The Lp electrostatic q-capacitary Minkowski problem for general measures

J. Math. Anal. Appl. 487 (2020) 123959

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The Lp electrostatic q-capacitary Minkowski problem for general measures ✩ Yibin Feng a , Yanping Zhou b,∗ , Binwu He c a b c

School of Mathematics and Statistics, Hexi University, Zhangye, 734000, China Department of Mathematics, China Three Gorges University, Yichang, 443002, China Department of Mathematics, Shanghai University, Shanghai, 200444, China

a r t i c l e

i n f o

Article history: Received 8 August 2019 Available online 17 February 2020 Submitted by J. Bastero Keywords: Lp Minkowski problem Electrostatic q-capacity Lp electrostatic q-capacitary measure Lp electrostatic q-capacitary Minkowski problem

a b s t r a c t Existence of solutions to the Lp Minkowski problem of electrostatic q-capacity for discrete measures was very recently given by G. Xiong, J. Xiong and L. Xu (JFA, 2019) when 0 < p < 1 and 1 < q < 2. In this paper, an existence result of solutions to this problem for general measures is proved when 0 < p < 1 and 1 < q < n, which is a supplement to the previous result. © 2020 Elsevier Inc. All rights reserved.

1. Introduction One of the most important problems in convex geometric analysis is the Minkowski problem, which has attracted great attention from many scholars, see e.g., [2,12,15,30,35–38]. Its generalization is the Lp Minkowski problem: Find necessary and sufficient conditions for a given sphere Borel measure so that it is the Lp surface area measure, see [32], of a convex body (a compact convex set with nonempty interior in the n-dimensional Euclidean space Rn ). The Lp Minkowski problem was first proposed by Lutwak [32]. Since then, it has been extensively studied, see e.g., [3–8,10,11,13,17,18,20–25,28,29,31,33,34,40,41,43–47]. The aim of this work is to focus on the Minkowski problem for capacitary measures. Let E be a compact set in the n-dimensional Euclidean space Rn . For 1 < q < n, the electrostatic q-capacity, Cq (E), of E was defined in [14] by ⎧ ⎫ ⎨ ⎬ Cq (E) = inf |∇u|q dx : u ∈ Cc∞ (Rn ) and u ≥ 1 on E , ⎩ ⎭ Rn

✩

This work was supported by the National Natural Science Foundation of China (Grant No. 11561020 and No. 11901346).

* Corresponding author. E-mail addresses: [email protected] (Y. Feng), [email protected] (Y. Zhou), [email protected] (B. He). https://doi.org/10.1016/j.jmaa.2020.123959 0022-247X/© 2020 Elsevier Inc. All rights reserved.

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Y. Feng et al. / J. Math. Anal. Appl. 487 (2020) 123959

where Cc∞ (Rn ) is the set of smooth functions with compact supports. When q = 2, the electrostatic qcapacity becomes the classical electrostatic capacity C2 (E). The electrostatic q-capacitary measure μq (Ω, ·), see [14], of a bounded open convex set Ω in Rn is the measure on the unit sphere S n−1 defined for ω ⊂ S n−1 and 1 < q < n by  |∇U |q dHn−1 ,

μq (Ω, ω) = g −1 (ω)

where g −1 : S n−1 → ∂Ω (the set of boundary points of Ω) denotes the inverse Gauss map, Hn−1 the (n − 1)-dimensional Hausdorff measure, and U the q-equilibrium potential of Ω (see [14, p. 1515]). In [14], the authors showed the following Hadamard variational formula for the electrostatic q-capacity: For two bounded open convex sets Ω, Ω1 ⊂ Rn and 1 < q < n,  d Cq (Ω + tΩ1 ) + = (q − 1) dt t=0

hΩ1 (u)dμq (Ω, u).

(1.1)

S n−1

Here, Ω+tΩ1 is the Minkowski addition: Ω+tΩ1 = {x+ty : x ∈ Ω, y ∈ Ω1 } and hΩ1 (u) = max {x · u : x ∈ Ω1 } is the support function of Ω1 with x · u denoting the standard inner product of x and u. Variational formula (1.1) leads to the following Poincaré q-capacity formula: Cq (Ω) =

q−1 n−q

 hΩ (u)dμq (Ω, u). S n−1

An important and natural problem prescribing the electrostatic q-capacitary measure is the Minkowski type problem posed in [14]: Suppose 1 < q < n and μ is a finite Borel measure on the unit sphere S n−1 . Under what necessary and sufficient conditions does there exist a bounded open convex set Ω such that μq (Ω, ·) = μ? When q = 2, the Minkowski type problem is the classical electrostatic capacitary Minkowski problem. In the paper [26], Jerison established the existence of a solution to the electrostatic capacitary Minkowski problem. In a subsequent paper [27], he gave a new proof of this result using a variational approach. The uniqueness of its solutions was proved by Caffarelli, Jerison and Lieb [9], and the regularity was given in [26]. In [14], the existence and regularity for 1 < q < 2 and the uniqueness for 1 < q < n of the solutions to the Minkowski type problem were proved, and in [1], the existence for 2 < q < n was very recently solved. The Lp electrostatic q-capacitary measure was introduced by Zou and Xiong [48]: Let p ∈ R and 1 < q < n. For a convex body K in Rn that contains the origin in its interior, the Lp electrostatic q-capacitary measure, μp,q (K, ·), of K is a finite Borel measure on S n−1 defined for a Borel ω ⊂ S n−1 by  h1−p K (u)dμq (K, u).

