The isoperimetric problem in the Heisenberg group $\pmb {\mathbb {H}^n}$ with density

Abstract

We study the isoperimetric problem for the axially symmetric sets in the Heisenberg group $\mathbb {H}^n$ with density $|z|^p$. At first, we prove the existence of weighted isoperimetric sets. Then, we characterize weighted isoperimetric sets uniquely as bubble sets. Finally, we deduce an interesting result that, up to a constant multiplicator, $|z|^p$ is the only horizontal radial density for which bubble sets can be weighted isoperimetric sets.

1 Introduction

The isoperimetric problem in Euclidean spaces $\mathbb {R}^n$ with density, a natural generalization of the classical Gaussian isoperimetric problem [8, 9, 11, 13], has been an active field in the past few years; see [4, 6, 12, 15,16,17, 25, 30, 31, 37] and the references therein. Given a positive function f, usually called “density”, the weighted volume $V_{f}$ and weighted perimeter $P_{f}$ of a set $E\subset \mathbb {R}^n$ with smooth boundary are given by

$$\begin{aligned} V_{f}(E)=\int _{E}f(x){d}v,\quad P_{f}(E)=\int _{\partial {E}}f(x){d}a, \end{aligned}$$

where dv and da are elements of Euclidean volume and area. In general, the weighted isoperimetric problem in $\mathbb {R}^n$ is that one studies the following problem:

$$\begin{aligned} \min \{P_f(E):E\in \mathscr {A}\;\;\text{ such } \text{ that }\;\;V_f(E)=v\} \end{aligned}$$

(1.1)

for a given volume $v>0$ and for a given admissible sets $\mathscr {A}$. Minimizers of (1.1) are called weighted isoperimetric sets. The weighted isoperimetric problem in the Euclidean space becomes more challenging when perimeter and volume are relative to different densities, see [1,2,3, 7, 34].

Recently, the weighted isoperimetric-type and Sobolev-type inequalities for hypersurfaces in Carnot group with density have been obtained in [21]. And the isoperimetric problem with density in Grushin space with density has been studied in [22]. Motivated by the above work, in this paper we are interested in the isoperimetric problem in the Heisenberg group $\mathbb {H}^n$ which is endowed with a certain density.

An n-dimensional Heisenberg group is a manifold $\mathbb {H}^{n}=\mathbb {C}^{n}\times \mathbb {R}\cong \mathbb {R}^{2n+1}$ with the group product

$$\begin{aligned} (z,t)*(z^{\prime },t^{\prime })=(z+z^{\prime },t+t^{\prime }+2I_{m}z\bar{{z}^{\prime }}), \end{aligned}$$

where $t,t^{\prime }\in \mathbb {R},z=x+iy,z^{\prime }=x^{\prime }+iy^{\prime }\in \mathbb {C}^{n}$ with $x,y,x^{\prime },y^{\prime }\in \mathbb {R}^{n}$ and $z\bar{{z}^{\prime }}=\sum _{j=1}^{n}z_{j}\bar{{z}_{j}^{\prime }}$. The Heisenberg group is a noncommutative Lie group and its center is $Z=\{(z,t)\in \mathbb {H}^{n}:z=0\}$. A left translation by $p\in \mathbb {H}^{n}$ is the mapping $L_{p}:\mathbb {H}^{n}\rightarrow \mathbb {H}^{n}, q\rightarrow L_{p}(q)=p*q$. For any $\lambda >0$, the mapping $\delta _{\lambda }:\mathbb {H}^{n}\rightarrow \mathbb {H}^{n},p\rightarrow \delta _{\lambda }(z,t)=(\lambda z,\lambda ^{2}t)$ is called a dilation. The Lie algebra of the Heisenberg group is spanned by the following left-invariant vector fields:

$$\begin{aligned} X_{i}=\frac{\partial }{\partial x_{i}}+2y_{i}\frac{\partial }{\partial t},\;\; Y_{i}=\frac{\partial }{\partial y_{i}}-2x_{i}\frac{\partial }{\partial t},\;\;i=1,\ldots ,n;\;\;T=\frac{\partial }{\partial t}. \end{aligned}$$

The horizontal distribution at a point $p\in \mathbb {H}^n$ is defined by

$$\begin{aligned} H_p=\mathrm{span}\{{X_i}(p),{Y_i}(p):i=1,\ldots ,n\}. \end{aligned}$$

The horizontal distribution is nonintegrable. In fact there holds $[X_{i},Y_{i}]=-4T\ne 0$ for any $i=1,\ldots ,n$. All other commutators vanish.

Let g be a (left-invariant) Riemannian metric which makes the basis $\{X_{i},Y_{i},T:i=1,\ldots ,n\}$ be an orthornormal frame. The natural volume in $\mathbb {H}^n$ is the Haar measure, which coincides with Lebesgue measure in $\mathbb {R}^{2n+1}$. Let $\Omega \subset \mathbb {H}^n$ be an open set. The horizontal divergence of a vector field $\varphi \in C^1(\Omega ;\mathbb {R}^{2n})$ is defined by

$$\begin{aligned} \mathrm{div}_{H}\varphi =\sum _{i=1}^{n}(X_i\varphi _i+Y_i\varphi _{n+i}). \end{aligned}$$

The H-perimeter of a Lebesgue measurable set $E\subset \mathbb {H}^n$ in $\Omega$ is defined as

$$\begin{aligned} P_{H}(E;\Omega )=\sup \left\{ \int _{E}\mathrm{div}_{H}\varphi {d}z{d}t:\varphi \in C_{c}^{1}(\Omega ;\mathbb {R}^{2n}),||\varphi ||_{\infty }\le 1 \right\} . \end{aligned}$$

If $P_{H}(E;\Omega )<\infty$, we say that E has finite H-perimeter in $\Omega$. In the case of $\Omega =\mathbb {H}^{n}$, we let $P_{H}(E)=P_{H}(E;\mathbb {H}^{n})$.

The isoperimetric problem in Heisenberg group, roughly speaking, is that seeking sets with the least H-perimeter enclosing a fixed amount of volume among all admissible sets. In 1983, Pansu [33] gave a conjecture about the possible solution of the Heisenberg isoperimetric problem. The conjecture can be stated as the following way: up to a negligible set, a left translation, and a dilation, the isoperimetric set in $\mathbb {H}^{1}$ is the set

$$\begin{aligned} E_\mathrm{isop}=\{(z,t)\in \mathbb {H}^{n}:|t|<\arccos |z|+|z|\sqrt{1-|z|^{2}},|z|<1\}. \end{aligned}$$

(1.2)

Pansu’s conjecture can be naturally extended to the higher dimensional Heisenberg group $\mathbb {H}^{n}$, $n\ge 2$. The set $E_\mathrm{isop}$ is called a bubble set or a Heisenberg bubble. The boundary of a bubble set is called the Pansu sphere. There are many celebrated works about the isoperimetric problem in the Heisenbeg group. We can see [10, 14, 19, 20, 23, 24, 26,27,28, 35, 36].

In this paper we will study the weighted isoperimetric problem for axially symmetric sets in the Heisenberg group $\mathbb {H}^{n}$. A set $E\subset \mathbb {H}^n$ is axially symmetric if $(z,t)\in E$ implies that $(\xi ,t)\in E$ for all $\xi \in \mathbb {C}^{n}$ such that $|\xi |=|z|$. The set $F\subset \mathbb {R}_{+}^{2}=\mathbb {R}^{+}\times \mathbb {R}$ such that the $E{\setminus } Z=\{(z,t)\in \mathbb {H}^n:(|z|,t)\in F\}$ is called the generating set of E.

Noticing that the existence of isoperimetric sets in Euclidean space $\mathbb {R}^n$ with density depends on the density; see [32, 37], we think naturally that the existence of isoperimetric sets in $\mathbb {H}^n$ with density should depend on the density. Consequently, how to choose a suitable density function is a key in studying the weighted isoperimetric problems in $\mathbb {H}^n$.

In the present paper, we endow the Heisenberg group $\mathbb {H}^n$ with density $f=|z|^{p}, p>-2n+1$, where $|z|=\sqrt{\sum _{i=1}^{n}(x_{i}^{2}+y_{i}^{2})}$ is called the horizontal norm of the point $(z,t)\in \mathbb {H}^{n}$. Though the function $|z|^{p}$ is one of the simplest horizontal radial density functions, it has some fine properties. Firstly, since the function $|z|^{p}$ is homogeneous in degree p, one can deduce that the weighted H-perimeter and the weighted volume are also homogeneous with respect to dilations. It means that if E is a weighted isoperimetric set, then $\delta _\lambda (E)$ is also a weighted isoperimetric set. Secondly, up to a constant multiplicator, $|z|^{p}$ is the only horizontal radial density for which bubble sets are isoperimetric sets (see Sect. 5).

In the following we give the definitions of the weighted H-perimeter and the weighted volume in $\mathbb {H}^n$ with density.

