The L p Minkowski problem for q -capacity

In the present paper, we first introduce the concepts of the L p q -capacity measure and L p mixed q -capacity and then prove some geometric properties of L p q -capacity measure and a L p Minkowski inequality for the q -capacity for any fixed p ⩾ 1 and q > n . As an application of the L p 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 L p Minkowski inequality mentioned above, we prove the existence and uniqueness of the solution for the L p 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 L p Minkowski inequality rather than the direct argument which was adopted by Akman ( Adv. Calc. Var. (in press)). Moreover, as a consequence of L p Minkowski inequality for q -capacity, we get an interesting isoperimetric inequality for q -capacity.