μp,q (K, ω) =

(1.2)

ω

The problem characterizing the Lp electrostatic q-capacitary measure is the following (see [48]). The Lp electrostatic q-capacitary Minkowski problem: Let p ∈ R and 1 < q < n. Given a finite Borel measure μ on the unit sphere S n−1 , what are the necessary and sufficient conditions so that μ = μp,q (K, ·) for some convex body K in Rn ? The Lp electrostatic q-capacitary Minkowski problem for the case p > 1 and 1 < q < n was essentially solved by Zou and Xiong [48]. However, the critical cases p < 1 and 1 < q < n remain open. Very recently,

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when the given measure is discrete, a breakthrough for the case 0 < p < 1 and 1 < q < 2 was made by G. Xiong, J. Xiong and L. Xu [42]. In this paper, we will supplement the result for 0 < p < 1 and 1 < q < 2, i.e., the Lp electrostatic q-capacitary Minkowski problem when 0 < p < 1 and 1 < q < n is solved in the general measures. Thus, the following result is obtained. Theorem 1.1. Let 0 < p < 1 and 1 < q < n, and let μ be a finite Borel measure on S n−1 which is not concentrated on any closed hemisphere. If p + q = n, then there exists a convex body K in Rn such that μ = μp,q (K, ·); if p + q = n, then there exist a convex body K in Rn and a constant λ > 0 such that μ = λμp,q (K, ·). The rest of this paper is organized as follows. In Section 2, some necessary facts about convex bodies and capacity are provided. In Section 3, a minimizing problem related to the Lp electrostatic q-capacitary Minkowski problem is considered and its corresponding solution is given. In Section 4, we will show that the solution of the minimizing problem is exactly the solution to the related Minkowski problem in the discrete case. Then we use approximation to complete the proof of Theorem 1.1. 2. Preliminaries 2.1. Basics regarding convex bodies One can consult [16,19,39] for general references about convex bodies. √ We will work in Rn equipped with the standard Euclidean norm. For x ∈ Rn , |x| = x · x denotes the Euclidean norm of x. The unit ball will be denoted by B = {x ∈ Rn : |x| ≤ 1}. We will write C(S n−1 ) for the set of continuous functions on S n−1 and C + (S n−1 ) for the set of positive functions in C(S n−1 ). The Hausdorff distance of two convex bodies K, L in Rn is defined by δ(K, L) = max |hK (u) − hL (u)|. n−1 u∈S

Suppose Ki is a sequence of convex bodies in Rn . We say Ki converges to a convex body K0 ⊂ Rn if δ(Ki , K0 ) → 0, as i → ∞. For g ∈ C + (S n−1 ) and a closed subset Ω ⊂ S n−1 not contained in any closed hemisphere, the Aleksandrov body associated to (g, Ω), denoted by [g], is the convex body defined by

[g] = {ξ ∈ Rn : ξ · u ≤ g(u)}. (2.1) u∈Ω

It is easy to see that h[g] ≤ g and for every convex body K ⊂ Rn containing the origin in its interior, then [hK ] = K if Ω = S n−1 . For a convex body K in Rn , the diameter of K is defined by D(K) = max{|x − y| : x, y ∈ K}. The following lemma is from [39, Theorem 1.8.8]. Lemma 2.1. For convex bodies Ki and K in Rn , the convergence limi→∞ Ki = K is equivalent to the following conditions taken together: (1) each point in K is the limit of a sequence {xi } with xi ∈ Ki for i ∈ N; (2) the limit of any convergent sequence {xij } with xij ∈ Kij for j ∈ N belongs to K.

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2.2. Electrostatic q-capacity and q-capacitary measure Here, we collect some necessary facts on electrostatic q-capacity and q-capacitary measure (see [14,42,48]). The following lemma gives some basic properties of the electrostatic q-capacity. Lemma 2.2. Let E and F be two compact sets in Rn and 1 < q < n. (i) If E ⊂ F , then Cq (E) ≤ Cq (F ). (ii) For λ > 0, Cq (λE) = λn−q Cq (E). (iii) For x0 ∈ Rn , Cq (E + x0 ) = Cq (E). (iv) The functional Cq (·) is continuous with respect to the Hausdorff metric. The following lemma provides some basic properties of the electrostatic q-capacitary measure. Lemma 2.3. Let K be a convex body in Rn and 1 < q < n. (i) For λ > 0, μq (λK, ·) = λn−q−1 μq (K, ·). (ii) For x0 ∈ Rn , μq (K + x0 , ·) = μq (K, ·). (iii) For convex bodies Kj , K, if Kj → K, then μq (Kj , ·) → μq (K, ·) weakly as j → +∞. (iv) The measure μq (K, ·) is absolutely continuous with respect to the surface area measure, see [39], S(K, ·). The following variational formula given in [14] of electrostatic q-capacity plays a crucial role in the paper. Lemma 2.4. Let I ⊂ R be an interval containing 0 in its interior, and let ht (u) = h(t, u) : I ×S n−1 → (0, ∞) be continuous such that the convergence in h (0, u) = lim

t→0

h(t, u) − h(0, u) t

is uniform on S n−1 . Then  dCq ([ht ]) = (q − 1) dt t=0

S n−1

h (0, u)dμq ([h0 ], u).

(2.2)

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3. The minimization problem In this section, we solve a minimization problem. Its solution also solves the Lp electrostatic q-capacitary Minkowski problem for discrete measures. Suppose that μ is a finite discrete measure on S n−1 which is not concentrated on any closed hemisphere m of S n−1 , that is, μ = i=1 αi δui where δui is Kronecker delta, α1 , . . . , αm > 0, and u1 , . . . , um ∈ S n−1 are not concentrated on any closed hemisphere. Let g ∈ C + (S n−1 ). Then for the Aleksandrov body associated to (g, supp(μ)), we will denote it by [g]μ . Thus, for 0 < p < 1, we define the function, Gg,μ : [g]μ → R, by 

 (g(u) − ξ · u)p dμ(u) =

Gg,μ (ξ) = S n−1

(g(u) − ξ · u)p dμ(u),

(3.1)

supp(μ)

and consider the following minimizing problem:

max Gg,μ (ξ) : g ∈ C (S +

inf

ξ∈[g]μ

n−1

) and Cq ([g]μ ) = 1 .