Definition 1.1

Let $\mathbb {H}^n$ be endowed with density $f=|z|^p$. The weighted H-perimeter of a $\mathcal {L}^{2n+1}$-measurable set $E\subset \mathbb {H}^n$ is defined as

$$\begin{aligned} P_{H,f}(E)=\sup \left\{ \int _{E}div_{H}(|z|^{p}\varphi (z,t)){d}z{d}t:\varphi \in C_{c}^{1}(\mathbb {H}^{n};\mathbb {R}^{2n}),||\varphi ||_{\infty }\le 1\right\} . \end{aligned}$$

If $P_{H,f}(E)<+\infty$, we say that E has finite weighted H-perimeter.

The weighted volume of $E\subset H^n$ is defined as

$$\begin{aligned} V_{f}(E)=\int _{E}|z|^{p}{d}z{d}t. \end{aligned}$$

The weighted H-perimeter and the weighted volume are invariant under a vertical translation, i.e., a left translation by some $(0,t_0)\in \mathbb {H}^n$. Moveover, they are homogeneous with respect to dilations. In fact, for any measurable set $E\subset \mathbb {H}^n$ and for any $\lambda >0$, we have

$$\begin{aligned} P_{H,f}(\delta _{\lambda }(E))=\lambda ^{d-1}P_{H,f}(E),\quad V_{f}(\delta _{\lambda }(E))=\lambda ^{d}V_{f}(E), \end{aligned}$$

where the homogenous dimension $d=2n+2+p$. Then the isoperimetic ratio of $E\subset \mathbb {H}^n$

$$\begin{aligned} \mathscr {I}_{H,f}(E)=\frac{P_{H,f}(E)^{d}}{V_{f}(E)^{d-1}} \end{aligned}$$

is homogeneous in degree 0.

Let $\mathscr {A}$ denote the family of all sets $E\subset \mathbb {H}^n$ which are axially symmetric. We consider the weighted isoperimetric problem among the class of $\mathscr {A}$. Namely we study the following problem:

$$\begin{aligned} \inf \{\mathscr {I}_{H,f}(E): E\in \mathscr {A} \;\; \text{ with }\;\; V_{f}(E)<+\infty \}. \end{aligned}$$

(1.3)

A set $E\subset \mathbb {H}^n$ which attains the infimum in (1.3) is called an weighted axially symmetric isoperimetric set.

By the homogeneity of $P_{H,f}$ and $V_{f}$, the weighted isoperimetric problem (1.3) can be also formulated as following:

$$\begin{aligned} C_{I,f}=\inf \{P_{H,f}(E):E\in \mathscr {A}\;\; \text{ with }\;\;V_{f}(E)=V\}, \end{aligned}$$

(1.4)

where V is a given constant.

Our main results in this paper are the following:

Theorem 1.1

In the Heisenberg group $\mathbb {H}^n$ with density $|z|^p$, $p>-2n+1$, the infimum in (1.3) is attained. Up to a dilation, a vertical translation and a negligible set, any weighted axially symmetric isoperimetric set coincides with a bubble set

$$\begin{aligned} E_\mathrm{isop}=\Big \{(z,t)\in \mathbb {H}^n:|t|<\arccos |z|+|z|\sqrt{{1-|z|}^{2}},|z|<1\Big \}. \end{aligned}$$

Remark 1.1

When p is zero, Theorem 1.1 in the present paper is exactly Theorem 1.2 in [27].

Theorem 1.2

If bubble sets are weighted isoperimetric in the Heisenberg group $\mathbb {H}^n$ with a horizontal radial density f(|z|), then, up to a constant multiplicator, the density f(|z|) is only of the form $|z|^p$.

The paper is organized as follows: In Sect. 2, we give the representation formula for the weighted H-perimeter of sets with Lipschitz boundary, and deduce reduced formulae for the weighted H-perimeter and the weighted volume of axially symmetric sets in $\mathbb {H}^n$ with density $|z|^p$. In Sect. 3, we improve the symmetry of admissible sets by the rearrangement theorems in [27], and prove the existence of weighted axially isoperimetric sets. In Sect. 4, we use a variational approach to show that bubble sets are uniquely solution of the weighted axially isoperimetric problem. In Sect. 5, we derive that the horizontal radial density is only of the form $|z|^p$ if bubble sets are weighted isoperimetric.

2 The representation and reduction formulae

In this section, we give the representation formula of the weighted H-perimeter for sets with Lipschitz boundary. For axially symmetric sets we deduce reduction formulae of the weighted H-perimeter and the weighted volume.

For any open set $A\subset \mathbb {H}^{n}$ and $m\in \mathbb {N}$, let us denote the family of test functions by

$$\begin{aligned} \mathscr {F}_{m}(A)=\big \{\varphi =(\varphi _{1},\ldots ,\varphi _{m}) \in C_{c}^{1}(A;\mathbb {R}^{m}):||\varphi ||_{\infty }=\max _{(x,y)\in A}|\varphi (x,y)|\le 1\big \}. \end{aligned}$$

2.1 The representation formula of the weighted H-perimeter

For an open set $E\subset \mathbb {H}^n$ with Lipschitz boundary, the outer unit normal $N^E:\partial E\rightarrow \mathbb {R}^{2n+1}$ is defined at $\mathcal {H}^{2n}$-a.e. point of $\partial E$, where $\mathcal {H}^{2n}$ is the standard 2n-dimensional Hausdorff measure. We define the 2n-dimensional vector field $N_{H}^{E}:\partial E\rightarrow \mathbb {R}^{2n}$, $N_{H}^{E}=(\langle X_{1},N^E\rangle ,\ldots ,\langle X_{n},N^E\rangle ,\langle Y_{1},N^E\rangle ,\ldots ,\langle Y_{n},N^E\rangle )$ , where $\langle \cdot ,\cdot \rangle$ is the standard scalar product of Euclidean space and $X_{i},Y_{i}\;\;(i=1,\ldots ,n)$ are thought of as elements of $\mathbb {R}^{2n+1}$ with respect to the standard basis $\frac{\partial }{\partial x_{1}},\ldots , \frac{\partial }{\partial x_{n}},\frac{\partial }{\partial y_{1}},\ldots , \frac{\partial }{\partial y_{n}},\frac{\partial }{\partial t}$.

Proposition 2.1

Let $\mathbb {H}^n$ be endowed with density $|z|^{p}$. If $E\subset \mathbb {H}^n$ is a measurable set with Lipschitz boundary, then the weighted H-perimeter of E is

$$\begin{aligned} P_{H,f}(E)=\int _{\partial E}|N_{H}^{E}||z|^{p}{d} \mathcal {H}^{2n}, \end{aligned}$$

where $|N_{H}^{E}|$ is the Euclidean norm.

Proof

The proof is standard. For any $\varphi \in \mathscr {F}_{2n}(\mathbb {H}^n)$, we have

$$\begin{aligned} \int _{E}\mathrm{div}_{H}(|z|^{p}\varphi ){d}z{d}t=\int _{\partial E}\langle |z|^{p}\varphi ,N_{H}^{E}\rangle {d}\mathcal {H}^{2n}\le \int _{\partial E}|N_{H}^{E}||z|^{p}|\varphi |{d}\mathcal {H}^{2n}. \end{aligned}$$

Taking the supremum with $||\varphi ||_{\infty }\le 1$, it follows that

$$\begin{aligned} P_{H,f}(E)\le \int _{\partial E}|N_{H}^{E}||z|^{p}{d}\mathcal {H}^{2n}. \end{aligned}$$

The opposite inequality follows by approximating $N_{H}^{E}/|N_{H}^{E}|$ with functions in $\mathscr {F}_{2n}(\mathbb {H}^n)$. By Lusin’s theorem, for any $\varepsilon >0$ there exists a compact set $K\subset \partial E$ such that

$$\begin{aligned} \int _{\partial E\backslash K}|N_{H}^{E}||z|^{p}{d}\mathcal {H}^{2n}<\varepsilon \end{aligned}$$

and $N_{H}^{E}:K\rightarrow \mathbb {R}^{2n}$ is continuous and nonzero.

By Titze’s theorem there exists $\varphi \in C_{c}(\mathbb {H}^{n};\mathbb {R}^{2n})$ such that $||\varphi ||_{\infty }\le 1$ and $\varphi =N_{H}^{E}/|N_{H}^{E}|$ on K. By mollification there exists $\psi \in C_{c}^{1}(\mathbb {H}^n;\mathbb {R}^{2n})$ such that $||\psi ||_{\infty }\le 1$ and $||\varphi -\psi ||_{\infty }<\varepsilon$. For such a test function $\psi$ we have

$$\begin{aligned} \begin{aligned} P_{H,f}(E)&\ge \int _{E}\mathrm{div}_{H}(|z|^{p}\psi ){d}z{d}t=\int _{\partial E}\langle |z|^{p}\psi ,N_{H}^{E}\rangle {d}\mathcal {H}^{2n}\\&=\int _{\partial E}|z|^{p}\langle \psi -\varphi ,N_{H}^{E}\rangle {d}\mathcal {H}^{2n}+\int _{\partial E}|z|^{p}\langle \varphi ,N_{H}^{E}\rangle {d}\mathcal {H}^{2n}\\&\ge -\varepsilon \int _{\partial E}|z|^{p}|N_{H}^{E}|{d}\mathcal {H}^{2n}+\int _{\partial E\backslash K}|z|^{p}\langle \varphi ,N_{H}^{E}\rangle {d}\mathcal {H}^{2n}\\&\quad +\int _{K}|z|^{p}\langle N_{H}^{E}/|N_{H}^{E}|,N_{H}^{E}\rangle {d}\mathcal {H}^{2n}\\&\ge (1-\varepsilon )\int _{\partial E}|z|^{p}|N_{H}^{E}|{d}\mathcal {H}^{2n}-2\varepsilon .\\ \end{aligned} \end{aligned}$$

