which extends some beautiful results of Jerison (1996, Acta Math.176, 1–47), Colesanti et al. (2015, Adv. Math.285, 1511–588), Akman et al. (Mem. Amer. Math. Soc. (in press)) and Akman et al. (Adv. Calc. Var. (in press)). It is worth mentioning that our proof of Hadamard variational formula is based on Lp Minkowski inequality rather than the direct argument which was adopted by Akman (Adv. Calc. Var. (in press)). Moreover, as a consequence of Lp Minkowski inequality for q-capacity, we get an interesting isoperimetric inequality for q-capacity.

中文翻译：

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q 容量的 Lp Minkowski 问题

在本文中，我们首先介绍了大号pq-容量测量和大号p混合q-容量，然后证明一些几何性质大号pq-容量测量和大号pMinkowski 不等式q- 任何固定的容量p⩾ 1 和q>n. 作为应用大号p上面提到了 Minkowski 不等式，我们建立了一个 Hadamard 变分公式q-容量下p-sum 任何固定的p⩾ 1 和q>n，扩展了 Akman 的结果等。(进阶。计算。变量。（在新闻））。用 Hadamard 变分公式、变分方法和大号p上面提到了 Minkowski 不等式，我们证明了解的存在唯一性大号p闵可夫斯基问题q- 扩展了杰里森（1996 年，数学学报。176, 1–47), 科莱桑蒂等。（2015 年，进阶。数学。285, 1511–588), 阿克曼等。(内存。阿米尔。数学。社会党。（印刷中））和阿克曼等。(进阶。计算。变量。（在新闻））。值得一提的是，我们对 Hadamard 变分公式的证明是基于大号pMinkowski 不等式而不是 Akman 采用的直接论证（进阶。计算。变量。（在新闻））。此外，由于大号pMinkowski 不等式q-容量，我们得到一个有趣的等周不等式q-容量。