(3.2)

We shall require the following two lemmas to solve problem (3.2). Lemma 3.1. Let 0 < p < 1. If μ is a finite discrete measure on S n−1 which is not concentrated in any closed hemisphere of S n−1 , then Gg,μ is strictly concave on [g]μ for g ∈ C + (S n−1 ). Proof. Note that for ξ ∈ [g]μ and u ∈ supp(μ), g(u) − ξ · u ≥ h[g]μ (u) − ξ · u ≥ 0. Since tp , 0 < p < 1, is strictly concave on [0, +∞), we get that for 0 < λ < 1 and ξ1 , ξ2 ∈ [g]μ , Gg,μ (λξ1 + (1 − λ)ξ2 )  = (g(u) − (λξ1 + (1 − λ)ξ2 ) · u)p dμ(u) S n−1



p

(λ(g(u) − ξ1 · u) + (1 − λ)(g(u) − ξ2 · u)) dμ(u)

= S n−1





≥λ

(g(u) − ξ1 · u)p dμ(u) + (1 − λ)

S n−1

(g(u) − ξ2 · u)p dμ(u)

S n−1

= λGg,μ (ξ1 ) + (1 − λ)Gg,μ (ξ2 ), with equality if and only if g(u) − ξ1 · u = g(u) − ξ2 · u for any u ∈ supp(μ), i.e., (ξ1 − ξ2 ) · u = 0. Since μ is not concentrated on any closed hemisphere, supp(μ) spans the whole space Rn . Hence,

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ξ1 = ξ2 , which implies that Gg,μ is strictly concave on [g]μ . Lemma 3.2. Let 0 < p < 1 and g ∈ C + (S n−1 ). If μ is a finite discrete measure on S n−1 which is not concentrated in any closed hemisphere of S n−1 , then there exists a unique ξg ∈ int[g]μ such that Gg,μ (ξg ) = max Gg,μ (ξ), ξ∈[g]μ

and ξg depends continuously on g. Proof. Our proof is based on the techniques in [28]. Since Gg,μ is continuous and strictly concave on [g]μ , there exists a unique ξg ∈ [g]μ such that Gg,μ (ξg ) = max Gg,μ (ξ). ξ∈[g]μ

We next prove ξg ∈ int[g]μ by contradiction. Let ξg ∈ ∂[g]μ . Then recalling [g]μ =



{ξ ∈ Rn : ξ · u ≤ g(u)},

u∈supp(μ)

we claim that there exists u ∈ supp(μ) such that ξg · u = g(u).

(3.3)

Otherwise, ξg · u < g(u) for all u ∈ supp(μ). Thus, for some δ1 > 0 and all u ∈ supp(μ), we have ξg · u + δ1 < g(u). That is, (ξg + δ1 u) · u < g(u). This implies that ξg ∈ int[g]μ which is a contradiction. Let supp(μ) = A ∪ B,

(3.4)

where A := {u ∈ supp(μ) : ξg · u = g(u)} and B := {u ∈ supp(μ) : ξg · u < g(u)}. Then, from (3.3), and noting that μ is not concentrated in any closed hemisphere of S n−1 , we see that A and B are two disjoint nonempty sets. By the definition of A, and noting g ∈ C + (S n−1 ), we can find a unit vector u0 ∈ S n−1 such that for all u ∈ A,

Y. Feng et al. / J. Math. Anal. Appl. 487 (2020) 123959

u0 · u < 0.

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(3.5)

From these facts we obtain that B is a closed subset of S n−1 and ξg · u − g(u), u ∈ S n−1 , is continuous. Therefore, there exists a constant a > 0 such that for any u ∈ B ξg · u + 2a < g(u).

(3.6)

Thus, we have that for any 0 < λ < 2a and any u ∈ supp(μ), we have (ξg + λu0 ) · u < g(u). This implies that there exists some δ2 > 0 such that for all u ∈ supp(μ), (ξg + λu0 + δ2 u) · u < g(u), i.e., ξ(λ) := ξg + λu0 ∈ int[g]μ . Hence, it follows from (3.1) and (3.4) that Gg,μ (ξ(λ)) − Gg,μ (ξg )   = (g(u) − ξ(λ) · u)p dμ(u) − (g(u) − ξg · u)p dμ(u) A∪B

A∪B



(g(u) − ξ(λ) · u)p dμ(u)

= A

 (g(u) − ξ(λ) · u)p − (g(u) − ξg · u)p dμ(u).

+

(3.7)

B

For all u ∈ A and some constant δ3 > 0, inequality (3.5) can be strengthened by u0 · u < −δ3 < 0. Thus, for all u ∈ A, g(u) − ξ(λ) · u = −λu0 · u > λδ3 . From this, we see 

 (g(u) − ξ(λ) · u)p dμ(u) >

A

(λδ3 )p dμ(u) = (λδ3 )p μ(A). A

By (3.6), it follows that for 0 < λ < a and any u ∈ B, g(u) − ξ(λ) · u = g(u) − ξg · u − λu0 · u > 2a − λ > a.