$\square$

Remark 2.1

When E has Lipschiz boundary $\partial E$, for a general density f we still have

$$\begin{aligned} P_{H,f}(E)=\int _{\partial E}|N_{H}^{E}|f{d}\mathcal {H}^{2n}. \end{aligned}$$

2.2 The reduction formula of the weighted H-perimeter for axially symmetric sets

Proposition 2.2

Let $E\subset \mathbb {H}^n$ be a measurable axially symmetric set with the generating set $F\subset \mathbb {R}_{+}^2$, where $\mathbb {H}^n$ is endowed with density $|z|^{p}$. Then we have

$$\begin{aligned} P_{H,f}(E)=c_{2n-1}\sup _{\psi \in \mathscr {F}_{2}(\mathbb {R}_{+}^{2})}\left\{ \int _{F}\Big [\partial _{r}(r^{2n-1+p}\psi _{1})+2r^{2n+p}\partial _{t}\psi _{2}\Big ]{d}r{d}t\right\} . \end{aligned}$$

(2.1)

Above, $c_{2n-1}=\mathcal {H}^{2n-1}(\mathbb {S}^{2n-1})$ is the standard surface measure of the $(2n-1)$-dimensional unit sphere.

In particular, if E has Lipschitz boundary, then we have

$$\begin{aligned} P_{H,f}(E)=c_{2n-1}\int _{\partial F}\big |(N_{r}^{F},2rN_{t}^{F})\big |r^{2n-1+p}{d}\mathcal {H}^{1}(r,t), \end{aligned}$$

(2.2)

where $N^{F}=(N_{r}^{F},N_{t}^{F})$ is the outer unit normal to the boundary $\partial F$.

Proof

Setting

$$\begin{aligned} Q(F)=\sup _{\psi \in \mathscr {F}_{2}(\mathbb {R}_{+}^{2})}\left\{ \int _{\partial F}\Big [\partial _{r}(r^{2n-1+p}\psi _{1})+2r^{2n+p}\partial _{t}\psi _{2}\Big ]{d}r{d}t\right\} . \end{aligned}$$

Step 1. We claim that if E is of finite weighted H-perimeter and axially symmetric with generating set $F\subset \mathbb {R}_{+}^{2}$, then we have $P_{H,f}(E)\ge c_{2n-1}Q(F)$.

For any $\psi \in \mathscr {F}_{2}(\mathbb {R}_{+}^2)$, we define $\varphi \in \mathscr {F}_{2n}(\mathbb {H}^{n})$ by letting

$$\begin{aligned} \varphi (z,t)=\frac{1}{|z|}\Big \{\psi _{1}(|z|,t)z-\psi _{2}(|z|,t)J(z)\Big \}, \end{aligned}$$

for $|z|\ne 0$, where $J(z)=iz$, when $z=x+iy\in \mathbb {C}^n$. And let $\varphi (0,t)=0$ for $|z|=0$.

By a computation, we have

$$\begin{aligned} \begin{aligned} \mathrm{div}_{H}(|z|^{p}\varphi (z,t))&=|z|^{p}\mathrm{div}_{H}\varphi +\langle \varphi ,\mathrm{grad}_{H}|z|^{p}\rangle \\&=|z|^{p}\Big [\partial _ {r}\psi _{1}(|z|,t)+\frac{2n-1}{|z|}\psi _{1}(|z|,t)+2|z|\partial _{t}\psi _{2}(|z|,t)\Big ]\\&\quad +p|z|^{p-1}\langle \varphi ,\mathrm{grad}_{H}|z|\rangle \\&=|z|^{p}\Big [\partial _{r} \psi _{1}(|z|,t)+\frac{2n-1+p}{|z|}\psi _{1}(|z|,t)+2|z|\partial _{t}\psi _{2}(|z|,t)\Big ],\\ \end{aligned} \end{aligned}$$

(2.3)

where $\mathrm{grad}_H:C^{1}(\mathbb {H}^n)\rightarrow \mathbb {R}^{2n},\;\;f\rightarrow (X_{1}f,\ldots ,X_{n}f,Y_{1}f,\ldots ,Y_{n}f)$.

For any $t\in \mathbb {R}$, we define the section $F_{t}=\{r>0:(r,t)\in F\}$. Using Fubini’s theorem, the symmetry of E, the Coarea formula and (2.3) we obtain

$$\begin{aligned} \begin{aligned}&\int _{E}\mathrm{div}_{H}(|z|^{p}\varphi ){d}z{d}t\\&\quad =\int _{\mathbb {R}}\int _{F_{t}}\int _{|z|=r}r^{p}\Big [\partial _{r}\psi _{1}(r,t) +\frac{2n-1+p}{r}\psi _{1}(r,t)+2r\partial _{t}\psi _{2}(r,t)\Big ]{d}\mathcal {H}^{2n-1}(z){d}r{d}t\\&\quad =\int _{\mathbb {R}}\int _{F_{t}}c_{2n-1}r^{2n-1+p}\Big [\partial _{r}\psi _{1}(r,t)+\frac{2n-1+p}{r}\psi _{1}(r,t)+2r\partial _{t}\psi _{2}(r,t)\Big ]{d}r{d}t\\&\quad =c_{2n-1}\int _{F}\Big [\partial _{r}(r^{2n-1+p}\psi _{1}(r,t))+2r^{2n+p}\partial _{t}\psi _{2}(r,t)\Big ]{d}r{d}t.\\ \end{aligned} \end{aligned}$$

(2.4)

Taking the supremum over all $\psi \in \mathscr {F}_{2}(\mathbb {R}_+^{2})$ of the right side of (2.4), we obtain

$$\begin{aligned} P_{H,f}(E)\ge c_{2n-1}Q(F). \end{aligned}$$

Step 2. We claim that when $E\subset \mathbb {H}^n$ is a measurable axially symmetric set with smooth boundary, we have $P_{H,f}(E)\le c_{2n-1}Q(F)$.

Let $N^F=(N_{r}^{F},N_{t}^{F})$ and $N^E=(N_{z}^{E},N_{t}^{E})$ be the outer unit normal to $\partial F$ and $\partial E$, respectively. Then we have the relation $N_{z}^{E}=\frac{z}{|z|}N_{r}^{F}$ for $z\ne 0$. Furthermore we have $N_{H}^{E}=N_{z}^{E}-2N_{t}^{E}J(z)$. Obviously $N_{H}^{E}$ is continuous on $\partial E\backslash Z$.

By Lusin’s Theorem, for any $\varepsilon >0$ there exists a compact set $K\subset \partial E\backslash Z$ such that

$$\begin{aligned} \int _{\partial E\backslash K}|z|^{p}|N_{H}^{E}|{d}\mathcal {H}^{2n}<\varepsilon . \end{aligned}$$

Let $\widetilde{K} \subset \mathbb {R}_{+}^{2}$ be the generating set of K. By standard extension theorems, there exists $\psi \in \mathscr {F}_{2}(\mathbb {R}_{+}^{2})$ such that

$$\begin{aligned} \psi (r,t)=\frac{(N_{r}^{F},2rN_{t}^{F})}{|N_{r}^{F},2rN_{t}^{F}|},\; \forall (r,t)\in \widetilde{K}. \end{aligned}$$

Set $\varphi =\frac{1}{|z|}\Big \{\psi _{1}(|z|,t)z-\psi _{2}(|z|,t)J(z)\Big \}$. Then we know $\varphi \in \mathscr {F}_{2n}(\mathbb {H}^n)$. And we have

$$\begin{aligned} \varphi (z,t)=\frac{N_{z}^{E}-2N_{t}^{E}J(z)}{|N_{z}^{E}-2N_{t}^{E}J(z)|} =\frac{N_{H}^{E}}{|N_{H}^{E}|},\; \forall (z,t)\in K. \end{aligned}$$

By the definition of Q(F), we know

$$\begin{aligned} \begin{aligned} c_{2n-1}Q(F)&\ge c_{2n-1}\int _{F}\Big [\partial _{r}(r^{2n-1+p}\psi _{1}(r,t))+2r^{2n+p} \partial _{t}\psi _{2}(r,t)\Big ]{d}r{d}t\\&=\int _{E}\mathrm{div}_{H}(|z|^{p}\varphi (z,t)){d}z{d}t =\int _{\partial E}|z|^{p}\langle \varphi ,N_{H}^{E}\rangle {d}\mathcal {H}^{2n}\\&=\int _{K}|z|^{p}\langle \frac{N_{H}^{E}}{|N_{H}^{E}|},N_{H}^{E}\rangle {d}\mathcal {H}^{2n} +\int _{\partial E\backslash K}|z|^{p}\langle \varphi ,N_{H}^{E}\rangle {d}\mathcal {H}^{2n}\\&\ge \int _{\partial E}|z|^{p}|N_{H}^{E}|{d}\mathcal {H}^{2n}-2\varepsilon \\&=P_{H,f}(E)-2\varepsilon . \end{aligned} \end{aligned}$$

Letting $\varepsilon \rightarrow 0$, we get $c_{2n-1}Q(F)\ge P_{H,f}(E)$.

Step 3. The general case follows by the approximation. We can assume that $Q(F)<+\infty$. By Corollary 2.3.6 in [18], there exists a sequence $\{F_{j}\}_{j\in \mathbb {N}}$ of $\mathbb {R}_{+}^{2}$ with boundary of class $C^{\infty }$ such that

$$\begin{aligned}&(i) \lim _{j\rightarrow +\infty }\mathcal {L}^2((F_{j}\triangle F)\cap K)=0, \text{ for } \text{ any } \text{ compact } \text{ set }\;\; K\subset \mathbb {R}_{+}^{2}; \quad \\&(ii) \lim _{j\rightarrow +\infty }Q(F_{j})=Q(F). \end{aligned}$$

For any $j\in \mathbb {N},E_{j}\subset \mathbb {H}^n$ generated by $F_{j}$ denotes the axially symmetric set with boundary of class $C^\infty$ in $\mathbb {H}^n\backslash Z$. From i), it follows that for compact set $K^{\prime }\subset \mathbb {H}^{n}\backslash Z$ we have

$$\begin{aligned} \lim _{j\rightarrow +\infty }\mathcal {L}^{2n+1}((E_{j}\triangle E)\cap K^{\prime })=0. \end{aligned}$$

By the lower semicontinuity of weighted H-perimeter and the conclusion of Step 2, it follows that

$$\begin{aligned} P_{H,f}(E)&=P_{H,f}(E;\mathbb {H}^n\backslash Z)\le \lim _{j\rightarrow +\infty }\inf P_{H,f}(E_{j};\mathbb {H}^n\backslash Z)=\lim _{j\rightarrow +\infty }\omega _{2n-1}Q(F_{j})\\&=\omega _{2n-1}Q(F). \end{aligned}$$

Thus we obtain the formula (2.1).

Starting from (2.1), formula (2.2) is obtained with the same argument in the proof of Proposition 2.1. $\square$

2.3 The reduction formula of the weighted volume for axially symmetric sets

Using the Coarea formula, we find the representation formula of the weighted volume for axially symmetric sets E in $\mathbb {H}^n$ with density $|z|^{p}$ as follows:

$$\begin{aligned} \begin{aligned} V_{f}(E)&=\int _{E}|z|^{p}{d}z{d}t =\int _{-\infty }^{+\infty }\int _{0}^{+\infty }\int _{|z|=r}|z|^{p}{d}\mathcal {H}^{2n-1}(z){d}r{d}t\\&=\int _{-\infty }^{+\infty }\int _{0}^{+\infty }c_{2n-1}r^{2n-1+p}{d}r{d}t =c_{2n-1}\int _{F}r^{2n-1+p}{d}r{d}t. \end{aligned} \end{aligned}$$

Letting $V(F)=\int _{F}r^{2n-1+p}{d}r{d}t$ , then we have $V_{f}(E)=c_{2n-1}V(F)$. V(F) is called V-volume of F. Moreover, we have

$$\begin{aligned} \frac{P_{f}(E)^{d}}{V_{f}(E)^{d-1}}=c_{2n-1}\frac{Q(F)^{d}}{V(F)^{d-1}}. \end{aligned}$$

Thus (1.3) turns into

$$\begin{aligned} c_{2n-1}\inf \Big \{\mathscr {I}(F)=\frac{Q(F)^{d}}{V(F)^{d-1}}:F\subset \mathbb {R}_{+}^{2}\;\; \text{ with } \;\; 0<V(F)<+\infty \Big \}. \end{aligned}$$

(2.5)

A set $F\subset \mathbb {R}_{+}^{2}$ which attains the infimum in (2.5) is called the Q-isoperimetric set.

It is easily verified that Q(F) and V(F) are invariant under vertical translations. And Q(F) and V(F) are homogeneous by dilations $\delta _{\lambda }(r,t)=(\lambda r,\lambda ^{2}t)$ for any $\lambda >0$. Namely, we have

$$\begin{aligned} Q(\delta _{\lambda }(F))=\lambda ^{d-1}Q(F), \quad V(\delta _{\lambda }(F))=\lambda ^{d}V(F). \end{aligned}$$

3 Existence of weighted axially isoperimetric sets

From Sect. 2 we know that the isoperimetric problem for axially symmetric sets in $\mathbb {H}^n$ with density $|z|^{p}$ can turn into the Q-isoperimetric problem of corresponding generating sets in $\mathbb {R}_{+}^{2}$. By the homogeneity of Q(F) and V(F), we define the constant

$$\begin{aligned} C_{I}^{\prime }=\inf \big \{Q(F):F\subset \mathbb {R}_{+}^{2} \;\;\text{ with } \;\; V(F)=V_0 \big \}. \end{aligned}$$

(3.1)

3.1 Rearrangements

In this section we rearrange the generating sets in $\mathbb {R}_{+}^{2}$ by using the rearrangement technique introduced in [27]. Furthermore we improve the symmetry of axially symmetric sets in $\mathbb {H}^n$ with density $|z|^{p}$. A set E is z-Schwarz symmetric if for all $t\in \mathbb {R}$ we have $E_{t}=\{z\in \mathbb {C}^{n}:(z,t)\in E\}=\{z\in \mathbb {C} ^{n}:|z|<\rho (t)\}$ for some function $\rho :\mathbb {R}\rightarrow [0,+\infty ]$. A set E is called t-symmetric if $(r,t)\in E$ implies $(r,-t)\in E$. A set E is t-convex if sections $E_{z}=\{t\in \mathbb {R}:(z,t)\in E\}$ are intervals for every $z\in \mathbb {\mathbb {C}}^{n}$.

Theorem 3.1

For any set $F\subset \mathbb {R}_{+}^{2}$ such that $Q(F)<+\infty$ and $0<V(F)<+\infty$, there exists a t-symmetric and t-convex set $F^{\sharp }=\{(r,t)\in \mathbb {R}_{+}^{2} :r<g(t)\}$ where $g:\mathbb {R} \rightarrow [0,+\infty ]$, such that $Q(F^{\sharp })\le Q(F)$ and $V(F^{\sharp })=V(F)$. Moreover, if $Q(F^{\sharp })= Q(F)$, then we have

• (i) $F=F^{\sharp }$, and sections $F_{r}$ are intervals for $\mathcal {L}^{1}$-a.e. $r\in \mathbb {R}^{+}$.

• (ii) F is contained in a bounded rectangle. Namely

  $$\begin{aligned} F\subset \Big \{(r,t)\in \mathbb {R}_{+}^{2}:0\le r \le a_{n,p}Q(F)^{\frac{1}{2n+1+p}},|t-t_{0}|\le b_{n,p}\frac{Q(F)^{2n+p}}{V(F)^{2n-1+p}}\Big \} \end{aligned}$$

  for some $t_{0}\in \mathbb {R}$ and for constants $a_{n,p}=(\frac{2n+1+p}{4})^{\frac{1}{2n+1+p}}>0,b_{n,p}=(\frac{2}{2n+p})^{2n+1+p}>0$.

Proof

Because of $V(F)<+\infty$, the function $h:\mathbb {R}^{+}\rightarrow [0,+\infty ]$, $h(r)=\frac{1}{2}\mathcal {L}^{1}(F_{r})$ belongs to $L_{loc}^1(\mathbb {R}^+)$, where $F_{r}=\{t\in \mathbb {R}:(r,t)\in F\}$. So we can rearrange the set F using the Steiner symmetrization in direction t. Namely, we let

$$\begin{aligned} F^{*}=\{(r,t)\in \mathbb {R}_{+}^{2}:|t|<h(r) \}. \end{aligned}$$

By Theorem 3.2 in [27], we have

$$\begin{aligned} Q(F^{*})\le Q(F), \end{aligned}$$

and the equality $Q(F^{*})=Q(F)$ implies that $F_{r}$ are intervals for $\mathcal {L}^1$-a.e. $r\in \mathbb {R}^{+}$.

The V-volume of $F^{*}$ is

$$\begin{aligned} \begin{aligned} V(F^{*})&=\int _{F^{*}}r^{2n-1+p}{d}r{d}t=\int _{0}^{+\infty }{d}r\int _{F_{r}^{*}}r^{2n-1+p}{d}t\\&=\int _{0}^{+\infty }r^{2n-1+p}\mathcal {L}^{1}(F_{r}){d}r\\ {}&=V{(F)}.\\ \end{aligned} \end{aligned}$$

From $Q(F^{*})\le Q(F)<+\infty$, we deduce that $F^{*}$ is “$\tau$-rearrangeable” (see [27]), i.e., we can rearrange $F^{*}$ in the coordinate r using the function $r^{2n+p}$. Namely, we define the function $g: \mathbb {R}\rightarrow [0,+\infty ]$ via the identity

$$\begin{aligned} \frac{1}{2n+1+p}g(t)^{2n+1+p}=\int _{0}^{g(t)}r^{2n+p}{d}r=\int _{F_{t}^{*}}r^{2n+p}{d}r. \end{aligned}$$

(3.2)

Let $F_{1}^{\sharp }=\{(r,t)\in \mathbb {R}_{+}^{2}:r<g(t)\}$ be the arrangement of $F^{*}$. By Theorem 1.5 in [27], we know $Q(F_{1}^{\sharp })\le Q (F^{*})$, and moreover, if $Q(F_{1}^{\sharp })= Q(F^{*})$, then $F^{*}=F_{1}^{\sharp }$, up to a $\mathcal {L}^{2}$-negligible set.