(3.8)

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Hence, for 0 < p < 1, |(g(u) − ξ(λ) · u)p − (g(u) − ξg · u)p | < pap−1 | − λu0 · u| ≤ λpap−1 . Thus,  (g(u) − ξ(λ) · u)p − (g(u) − ξg · u)p dμ(u) B

 ≤ (g(u) − ξ(λ) · u)p − (g(u) − ξg · u)p dμ(u) B  ≤ |(g(u) − ξ(λ) · u)p − (g(u) − ξg · u)p | dμ(u) B

< λpap−1 μ(B).

(3.9)

Combining (3.7), (3.8) and (3.9), it follows that Gg,μ (ξ(λ)) − Gg,μ (ξg ) > (λδ3 )p μ(A) − λpap−1 μ(B) = λp (δ3p μ(A) − pλ1−p ap−1 μ(B)). This implies that there exists a small enough λ0 with 0 < λ0 < a such that ξ(λ0 ) ∈ int[g]μ and Gg,μ (ξ(λ0 )) − Gg,μ (ξg ) > λp0 (δ3p μ(A) − pλ1−p ap−1 μ(B)) > 0. 0 Namely, Gg,μ (ξ(λ0 )) > Gg,μ (ξg ) which is a contradiction. Therefore, ξg ∈ int[g]μ . Let g ∈ C + (S n−1 ), and {gk } ⊂ C + (S n−1 ) be any sequence of functions uniformly converging to g on n−1 S . The following is to show that ξgk converges to ξg in Rn . Since ξgk ∈ [gk ]μ and [gk ]μ → [g]μ when gk → g uniformly on S n−1 (see [39, Lemma 7.5.2]), ξgk is bounded. Thus, we let {ξgki } ⊂ {ξgk } be any convergent subsequence and ξgki → ξ0 as i → +∞. We next prove ξ0 = ξg . Let ξ ∈ [g]μ . Since [gki ]μ → [g]μ as i → +∞, it follows from Lemma 2.1 that there exists a sequence of ξki ∈ [gki ]μ such that ξki → ξ as i → +∞. Thus,  (g(u) − ξ · u)p dμ(u)

Gg,μ (ξ) = S n−1



= lim

i→+∞ S n−1

(gki (u) − ξki · u)p dμ(u)



≤ lim

i→+∞ S n−1

(gki (u) − ξgki · u)p dμ(u)



(g(u) − ξ0 · u)p dμ(u)

= S n−1

= Gg,μ (ξ0 ).

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Therefore, max Gg,μ (ξ) = Gg,μ (ξ0 ).

ξ∈[g]μ

From the uniqueness of ξg , we have ξg = ξ0 . This proves ξgk → ξg . We now solve problem (3.2). Theorem 3.1. Let 0 < p < 1 and 1 < q < n. If μ is a finite discrete measure on S n−1 which is not concentrated in any closed hemisphere of S n−1 , then there exists a function h ∈ C + (S n−1 ) with ξh = o and Cq ([h]μ ) = 1 such that

max Gg,μ (ξ) : g ∈ C (S +

Gh,μ (o) = inf

ξ∈[g]μ

n−1

) and Cq ([g]μ ) = 1 .

(3.10)

Proof. Let {gk } ⊂ C + (S n−1 ), Cq ([gk ]μ ) = 1, and lim Ggk ,μ (ξgk ) = inf

k→+∞

max Gg,μ (ξ) : g ∈ C + (S n−1 ) and Cq ([g]μ ) = 1 .

ξ∈[g]μ

(3.11)

Define hk = h[gk ]μ . Then we know that for u ∈ supp(μ), hk (u) ≤ gk (u) and [hk ]μ = [h[gk ]μ ]μ = [gk ]μ . Since o ∈ int[gk ]μ , we have hk ∈ C + (S n−1 ). Thus, for any ξ ∈ [hk ]μ = [gk ]μ , we get  (hk (u) − ξ · u)p dμ(u)

Ghk ,μ (ξ) = supp(μ)



≤

(gk (u) − ξ · u)p dμ(u) supp(μ)

= Ggk ,μ (ξ). From this, we have max Ghk ,μ (ξ) ≤ max Ggk ,μ (ξ).

ξ∈[hk ]μ

ξ∈[gk ]μ

Therefore, lim

max Ghk ,μ (ξ) ≤ lim

k→+∞ ξ∈[hk ]μ

max Ggk ,μ (ξ).

k→+∞ ξ∈[gk ]μ

(3.12)

Since hk ∈ C + (S n−1 ) and Cq ([hk ]μ ) = Cq ([gk ]μ ) = 1, from (3.11) we have lim

max Ghk ,μ (ξ) ≥ lim

k→+∞ ξ∈[hk ]μ

max Ggk ,μ (ξ).

k→+∞ ξ∈[gk ]μ

From (3.12) and (3.13), it follows that lim

max Ghk ,μ (ξ) = lim

k→+∞ ξ∈[hk ]μ

max Ggk ,μ (ξ).

k→+∞ ξ∈[gk ]μ

(3.13)

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By (3.11), this has lim Ghk ,μ (ξhk ) = inf

k→+∞

max Gg,μ (ξ) : g ∈ C (S +

n−1

ξ∈[g]μ

) and Cq ([g]μ ) = 1 .

Recalling [hk ]μ = [gk ]μ , we have hk = h[gk ]μ = h[hk ]μ , i.e., hk is also the support function of [hk ]μ . For x ∈ Rn , we calculate Gh([hk ]μ +x) ,μ (ξhk + x) = Ghk ,μ (ξhk ). Thus, we can choose a sequence, also denoted by {hk } ⊂ C + (S n−1 ), with Cq ([hk ]μ ) = 1 and ξhk = o such that

lim Ghk ,μ (o) = inf max Gg,μ (ξ) : g ∈ C + (S n−1 ) and Cq ([g]μ ) = 1 . (3.14) k→+∞

ξ∈[g]μ

We claim that {hk } is uniformly bounded on S n−1 . Otherwise, we can take a subsequence of {hk }, again denoted by {hk }, such that lim

max hk (u) = +∞.

k→+∞ u∈S n−1

Let Rk = maxu∈S n−1 hk (u) = hk (uk ) for some uk ∈ S n−1 . Since {uk } ⊂ S n−1 , it follows from the compactness of S n−1 that there exists a convergent subsequence, say {uk }, assuming lim uk = u0 ∈ S n−1 .

k→+∞

Since supp(μ) is not concentrated on any closed hemisphere, there exists some u ∈ supp(μ) such that u · u0 > 0. Let b = 12 (u · u0 ) > 0. Then there exists k0 ∈ N such that when k ≥ k0 , u · uk > b. Note that Rk uk ∈ [hk ]μ . Thus, when k ≥ k0 , hk (u ) ≥ Rk (u · uk ) > Rk b. Since μ is a finite discrete measure, we get that for k ≥ k0 and 0 < p < 1,  lim Ghk ,μ (o) = lim hpk (u)dμ(u) k→+∞

≥

k→+∞ S n−1 lim hpk (u )μ(u ) k→+∞

> lim (Rk b)p μ(u ) = +∞. k→+∞

Let h ∈ C + (S n−1 ) and Cq ([h ]μ ) = 1. Then,  lim Ghk ,μ (o) ≤ Gh ,μ (ξh ) =

k→+∞

(h (u) − ξh · u)p dμ(u) < +∞

S n−1

which contradicts to (3.15). Therefore, {hk } is uniformly bounded.

(3.15)

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By the Blaschke selection theorem, {hk } has a convergent subsequence, likewise denoted by {hk }, letting hk → h on S n−1 as k → +∞. This yields h ≥ 0, [hk ]μ → [h]μ and h = h[h]μ . Thus, from (iv) of Lemma 2.2 there is Cq ([h]μ ) = 1. Moreover, it follows from Lemma 3.2 that o = lim ξhk = ξh ∈ int[h]μ . k→+∞

That is, h > 0. Hence, from (3.14) we have Gh,μ (o) = inf

max Gg,μ (ξ) : g ∈ C (S +

n−1

ξ∈[g]μ

) and Cq ([g]μ ) = 1 .

This completes the proof. 4. Solving the Lp electrostatic q-capacitary Minkowski problem In this section, we will use approximation to complete the proof of the main theorem. Therefore, we first need to prove the following result. Theorem 4.1. Let 0 < p < 1 and 1 < q < n. If μ is a finite discrete measure on S n−1 which is not concentrated in any closed hemisphere of S n−1 , then there exist a function h ∈ C + (S n−1 ) and a constant c > 0 such that μ = cμp,q ([h]μ , ·), where c=

q−1 n−q

 hp (u)dμ(u). S n−1

Proof. From Theorem 3.1, it follows that there exists a function h ∈ C + (S n−1 ) with ξh = o and Cq ([h]μ ) = 1 such that

Gh,μ (o) = inf max Gg,μ (ξ) : g ∈ C + (S n−1 ) and Cq ([g]μ ) = 1 . ξ∈[g]μ

Let δ > 0 be sufficiently small. Then for any f ∈ C(S n−1 ) and t ∈ (−δ, δ), we define zt = h + tf such that zt ∈ C + (S n−1 ). By (2.2) and (iv) of Lemma 2.3, we have Cq ([zt ]μ ) − Cq ([h]μ ) t  = (q − 1) f (u)dμq ([h]μ , u) lim

t→0

supp(μ)



= (q − 1) S n−1

Let gt = γ(t)zt , where

f (u)dμq ([h]μ , u).

(4.1)

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γ(t) = Cq ([zt ]μ )− n−q . 1

Then gt ∈ C + (S n−1 ) and Cq ([gt ]μ ) = 1 follows by (ii) of Lemma 2.2. Since g0 = h, from (4.1) we have that gt − g0 lim = −h t→0 t



q−1 n−q

  f (u)dμq ([h]μ , u) + f.

(4.2)

S n−1

Let ξ(t) = ξgt and  Gμ (t) = max

ξ∈[gt ]μ S n−1

(gt (u) − ξ · u)p dμ(u)



(gt (u) − ξ(t) · u)p dμ(u).

=

(4.3)

S n−1

Since ξ(t) ∈ int[gt ]μ , from (4.3) we have that for i = 1, . . . , n,  (gt (u) − ξ(t) · u)p−1 ui dμ(u) = 0,

(4.4)

S n−1

where u = (u1 , . . . , un )T . Noting ξ(0) = ξh = o and taking t = 0 in (4.4), it follows that  hp−1 (u)ui dμ(u) = 0

(4.5)

hp−1 (u)udμ(u) = 0.

(4.6)

S n−1

for i = 1, . . . , n. Hence,  S n−1

Let  (gt (u) − (ξ1 u1 + · · · + ξn un ))p−1 ui dμ(u)

Fi (t, ξ1 , . . . , ξn ) = S n−1

for i = 1, . . . , n. Then, ∂Fi = (1 − p) ∂ξj

 (gt (u) − (ξ1 u1 + · · · + ξn un ))p−2 ui uj dμ(u).