Using example 2.5 in [27], we have

$$\begin{aligned} \Big [(2n+1+p)\int _{F_{t}^{*}}r^{2n+p}{d}r\Big ]^{\frac{1}{2n+1+p}}\ge \Big [(2n+p)\int _{F_{t}^{*}}r^{2n-1+p}{d}r\Big ]^{\frac{1}{2n+p}} \end{aligned}$$

(3.3)

for any measurable set $F_{t}^{*}\subset \mathbb {R}^{+}$, for any $p>-2n+1$. Moreover, we know the equality holds in (3.3) if and only if $F_{t}^{*}$ is an interval.

By (3.2) and (3.3) we get

$$\begin{aligned} \int _{F_{1t}^{\sharp }}r^{2n-1+p}{d}r\ge \int _{F_{t}^{*}}r^{2n-1+p}{d}r \end{aligned}$$

(3.4)

and (3.4) is a equality if and only if $F_{t}^{*}=F_{1t}^{\sharp }=(0,g(t))$, up to a $\mathcal {L}^1$-negligible set.

Now from (3.4), by Fubini’s Theorem it follows that $V(F_{1}^{\sharp })\ge V(F^{*})$ with equality if and only if $F_{1}^{\sharp }=F ^{*}$, up to a $\mathcal {L}^{2}$-negligible set.

Using a dilation $\delta _{\lambda }(r,t)=(\lambda r,\lambda ^{2}t)$, we let $F^{\sharp }=\delta _{\lambda }(F_{1}^{\sharp })$ with some $\lambda >0$, such that $V(F^{\sharp })=V(F)$. In fact we have $0<\lambda \le 1$ because of $V(F^{\sharp })=\lambda ^{d}V(F_{1}^{\sharp })$. Then by the scaling property of Q-perimeter we have

$$\begin{aligned} Q(F^{\sharp })=\lambda ^{d-1}Q(F_{1}^{\sharp })\le Q(F_{1}^{\sharp }) \le Q(F^{*})\le Q(F). \end{aligned}$$

This proves the first part of Theorem 3.1.

If $Q (F^{\sharp })=Q(F)$, then we have $\lambda =1$ and $Q(F_{1}^{\sharp })=Q(F^{*})=Q(F)$. From $Q(F^{*})=Q(F)$ we know $F_{r}$ are intervals for $\mathcal {L}^1$-a.e. $r>0$. From $\lambda =1$ and $Q(F)=Q(F_{1}^{\sharp })$ we know $F=F_{1}^{\sharp }=F^{\sharp }$.

Finally we claim that, up to a $\mathcal {L}^2$-negligible, we have

$$\begin{aligned} F\subset \left\{ (r,t)\in \mathbb {R}_{+}^{2}:0\le r \le a_{n,p}Q(F)^{\frac{1}{2n+1+p}},|t-t_{0}|\le b_{n,p}\frac{Q(F)^{2n+p}}{V(F)^{2n-1+p}}\right\} \end{aligned}$$

for some $t_{0}\in \mathbb {R}$ and for some constants $a_{n,p}>0,b_{n,p}>0$.

From i) we know, up to a $\mathcal {L}^{2}$-negligible set, the set F is of the form

$$\begin{aligned} F^{\sharp }=\{(r,t)\in \mathbb {R}_{+}^{2}:0<r<g(t),t\in \mathbb {R}\} \end{aligned}$$

(3.5)

for some function $g:\mathbb {R}\rightarrow [0,+\infty ]$ which is decreasing on $(t_0,+\infty )$ and increasing on $(-\infty ,t_0)$ for some $t_{0}\in \mathbb {R}$. Setting

$$\begin{aligned} M=\sup _{t\in \mathbb {R}}g(t). \end{aligned}$$

(3.6)

By the definition of Q(F), we have

$$\begin{aligned} \begin{aligned} Q(F)&\ge \sup _{\psi \in \mathscr {F}_{1}(\mathbb {R}_{+}^{2})}\int _{F}2r^{2n+p}\partial _{t}\psi (r,t) {d}r{d}t\\&\ge 2\sup _{\psi \in \mathscr {F}_{1}(\mathbb {R})}\int _{F}r^{2n+p}\partial _{t}\psi (t) {d}r{d}t\\&=2\int _{0}^{\infty }r^{2n+p}\sup _{\psi \in \mathcal {F}_{1}(\mathbb {R})}\int _{\{g(t)>r\}}\partial _{t}(\psi (t)){d}t{d}r\\&=2\int _{0}^\infty 2r^{2n+p}{d}r=\frac{4M^{2n+1+p}}{2n+1+p}.\\ \end{aligned} \end{aligned}$$

Then we get the estimate

$$\begin{aligned} M\le \Big [\frac{(2n+1+p)Q(F)}{4}\Big ]^{\frac{1}{2n+1+p}}. \end{aligned}$$

(3.7)

On the other hand, the set F in (3.5) is also of the form

$$\begin{aligned} F=\{(r,t)\in \mathbb {R}_{+}^{2}:k(r)<t<h(r),r\in \mathbb {R}^{+} \} \end{aligned}$$

for some functions $k,h:\mathbb {R}^{+}\rightarrow [-\infty ,+\infty ]$ such that h and $-k$ are decreasing, thanks to $F=F^{\sharp }$. Moreover, we can assume that $h(r)=k(r)=t_{0}$ for all $r>M$. Thus by the definition of Q(F), we have

$$\begin{aligned} \begin{aligned} Q(F)&\ge \sup _{\psi \in \mathscr {F}_{1}(\mathbb {R}_{+}^{2})}\int _{F}\partial _{r}(r^{2n-1+p}\psi (r,t)){d}r{d}t\\&=\sup _{\psi \in \mathscr {F}_{1}(\mathbb {R}_{+}^{2})}\int _{\mathbb {R}^{+}}\Big [\partial _{r}(r^{2n-1+p}\psi (r,t))\int _{k(r)}^{h(r)}{d}t\Big ]{d}r\\&\ge \sup _{\psi \in \mathscr {F}_{1}(\mathbb {R}^{+})}\int _{\mathbb {R}^{+}} (h(r)-k(r))\partial _{r}(r^{2n-1+p}\psi (r)){d}r\\&\ge \lim _{r\rightarrow 0^{+}}(h(r)-k(r))M^{2n-1+p}. \end{aligned} \end{aligned}$$

Then we get

$$\begin{aligned} |t-t_{0}|\le \lim _{r\rightarrow 0^{+}}(h(r)-k(r))\le \frac{Q(F)}{M^{2n-1+p}}. \end{aligned}$$

(3.8)

From (3.6) and (3.8), we know

$$\begin{aligned} F\subset R:=[0,M]\times \left[ t_{0}-\frac{Q(F)}{M^{2n-1+p}},t_{0}+\frac{Q(F)}{M^{2n-1+p}}\right] . \end{aligned}$$

(3.9)

Now from (3.9) we have

$$\begin{aligned} V(F)\le V(R)=\int _{R}r^{2n-1+p}{d}r{d}t=\frac{2Q(F)}{2n+p}M. \end{aligned}$$

Thus we get an estimate from below for M:

$$\begin{aligned} M\ge \frac{(2n+p)V(F)}{2Q(F)}. \end{aligned}$$

(3.10)

Finally from (3.7), (3.9) and (3.10) we get

$$\begin{aligned} F\subset \Big [0,a_{n,p}(Q(F))^{\frac{1}{2n+1+p}}\Big ]\times \Big [t_{0}-b_{n,p}\frac{Q(F)^{2n+p}}{V(F)^{2n-1+p}},t_{0}+b_{n,p}\frac{Q(F)^{2n+p}}{V(F)^{2n-1+p}}\Big ] \end{aligned}$$

for constants $a_{n,p}=(\frac{2n+1+p}{4})^{\frac{1}{2n+1+p}}>0,b_{n,p}=(\frac{2}{2n+p})^{2n+1+p}>0$. $\square$

From Theorem 3.1, it is easy to show that there holds the following

Theorem 3.2

For any axially symmetric sets E in $\mathbb {H}^n$ with density $|z|^{p}$ , $p>-2n+1$, such that $P_{f}(E)<+\infty$ and $0<V_{f}(E)<+\infty$, there exists a z-Schwarz symmetric, t-symmetric and t-convex set $E^{\sharp }\subset \mathbb {H}^n$ such that $P_{f}(E^{\sharp })\le P_{f}(E)$ and $V_{f}(E^{\sharp })=V_{f}(E)$.

Moreover, if $P_{f}(E^{\sharp })= P_{f}(E)$, then E is a z-Schwarz symmetric and t-convex.

3.2 Existence of the weighted axially symmetric isoperimetric sets

Theorem 3.3

Up to a $\mathcal {L}^{2}$-negligible set, there exists a bounded, t-symmetric and t-convex set F of the form $\{(r,t)\in \mathbb {R}_{+}^{2}:0<r<g(t)\}$ realizing the infimum in (3.1).