S n−1

Let F = (F1 , . . . , Fn ) and ξ = (ξ1 , . . . , ξn ). Thus, ⎛

∂F ⎝ ∂ξ

⎞ ⎠ (0,...,0)

n×n

 = (1 − p)

hp−2 (u)uuT dμ(u),

S n−1

where uuT is an n × n matrix. Since μ is not concentrated on any closed hemisphere, supp(μ) spans the whole space Rn . Thus, for any x ∈ Rn with x = 0, there exists a ui0 ∈ supp(μ) such that ui0 · x = 0. Hence, for 0 < p < 1 we have

Y. Feng et al. / J. Math. Anal. Appl. 487 (2020) 123959

⎛

⎞

T ⎝ ∂F x ∂ξ

⎠x

(0,...,0)

⎛

13

= xT ⎝(1 − p)

⎞



hp−2 (u)uuT dμ(u)⎠ x

S n−1

 = (1 − p)

hp−2 (u)(x · u)2 dμ(u)

S n−1

≥ (1 − p)hp−2 (ui0 )(x · ui0 )2 μ(ui0 ) > 0.  This implies that



∂F ∂ξ (0,...,0)

 is positive definite, which in particular implies that ⎛

⎞

∂F det ⎝ ∂ξ

⎠ = 0. (0,...,0)

i By this, the facts that for i = 1, . . . , n, Fi (0, . . . , 0) = 0 follows by equation (4.5) and ∂F ∂ξj is continuous on a neighborhood of (0, . . . , 0) for all 1 ≤ i, j ≤ n, and the implicit function theorem, it follows that

ξ  (0) = (ξ1 (0), . . . , ξn (0)) exists. Note that Gμ (0) = Gh,μ (o) and Gμ (t) = Ggt ,μ (ξgt ) with gt ∈ C + (S n−1 ) and Cq ([gt ]μ ) = 1. Then from Theorem 3.1, we see Gμ (t) ≥ Gμ (0), i.e., Gμ (0) is an extreme value of Gμ (t). Hence, by (4.2) and (4.6) we have 1  G (0) p μ ⎛     q−1 = hp−1 (u) ⎝−h(u) n−q

0=

S n−1

=−

q−1 n−q 



S n−1

S n−1



S n−1



hp−1 (u)f (u)dμ(u) − c S n−1



f (u)dμq ([h]μ , u) +

ξ  (0) · hp−1 (u)udμ(u)

−

=

f (u)dμq ([h]μ , u) + f (u) − ξ  (0) · u⎠ dμ(u)

S n−1

 hp (u)dμ(u)

S n−1

⎞

f (u)dμq ([h]μ , u),

S n−1

where q−1 c= n−q That is,

 hp (u)dμ(u) > 0. S n−1

hp−1 (u)f (u)dμ(u)

Y. Feng et al. / J. Math. Anal. Appl. 487 (2020) 123959

14



 h

p−1

(u)f (u)dμ(u) = c

S n−1

f (u)dμq ([h]μ , u)

S n−1

for all f ∈ C(S n−1 ). Note that h = h[h]μ . Then, dμ(u) = ch1−p [h]μ (u)dμq ([h]μ , u). From this and (1.2), we see dμ(u) = cdμp,q ([h]μ , u), i.e., μ = cμp,q ([h]μ , ·). Let us end up by proving Theorem 1.1. Proof of Theorem 1.1. Following the proof of [39, Theorem 8.2.2], for a given finite Borel measure μ on S n−1 which is not concentrated in any closed hemisphere we can construct a sequence of finite discrete measures {μj } on S n−1 such that μj (S n−1 ) = μ(S n−1 ) and μj → μ weakly as j → +∞. In particular, μj is not concentrated in any closed hemisphere for large enough j. By Theorem 4.1, for each μj there exists a function hj ∈ C + (S n−1 ) and a constant cj > 0 such that μj = cj μp,q ([hj ]μj , ·),

(4.7)

where q−1 cj = n−q

 hpj (u)dμj (u). S n−1

In addition, hj satisfies that ξhj = o, Cq ([hj ]μj ) = 1, and 

 max Gg,μj (ξ) : g ∈ C (S +

Ghj ,μj (o) = inf

n−1

ξ∈[g]μj

) and Cq ([g]μj ) = 1 ,

where [g]μj =



{ξ ∈ Rn : ξ · u ≤ g(u)},

u∈supp(μj )

and 

 (g(u) − ξ · u) dμj (u) =

(g(u) − ξ · u)p dμj (u).

p

Gg,μj (ξ) = S n−1

supp(μj )

The following proof is similar to the one in [28]. Let mj = Ghj ,μj (o). We will show that mj is uniformly bounded. For the Aleksandrov body associated to (1, supp(μj )), we write it as [1]μj . Let  gj =

1 Cq ([1]μj )

1  n−q

.

Y. Feng et al. / J. Math. Anal. Appl. 487 (2020) 123959

15

Then we see [ gj ]μj = gj [1]μj . Thus from (ii) of Lemma 2.2 there is Cq ([ gj ]μj ) = 1. Since μj (S n−1 ) = μ(S n−1 ), we have mj = Ghj ,μj (o)  ≤ max ( gj (u) − ξ · u)p dμj (u) ξ∈[ gj ]μj

 ≤

S n−1

D([ gj ]μj )p dμj (u) S n−1

= D([ gj ]μj )p μ(S n−1 ) = gjp D([1]μj )p μ(S n−1 ).