Proof

Let $\{F_{j}\}_{j\in \mathbb {N}}$ be a minimizing sequence for the infimum in (3.1). Namely, let

$$\begin{aligned} V(F_{j})=V_0, \quad \lim _{j\rightarrow \infty }Q(F_{j})=C_{I}^{\prime }>0. \end{aligned}$$

By Theorem 3.1, without loss of generality we can assume that every set $F_{j}$ is equal to not only $F_{j}^{\sharp }$ but also $F_{j}^{*}$. So we have $F_{j}=\{(r,t)\in \mathbb {R}_{+}^{2}:|t|<h_{j}(r)\}$ for functions $h_{j}:\mathbb {R}^{+}\rightarrow [0,+\infty ]$ which are decreasing on $(0,+\infty ]$. By Theorem 3.1 we know functions $h_{j}$ are uniformly bounded and moreover, there exists $r_{0}>0$ such $h_{j}(r)=0$ for all $r>r_{0}$ and for all $j\in \mathbb {N}$. By Helly theorem, there exists a subsequence of $\{h_{j}\}_{j\in \mathbb {N}}$, still denoted by $\{h_{j}\}$, such that $\lim _{j\rightarrow \infty }h_{j}(r)=h(r),\; \forall r>0$ and the function h(r) is also decreasing. Let

$$\begin{aligned} F=\{(r,t)\in \mathbb {R}_{+}^2:|t|<h(r),r\in \mathbb {R}^{+} \}. \end{aligned}$$

By the dominated convergence theorem, we have

$$\begin{aligned} V(F)=\lim _{j\rightarrow +\infty }V(F_{j})=V_0. \end{aligned}$$

Moreover, $\chi _{F_{j}}\rightarrow \chi _{F}$ in $L_{loc}^{1}(\mathbb {R}_{+}^{2})$. Then by the lower semicontinuity of the perimeter, we have

$$\begin{aligned} Q(F)\le \lim _{j\rightarrow \infty }\inf Q(F_{j})=C_{I}^{\prime }. \end{aligned}$$

By the definition of $C_{I}^{\prime }$, we know $Q(F)=C_{I}^{\prime }$. The boundedness of F is preserved by the $L^1$-convergence. By Theorem 3.1, we know $F=F^{\sharp }=\{(r,t)\in \mathbb {R}_{+}^{2}:0<r<g(t),t\in \mathbb {R}\}$, and the section $F_{r}$ is interval for $\mathcal {L}^1$-a.e. $r>0$. $\square$

Using the relation (2.5), we get the following

Theorem 3.4

There exists a bounded, z-Schwarz symmetric, t-symmetric and t-convex set $E\subset \mathbb {H}^n$ which attains the infimum in (1.3).

4 Solutions of the weighted axially isoperimetric problem

In this section, using the variational principle we get the solutions of the weighted axially symmetric isoperimetric problem in the Heisenberg group $\mathbb {H}^n$ with density $|z|^p$.

Theorem 4.1

Up to a dilation, a negligible set, and a vertical translation, any axially symmetric isoperimetric set in the Heisenberg group $\mathbb {H}^n$ with density $|z|^p$, $p>-2n+1$, coincides with

$$\begin{aligned} E_\mathrm{isop}=\{(z,t)\in H^{n}:|t|<{\hbox {arccos}}|z|+|z|\sqrt{1-|z|^{2}},|z|<1\}. \end{aligned}$$

Proof

By the characterization of the equality case in Theorem 3.1 and Theorem 3.2, any axially symmetric set E in $\mathbb {H}^{n}$ with density $|z|^{p}$, $p>-2n+1$, is z-Schwarz symmetric. And up to a vertical translation, there exists a function $h:(0,+\infty ]\rightarrow (0,+\infty ]$ such that

$$\begin{aligned} E=\{(z,t)\in \mathbb {H}^{n}:|t|<h(|z|)\}. \end{aligned}$$

The function h is decreasing. Let $F\subset \mathbb {R}_{+}^{2}$ be the generating set of E. Then F is of the form

$$\begin{aligned} \{(r,t)\in \mathbb {R}_{+}^{2}:|t|<h(r),r\in (0,r_{0})\} \end{aligned}$$

for some $r_{0}\in (0,+\infty ]$.

By the regularity theory of isperometric hypersurfaces [29], the boundary $\partial E$ is class of $C^{\infty }$ where $z\ne 0$. So by Proposition 2.2, we have

$$\begin{aligned} P_{f}(E)=c_{2n-1}\int _{\partial F}\big |(N_{r}^{F},2r N_{t}^{F})\big |r^{2n-1+p}{d} \mathcal {H}^{1}(r,t), \end{aligned}$$

where $(N_{r}^{F},N_{t}^{F})$ is the outer unit normal to $\partial F$, that is defined $\mathcal {H}^{1}$-a.e. on $\partial F$.

Thus we get

$$\begin{aligned} P_{f}(E)= & {} 2c_{2n-1}\int _{0}^{r_{0}}\sqrt{h'(r)^{2}+4r^{2}}r^{2n-1+p}{d}r,\\ V_{f}(E)= & {} c_{2n-1}\int _{F}r^{2n-1+p}{d}r{d}t=2c_{2n-1}\int _{0}^{r_{0}}h(r)r^{2n-1+p}{d}r. \end{aligned}$$

For any $\psi \in C_{c}^{\infty }(0,r_{0})$ and $\varepsilon \in \mathbb {R}$, we consider perturbation $h+\varepsilon \psi$, and define the set $E_{\varepsilon }=\{(r,t)\in \mathbb {R}_{+}^{2}:|t|<h(r)+\varepsilon \psi (r)\}$. Then we have

$$\begin{aligned} P_{f}(\varepsilon )= & {} 2c_{2n-1}\int _{0}^{r_{0}}\sqrt{(h^{\prime }+\varepsilon \psi ')^{2}+4r^{2}}r^{2n-1+p}{d}r,\\ V_{f}(\varepsilon )= & {} 2c_{2n-1}\int _{0}^{r_{0}}(h+\varepsilon \psi )r^{2n-1+p}{d}r. \end{aligned}$$

Since E is a weighted isoperimetric set, we have

$$\begin{aligned} \frac{d}{d\varepsilon }\Big |_{\varepsilon =0}\frac{P_{f}(\varepsilon )^{d}}{V_{f}(\varepsilon )^{d-1}}=\frac{d(P_{f})^{d-1}(V_{f})^{d-1}P'_{f}-(d-1)(P_{f})^{d} (V_{f})^{d-2}V'_{f}}{V_{f}^{2(d-1)}}\Big |_{\varepsilon =0}=0. \end{aligned}$$

It follows that

$$\begin{aligned} P'_{f}(0)-kV'_{f}(0)=0,\quad \text{ where } \quad k=\frac{d-1}{d}\frac{P_{f}(E)}{V_{f}(E)}. \end{aligned}$$

i.e.

$$\begin{aligned} \int _{0}^{r_{0}}\frac{h{'}\psi {'}}{\sqrt{(h{'})^2+4r^{2}}}r^{2n-1+p}{d}r =\int _{0}^{r_{0}}kr^{2n-1+p}\psi {d}r. \end{aligned}$$

By the divergence theorem and using the fact $\psi \in C_{c}^{\infty }(0,r_{0})$ is arbitrary, we deduce that h solves the following second order ordinary differential equation:

$$\begin{aligned} \left( \frac{h{'}r^{2n-1+p}}{\sqrt{(h')^2+4r^{2}}}\right) {'}=-kr^{2n-1+p}. \end{aligned}$$

(4.1)

We integrate the Eq. (4.1) on the interval [0, r] and get

$$\begin{aligned} \frac{h{'}}{\sqrt{(h')^{2}+4r^{2}}}=-\frac{kr}{2n+p}+\frac{D}{r^{2n-1+p}}, \end{aligned}$$

(4.2)

where D is some constant. Because of $p>-2n+1$, we deduce that $D=0$. In fact, the left-hand side of (4.2) is bounded as $r\rightarrow 0^{+}$, while the right-hand side diverges to $\pm \infty$ according to the sign $D\ne 0$. Thus the Eq. (4.2) turns into

$$\begin{aligned} \frac{h{'}}{\sqrt{(h')^{2}+4r^{2}}}=-\frac{kr}{2n+p}. \end{aligned}$$

(4.3)

By a dilation of the set E, we can assume that $k=2n+p$. Noting $h{'}\le 0$, so we have

$$\begin{aligned} h{'}(r)=-\frac{2r}{\sqrt{1-r^{2}}}. \end{aligned}$$

(4.4)

From (4.4) we know the function $h{'}$ is defined on the interval $(-1,1)$.