(4.8)

Hence, we need to prove that D([1]μj ) is uniformly bounded. If not, then there exists a sequence of {ξj } such that ξj ∈ [1]μj and lim |ξj | = +∞.

j→+∞

Let ξ j =

ξj |ξj |

∈ S n−1 . Then from the compactness of S n−1 , we can assume lim ξ j = ξ ∈ S n−1 .

j→+∞

Since supp(μ) is not concentrated in any closed hemisphere, there exists w ∈ supp(μ) such that ξ · w > 0. Let U (w) ⊂ S n−1 be any neighborhood of w. Then we have that lim inf μj (U (w)) ≥ μ(U (w)) > 0. j→+∞

Notice that when j is large enough, U (w)



supp(μj ) = ∅,

which implies that we can take a sequence {wji } such that wji ∈ supp(μji ) and

lim wji = w.

i→+∞

Remember that ξji ∈ [1]μji . Thus, ξji · wji ≤ h[1]μj (wji ) ≤ 1, i

i.e., ξ ji · w ji ≤ Taking the limit, this has

1 . |ξji |

(4.9)

Y. Feng et al. / J. Math. Anal. Appl. 487 (2020) 123959

16

ξ · w ≤ 0, which contradicts (4.9). Therefore, there exists a constant M > 0 such that for all j ∈ N, D([1]μj ) ≤ M.

(4.10)

Since B ⊂ [1]μj for each j ∈ N, from (i) of Lemma 2.2 we have for 1 < q < n,  gj ≤

1 Cq (B)

1  n−q

.

(4.11)

From (4.8), (4.10) and (4.11), it follows that for 0 < p < 1 and all j ∈ N, p

mj ≤ M p Cq (B)− n−q μ(S n−1 ).

(4.12)

Therefore, mj is uniformly bounded. We next show that {hj } is uniformly bounded on S n−1 . Otherwise, there exists a subsequence {hji } ⊂ {hj } such that lim

max hji (u) = +∞.

i→+∞ u∈S n−1

Let Rji = maxu∈S n−1 hji (u) = hji (uji ). Since {uji } ⊂ S n−1 , from the compactness of S n−1 , without loss of generality we can assume lim uji = u0 ∈ S n−1 .

i→+∞

Since supp(μ) is not concentrated in any closed hemisphere, there exists v0 ∈ supp(μ) such that v0 · u0 > 0. Let U (v0 ) be a small neighborhood of v0 such that for all u ∈ U (v0 ), we have that u · u0 > 0. Let δ(u) = 12 (u · u0 ) > 0 for u ∈ U (v0 ), and note that Rji uji ∈ [hji ]μji . Then when i is large enough, we see that for all u ∈ U (v0 ), u · uji > δ(u) and μji (U (v0 )) ≥ μ(U (v0 )) > 0. Thus, hji (u) ≥ Rji (u · uji ) > Rji δ(u). Hence, when i large enough we get

Y. Feng et al. / J. Math. Anal. Appl. 487 (2020) 123959

17

 hpji (u)dμji (u)

m ji = S n−1



hpji (u)dμji (u)

≥ U (v0 )



> Rjpi

δ(u)p dμji (u) U (v0 )



≥

Rjpi

δ(u)p dμ(u), U (v0 )

which implies that mji → +∞ as i → +∞. This contradicts (4.12). That is, {hj } is uniformly bounded on S n−1 . By the Blaschke selection theorem, the sequence {hj } has a convergent subsequence, also written as {hj }, assuming that hj → h on S n−1 when j → +∞. This means h ≥ 0, [hj ]μj → [h]μ as j → +∞, and ⎛ q−1 lim cj = lim ⎝ j→+∞ j→+∞ n−q

⎞



hpj (u)dμj (u)⎠

S n−1

q−1 = n−q

 hp (u)dμ(u) =: c0 ≥ 0. S n−1

From this, (1.2) and (iii) of Lemma 2.3, and taking the limit in (4.7), it follows that μ = c0 μp,q ([h]μ , ·). By this, we have c0 = 0. If p + q = n, then from (i) of Lemma 2.3 and (1.2) we have 1

μ = μp,q (c0n−p−q [h]μ , ·). 1

Let K = c0n−p−q [h]μ . Then our desired solution is given. If p + q = n, letting λ = c0 and K = [h]μ we have μ = λμp,q (K, ·), which yields the desired result. Acknowledgments The authors are very grateful to the anonymous referee for very valuable comments and suggestions which improved greatly the quality of this paper. References [1] M. Akman, J. Gong, J. Hineman, J. Lewis, A. Vogel, The Brunn-Minkowski inequality and a Minkowski problem for nonlinear capacity, preprint. [2] A.D. Aleksandrov, On the theory of mixed volumes. III. Extension of two theorems of Minkowski on convex polyhedra to arbitrary convex bodies, Mat. Sb. (N.S.) 3 (1938) 27–46. [3] G. Bianchi, K.J. Böröczky, A. Colesanti, The Orlicz version of the Lp Minkowski problem on S n−1 for −n < p < 1, Adv. Appl. Math. 111 (2019) 1–29. [4] G. Bianchi, K.J. Böröczky, A. Colesanti, D. Yang, The Lp Minkowski problem for −n < p < 1, Adv. Math. 341 (2019) 493–535.

18

[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48]