On the other hand, because $\partial F\cap \mathbb {R}_{+}^{2}$ is of class $C^{\infty }$, it must be $h(1)=0$. With this condition, the solution of the Eq. (4.4) is

$$\begin{aligned} h(r)=\arccos r+r\sqrt{1-r^{2}},r<1. \end{aligned}$$

So the axially symmetric isoperimetric set E in $\mathbb {H}^n$ with density $|z|^{p}$ is the form as follows

$$\begin{aligned} E_\mathrm{isop}=\{(z,t)\in H^{n}:|t|<\arccos |z|+|z|\sqrt{1-|z|^{2}},|z|<1\}. \end{aligned}$$

5 A characteristic of the density f(|z|) for bubble sets being isoperimetric

In this section, we consider the question that what horizontal radial densities can guarantee bubble sets to be weighted isoperimetric sets in $\mathbb {H}^n$. And we draw a conclusion that the horizontal radial density is only equal to $|z|^p$ if bubble sets are weighted isoperimetric.

First we consider critical surfaces for the weighted H-perimeter functional under a weighted volume constraint. Let $g=\langle \cdot ,\cdot \rangle$ be a (left invariant) Riemannian metric such that $\{X_i,Y_i,T:i=i,\ldots ,n\}$ is orthornomal. Denote by $\nabla$ the left levi-Civita connection on $(\mathbb {H}^n,g)$. Using the koszul identity, we have, for $k,j=1,\ldots ,n$

$$\begin{aligned} \begin{aligned}&\nabla _{X_{k}}X_{j}=\nabla _{Y_{k}}Y_{j}=\nabla _{T}T=0\\&\nabla _{X_{k}}Y_{j}=-2\delta _{k}^jT,\nabla _{X_{k}}T=2Y_{k},\nabla _{Y_{k}}T=-2X_{k}\\&\nabla _{Y_{k}}X_{j}=2\delta _{k}^jT,\nabla _{T}{X_{k}}=2Y_{k},\nabla _{T}{Y_{k}}=2X_{k}.\\ \end{aligned} \end{aligned}$$

(5.1)

Let $\Sigma \subset \mathbb {H}^{n}$ be a hypersurface of class $C^{2}$. Let N be a (Riemannian) unit normal vector field to $\Sigma$ and $N_{H}$ is the orthogonal projection of N to the horizontal distribution, i.e., $N_H=\sum _{i=1}^n(\langle N,X_i\rangle X_i+\langle N,Y_i\rangle Y_i)$. The characteristic set $\Sigma _{0}$ of $\Sigma$ is given by $\{q\in \Sigma :N_{H}(q)=0\}$. It is easily proved that $\Sigma _0=\{q\in \Sigma :T_q\Sigma =H_q\}$. Then a horizontal unit normal $v_{H}$ can be defined in $\Sigma \backslash \Sigma _{0}$ by

$$\begin{aligned} v_{H}=\frac{N_{H}}{|N_{H}|}. \end{aligned}$$

Let $\{e_{1},\ldots ,e_{2n-1}\}$ be an orthonormal basis of $T_p \Sigma \cap H_{p}$. The horizontal mean curvature of $\Sigma$ is the function, defined in $\Sigma \backslash \Sigma _{0}$, by

$$\begin{aligned} \mathscr {H} _{H}(p)=-\sum _{i=1}^{2n-1}\langle \nabla _{e_{i}}v_{H}(p),e_{i}\rangle . \end{aligned}$$

We recall that the horizontal divergence of $X\in C^{1}(\mathbb {H}^{n};H)$ is defined by

$$\begin{aligned} \mathrm{div}_{H}X=\sum _{i=1}^{n}(\langle \nabla _{X_i}X,X_i\rangle +\langle \nabla _{Y_{i}}X,Y_{i}\rangle ). \end{aligned}$$

Actually if the horizontal vector $X=\sum _{i=1}^{n}(a_{i}X_{i}+a_{n+i}Y_{i})$, then $\mathrm{div}_{H}X=\sum _{i=1}^{n}(X_{i}a_{i}+Y_{i}a_{n+i})$. It is verified that $\mathscr {H}_H=-\mathrm{div}_Hv_H$.

Assume that $\{e_{1},\ldots ,e_{2n}\}$ is an orthonormal basis of $T\Sigma$. Let X be a $C^1$ vector field defined on $\Sigma$. The divergence of X in $\Sigma$, $\mathrm{div}_{\Sigma }X$ , is defined by

$$\begin{aligned} \mathrm{div}_{\Sigma }X(p)=\sum _{i=1}^{2n}\langle \nabla _{e_{i}}X(p),e_{i}\rangle . \end{aligned}$$

At every point of $\Sigma \backslash \Sigma _{0}$, we may choose an orthornormal frame of $T\Sigma$ consisting of an orthonormal frame $\{e_{1},\ldots ,e_{2n-1}\}$ of $T\Sigma \cap H$ together with the vector

$$\begin{aligned} S:=\langle N,T\rangle v_{H}-|N_{H}|T, \end{aligned}$$

which is orthogonal to N and of modulus 1. Hence in $\Sigma \backslash \Sigma _{0}$ we obtain

$$\begin{aligned} \mathrm{div}_{\Sigma }v_{H}=-\mathscr {H}_{H}+\langle \nabla _{S}v_{H},S\rangle . \end{aligned}$$

Using (5.1), we can get $\langle \nabla _{S}v_{H},S\rangle =0$. Thus it follows that $\mathscr {H}_{H}=-\mathrm{div}_{\Sigma }v_{H}$.

Now we endow the Heisenberg group $\mathbb {H}^{n}$ with smooth density $f=e^{\psi }$. Let $\Omega \subset \mathbb {H}^{n}$ be a bounded region enclosed by a compact hypersurface $\Sigma$ of class $C^{2}$. Then the weighted H-perimeter and the weighted volume of $\Omega$ are

$$\begin{aligned} P_{f}(\Omega )=\int _{\Sigma }e^{f}|N_{H}|{d}\Sigma ,\quad V_{f}(\Omega )=\int _{\Omega }e^{f}{d}V, \end{aligned}$$

respectively, where ${d}\Sigma$ is the Riemannian area element and dV is the Riemannian volume element.

Consider a $C^{2}$ vector field U with compact support on $\Sigma$, and denote by $\{\varphi _{t}\}_{t\in \mathbb {R}}$ the associated group of diffeomorphisms. For t small, the family $\{\Sigma _t=\varphi _{t}(\Sigma )\}_{t\in \mathbb {R}}$ is the variation of $\Sigma$ induced by U. Let $\Omega _{t}$ be the region enclosed by $\Sigma _t$. We define

$$\begin{aligned} P_{f}(t):=P_{f}(\Omega _{t}),\;\; V_{f}(t):=V_{f}(\Omega _{t}). \end{aligned}$$

We say that the variation is weighted volume-preserving if $V_{f}(t)$ is constant for t small enough. We say that $\Sigma$ is weighted area-stationary under a volume constraint or weighted volume-preserving area-stationary if $P'_{f}(0)=0$ for any weighted volume-preserving variation of $\Sigma$. From [37], we have

$$\begin{aligned} V'_{f}(0)=\int _{\Sigma }ufd\Sigma , \end{aligned}$$

(5.2)

where $u=\langle U,N\rangle$ is the component of U with respect to the outer unit normal vector N.

Adapting the method of Lemma 4.3’s proof in [35], we have the following

Lemma 5.1

Let $\Sigma \subset \mathbb {H}^n$ be a compact hypersurface of class $C^{2}$ enclosing a bounded region $\Omega$, where $\mathbb {H}^n$ is endowed with smooth density $f=e^{\psi }$. Let U be a $C^2$ vector field on $\mathbb {H}^{n}$ with compact support on $\Sigma$ and associated with one-parameter family of diffeomorphisms $\{\varphi _{t}\}_{t\in \mathbb {R}}$. Denote by $u=\langle U,N\rangle$. If $\mathrm{div}_Hv_H\in L^1_{loc}(\Sigma )$, we have

$$\begin{aligned} P'_{f}(0)=-\int _{\Sigma }\mathscr {H}_{H,f}ufd\Sigma -\int _{\Sigma }\mathrm{div}_{\Sigma }(ufv_{H}^\top ){d}\Sigma , \end{aligned}$$

(5.3)

where $\mathscr {H}_{H,f}=\mathscr {H}_{H}-\langle {\hbox {grad}}_{H}\psi ,v_{H}\rangle$ is called the weighted horizontal curvature of $\Sigma$ and $v_{H}^\top$ is the tangent projection of $v_H$ to $\Sigma$.

Moreover, if $\Sigma$ is weighted volume-preserving area-stationary, then

$$\begin{aligned} P'_{f}(0)=-\int _{\Sigma }\mathscr {H}_{H,f}ufd\Sigma . \end{aligned}$$

(5.4)

Proof

First from [5] we know that the Riemannian area of the characterize set $\Sigma _0$ of $\Sigma$ is zero.

Let $\Sigma _t=\varphi _t(\Sigma )$ be the variation of $\Sigma$ associated to U. Using the Coarea formula, we have

$$\begin{aligned} P_{f}(t)=\int _{\Sigma _t}e^{\psi }|N_{H}|{d}\Sigma _=\int _{\Sigma }(e^{\psi }|N_{H}|)\circ \varphi _ {t})\mathrm{Jac}\varphi _{t}{d}\Sigma , \end{aligned}$$

where $\mathrm{Jac}\varphi _{t}$ is the Jacobian determinant of the mapping $\varphi _{t}:\Sigma \rightarrow \Sigma _{t}$. And we know $\frac{d}{{d}t}\Big |_{t=0}\mathrm{Jac}\varphi _{t}=\mathrm{div}_{\Sigma }U$. So we get

$$\begin{aligned} \begin{aligned} P'_{f}(0)&=\int _{\Sigma }\frac{d}{{d}t}\Big |_{t=0}[(e^{\psi }|N_{H}|)\circ \varphi _t] d \Sigma +\int _{\Sigma }e^{\psi }|N_{H}|\frac{d}{{d}t}\Big |_{t=0}\mathrm{Jac}\varphi _{t}{d}\Sigma \\&=\int _{\Sigma }U(e^{\psi }|N_{H}|){d}\Sigma +\int _{\Sigma }e^{\psi }|N_{H}|\mathrm{div}_{\Sigma }Ud\Sigma \\&=\int _{\Sigma }U^{\bot }(e^\psi |N_{H}|){d}\Sigma +\int _{\Sigma }\mathrm{div}_{\Sigma }(e^{\psi }|N_{H}|U){d}\Sigma \\&=\int _{\Sigma }\Big [U^{\bot }(e^\psi |N_{H}|)+e^{\psi }|N_{H}|\mathrm{div}_{\Sigma }U^{\bot })\Big ]{d}\Sigma +\int _{\Sigma }\mathrm{div}_{\Sigma } (e^{\psi }|N_{H}|U^{\top }){d}\Sigma \\&=\int _{\Sigma }U^{\bot }(e^\psi )|N_{H}|{d}\Sigma +\int _{\Sigma }e^{\psi }\Big [U^{\bot }(|N_{H}|)+|N_{H}|div_{\Sigma }U^{\bot }\Big ]{d}\Sigma ,\\ \end{aligned} \end{aligned}$$

(5.5)

where $U^\bot$ denotes the normal component of U along $\Sigma$ and we used the divergence theorem to get $\int _{\Sigma }\mathrm{div}_{\Sigma } (e^{\psi }|N_{H}|U^{\top }){d}\Sigma =0$ under the condition that U has compact support on $\Sigma$.

From [35], we know in $\Sigma -\Sigma _{0}$

$$\begin{aligned} U^{\bot }(|N_{H}|)+|N_{H}|\mathrm{div}_{\Sigma }U^{\bot }=-\mathrm{div}_{\Sigma }(uv_{H}^{\top })+u\mathrm{div}_{\Sigma }v_{H}. \end{aligned}$$

As a consequence, (5.5) turns into

$$\begin{aligned} \begin{aligned} P'_{f}(0)&=\int _{\Sigma -\Sigma _0}U^{\bot }(e^{\psi })|N_{H}|{d}\Sigma +\int _{\Sigma -\Sigma _0} e^{\psi }\Big [-\mathrm{div}_{\Sigma }uv_{H}^{\top } +u\mathrm{div}_{\Sigma }v_{H}\Big ]{d}\Sigma \\&=\int _{\Sigma -\Sigma _0}u\Big [\langle N,grad\psi \rangle |N_{H}|+\langle v_{H}^{\top },\mathrm{grad}\psi \rangle -\mathscr {H} _{H}\Big ]e^{\psi }{d}\Sigma -\int _{\Sigma -\Sigma _0}\mathrm{div}_H(ue^\psi v_H^\top ){d}\Sigma \\&=\int _{\Sigma -\Sigma _0}-u(\mathscr {H}_{H}-\langle \mathrm{grad}_{H}\psi ,v_{H}\rangle )e^{\psi }{d}\Sigma -\int _{\Sigma -\Sigma _0}\mathrm{div}_H(ue^\psi v_H^\top ){d}\Sigma \\&=\int _{\Sigma }-u(\mathscr {H}_{H}-\langle \mathrm{grad}_{H}\psi ,v_{H}\rangle ) fd\Sigma -\int _{\Sigma }\mathrm{div}_H(uf v_H^\top ){d}\Sigma .\\ \end{aligned} \end{aligned}$$

Let $\mathscr {H}_{H,f}=\mathscr {H}_H-\langle \mathrm{grad}_{H}\psi ,v_{H}\rangle$, then we have

$$\begin{aligned} P'_{f}(0) =-\int _{\Sigma }u\mathscr {H}_{H,f}fd\Sigma -\int _{\Sigma }\mathrm{div}_\Sigma (uf v_H^\top ){d}\Sigma . \end{aligned}$$

Now we shall prove (5.4) for area-stationary surfaces under a weighted volume constraint. First inserting in (5.3) mean zero functions of class $C^1$ with compact support in $\Sigma -\Sigma _0$, we get that $\mathscr {H}_{H,f}$ is constant on $\Sigma -\Sigma _0$. If $u:\Sigma \rightarrow \mathbb {R}$ is any function, then we consider $v:\Sigma \rightarrow \mathbb {R}$ with compact support in $\Sigma -\Sigma _0$ such that $\int _\Sigma (u+v){d}\Sigma =0$. Then inserting the mean zero function $u+v$ in (5.3), taking into account that $\mathscr {H}_{H,f}$ is constant, we deduce that $\int _{\Sigma -\Sigma _0}\mathrm{div}_\Sigma (ue^\psi v_H^\top ){d}\Sigma =0$. So (5.3) is proved. $\square$

We say that $\Sigma$ has constant weighted horizontal mean curvature if $\mathscr {H}_{H,f}$ is constant on $\Sigma -\Sigma _0$.

Corollary 5.1

Let $\Omega \subset \mathbb {H}^n$ be a bounded region enclosed by a compact hypersurface $\Sigma$ of class $C^{2}$, where $\mathbb {H}^n$ is endowed with smooth density f. If $\Sigma$ is weighted area-stationary under a weighted volume constraint, then $\Sigma$ has constant weighted horizontal mean curvature.

Proof

Let U be any weighted volume-preserving variation on $\Sigma$. Setting $u=\langle U,N\rangle$, because $\Sigma$ is weighted area-stationary under a weighted volume constraint, then from (5.2) and (5.4) we get

$$\begin{aligned} \int _{\Sigma }ufd\Sigma =0, \int _{\Sigma }\mathscr {H}_{H,f}ufd\Sigma =0. \end{aligned}$$

A standard argument implies $\mathscr {H}_{H,f}=const.$$\square$

Corollary 5.2

The compact hypersurface of class $C^2$ enclosing a weighted isoperimetric set in $\mathbb {H}^{n}$ with density has constant weighted horizontal mean curvature.

Theorem 5.1

In the Heisenberg group $\mathbb {H}^n$ with density $f(|z|)=e^{\psi (|z|)}$, if bubble sets are weighted isoperimetric, then, up to a constant multiplicator, the density f(|z|) is only of the form $|z|^p$.

Proof

Up to a left translation and a dilation, any bubble set can be discribed as the following

$$\begin{aligned} E_\mathrm{isop}=\{(z,t)\in \mathbb {H}^{n}:|t|<\arccos |z|+|z|\sqrt{1-|z|^{2}},|z|<1\}. \end{aligned}$$

First we compute the weighted horizontal mean curvature of Pansu sphere $\partial E_\mathrm{isop}$. In fact, from the equation of $\partial E_\mathrm{isop}$ we deduce that the horizontal unit normal of $\partial E_\mathrm{isop}^+$ and $\partial E_\mathrm{isop}^-$, respectively, are the following:

$$\begin{aligned} v_{H}= & {} \sum _{i=1}^{n}\Big [(-x_{i}-\frac{\sqrt{1-|z|^{2}}}{|z|}y_{i})X_{i} +(-y_{i}+\frac{\sqrt{1-|z|^{2}}}{|z|}x_{i})Y_{i}\Big ],\\ v_{H}= & {} \sum _{i=1}^{n}\Big [(-x_{i}+\frac{\sqrt{1-|z|^{2}}}{|z|}y_{i})X_{i} +(-y_{i}-\frac{\sqrt{1-|z|^{2}}}{|z|}x_{i})Y_{i}\Big ]. \end{aligned}$$

Furthermore we obtain the horizontal mean curvature of $\partial E_\mathrm{isop}$

$$\begin{aligned} \mathscr {H}_{H}=-\mathrm{div}_{H}v_{H}=2n. \end{aligned}$$

Consequently, we get the weighted horizontal mean curvature of $\partial E_\mathrm{isop}$

$$\begin{aligned} \mathscr {H}_{H,f}=2n-\langle \mathrm{grad}_{H} \psi (|z|),v_{H}\rangle . \end{aligned}$$

From Corollary 5.3, we know the weighted horizontal mean curvature $\mathscr {H}_{H,f}$ of Pansu sphere is a constant, i.e.,

$$\begin{aligned} 2n-\langle \mathrm{grad}_{H} \psi (|z|),v_{H}\rangle =C_1, \end{aligned}$$

where $C_1$ is constant.

Futhermore we have

$$\begin{aligned} \psi '(|z|)|z|=-2n+C_1. \end{aligned}$$

(5.6)

Then the solution of the Eq. (5.6)

$$\begin{aligned} \psi (|z|)=\ln |z|^{-2n+C_1}+C_2. \end{aligned}$$

So we have

$$\begin{aligned} f(|z|)=e^{\psi (|z|)}=C|z|^{(-2n+C_1)}. \end{aligned}$$

This ends the proof.