Y. Feng et al. / J. Math. Anal. Appl. 487 (2020) 123959

K.J. Böröczky, F. Fodor, The Lp dual Minkowski problem for p > 1 and q > 0, J. Differ. Equ. 266 (2019) 7980–8033. K.J. Böröczky, H.T. Trinh, The planar Lp Minkowski problem for 0 < p < 1, Adv. Appl. Math. 87 (2017) 58–81. K.J. Böröczky, E. Lutwak, D. Yang, G. Zhang, The logarithmic Minkowski problem, J. Am. Math. Soc. 26 (2013) 831–852. K.J. Böröczky, P. Hegedűs, G. Zhu, On the discrete logarithmic Minkowski problem, Int. Math. Res. Not. 6 (2016) 1807–1838. L. Caffarelli, D. Jerison, E. Lieb, On the case of equality in the Brunn-Minkowski inequality for capacity, Adv. Math. 117 (1996) 193–207. S. Chen, Q.-R. Li, G. Zhu, On the Lp Monge-Ampère equation, J. Differ. Equ. 263 (2017) 4997–5011. W. Chen, Lp Minkowski problem with not necessarily positive data, Adv. Math. 201 (2006) 77–89. S.Y. Cheng, S.T. Yau, On the regularity of the solution of the n-dimensional Minkowski problem, Commun. Pure Appl. Math. 29 (1976) 495–516. K. Chou, X. Wang, The Lp Minkowski problem and the Minkowski problem in centroaffine geometry, Adv. Math. 205 (2006) 33–83. A. Colesanti, K. Nyström, P. Salani, J. Xiao, D. Yang, G. Zhang, The Hadamard variational formula and the Minkowski problem for p-capacity, Adv. Math. 285 (2015) 1511–1588. W. Fenchel, B. Jessen, Mengenfunktionen und konvexe Körper, Danske Vid. Selskab. Mat.-Fys. Medd. 16 (1938) 1–31. R.J. Gardner, Geometric Tomography, second ed., Cambridge Univ. Press, New York, 2006. R.J. Gardner, D. Hug, W. Weil, S. Xing, D. Ye, General volumes in the Orlicz Brunn-Minkowski theory and a related Minkowski problem I, Calc. Var. Partial Differ. Equ. 58 (2019) 1–35. R.J. Gardner, D. Hug, S. Xing, D. Ye, General volumes in the Orlicz Brunn-Minkowski theory and a related Minkowski problem II, Calc. Var. Partial Differ. Equ. 59 (2020) 1–33. P.M. Gruber, Convex and Discrete Geometry, Grundlehren der Mathematischen Wissenschaften, vol. 336, Springer, Berlin, 2007. P. Guan, C. Lin, On equation det(uij + δij u) = up f on S n , preprint. C. Haberl, E. Lutwak, D. Yang, G. Zhang, The even Orlicz Minkowski problem, Adv. Math. 224 (2010) 2485–2510. Q. Huang, B. He, On the Orlicz Minkowski problem for polytopes, Discrete Comput. Geom. 48 (2012) 281–297. Y. Huang, Q. Lu, On the regularity of the Lp Minkowski problem, Adv. Appl. Math. 50 (2013) 268–280. Y. Huang, Y. Zhao, On the Lp dual Minkowski problem, Adv. Math. 332 (2018) 57–84. D. Hug, E. Lutwak, D. Yang, G. Zhang, On the Lp Minkowski problem for polytopes, Discrete Comput. Geom. 33 (2005) 699–715. D. Jerison, A Minkowski problem for electrostatic capacity, Acta Math. 176 (1996) 1–47. D. Jerison, The direct method in the calculus of variations for convex bodies, Adv. Math. 122 (1996) 262–279. H. Jian, J. Lu, Existence of solutions to the Orlicz-Minkowski problem, Adv. Math. 344 (2019) 262–288. H. Jian, J. Lu, X.J. Wang, Nonuniqueness of solutions to the Lp Minkowski problem, Adv. Math. 281 (2015) 845–856. H. Lewy, On differential geometry in the large, I (Minkowski’s problem), Trans. Am. Math. Soc. 43 (1938) 258–270. J. Lu, X.-J. Wang, Rotationally symmetric solution to the Lp Minkowski problem, J. Differ. Equ. 254 (2013) 983–1005. E. Lutwak, The Brunn-Minkowski-Firey theory. I. Mixed volumes and the Minkowski problem, J. Differ. Geom. 38 (1993) 131–150. E. Lutwak, D. Yang, G. Zhang, On the Lp Minkowski problem, Trans. Am. Math. Soc. 356 (2004) 4359–4370. E. Lutwak, D. Yang, G. Zhang, Lp dual curvature measures, Adv. Math. 329 (2018) 85–132. H. Minkowski, Allgemeine Lehrsätze über die convexen Polyeder, Gött. Nachr. 1897 (1897) 198–219. H. Minkowski, Volumen und Oberfläche, Math. Ann. 57 (1903) 447–495. L. Nirenberg, The Weyl and Minkowski problems in differential geometry in the large, Commun. Pure Appl. Math. 6 (1953) 337–394. A.V. Pogorelov, The Minkowski Multidimensional Problem, V. H. Winston & Sons, Washington, D.C., 1978. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, second ed., Cambridge Univ. Press, New York, 2014. Y. Sun, Existence and uniqueness of solutions to Orlicz Minkowski problems involving 0 < p < 1, Adv. Appl. Math. 101 (2018) 184–214. Y.C. Wu, D.M. Xi, G.S. Leng, On the discrete Orlicz Minkowski problem, Trans. Am. Math. Soc. 371 (2019) 1795–1814. G. Xiong, J. Xiong, L. Xu, The Lp capacitary Minkowski problem for polytopes, J. Funct. Anal. 277 (2019) 3131–3155. G. Zhu, The logarithmic Minkowski problem for polytopes, Adv. Math. 262 (2014) 909–931. G. Zhu, The centro-affine Minkowski problem for polytopes, J. Differ. Geom. 101 (2015) 159–174. G. Zhu, The Lp Minkowski problem for polytopes for 0 < p < 1, J. Funct. Anal. 269 (2015) 1070–1094. G. Zhu, The Lp Minkowski problem for polytopes for p < 0, Indiana Univ. Math. J. 66 (2017) 1333–1350. G. Zhu, Continuity of the solution to the Lp Minkowski problem, Proc. Am. Math. Soc. 145 (2017) 379–386. D. Zou, G. Xiong, The Lp Minkowski problem for the electrostatic p-capacity, J. Differ. Geom. (2020), in press.