Laguerre BV spaces, Laguerre perimeter and their applications

1.   Introduction

The spaces BV of functions of bounded variation in Euclidean spaces have been a class of function space which can be used in the geometric measure theory. For example, when working with minimization problems, reflexivity or the weak compactness property involving the function space $W^{1, p}(\mathbb R^{d})$ for $p > 1$, in such cases, the space BV usually plays a crucial role. However, for the case of the space $W^{1, 1}(\mathbb R^{d})$, one possible approach to address its lack of reflexivity is to consider the space $BV(\mathbb R^{d})$. The importance of generalizing the classical notion of variation has been pointed out in several occasions by E. De. Giorgi in ^[1]. Recently, Huang, Li and Liu in ^[2] investigate the capacity and perimeters derived from $\alpha$-Hermite bounded variation. In a general framework of strictly local Dirichlet spaces with doubling measure, Alonso-Ruiz, Baudoin and Chen et al. in ^[3] introduce the class of bounded variation functions and proved the Sobolev inequality under the Bakry-$\acute{\text{E}}$mery curvature type condition. For further information on this topic, we refer the reader to ^[4,5,6] and the references therein.

One of the aims of this paper is intended to explore and analyze a number of fundamental inquiries in geometric measure theory that are associated with the Laguerre operator in Laguerre BV spaces. To begin with, we will provide a brief introduction to the Laguerre operator.

Given a multiindex $\alpha = (\alpha_1, \cdots, \alpha_d)$, $\alpha\in (-1, \infty)^d$, the Laguerre differential operator is defined by:

Let the probabilistic gamma measure $\mu_{\alpha}$ in $\mathbb{R}_{+}^{d} = (0, \infty)^{d}$ be defined as

As we know that $\mathcal{L}^{\alpha}$ is positive and symmetric in $L^2(\mathbb{R}_{+}^{d}, d\mu_{\alpha})$, and it has a closure which is selfadjoint in $L^2(\mathbb{R}_{+}^{d}, d\mu_{\alpha})$ and will be denoted by $\mathcal{L}^{\alpha}$. The $i$-th partial derivative associated with $\mathcal{L}^{\alpha}$ is defined as

see ^[7] or ^[8]. The operator $\mathcal{L}^{\alpha}$ has the following decomposition:

where

is the formal adjoint of $\delta_i$ in $L^2(\mathbb{R}_{+}^{d}, \mathrm{d}\mu_{\alpha})$. Throughout this paper, suppose that $\Omega \subset \mathbb R_{+}^{d}$ be an open set. For $u \in C^1(\mathbb R_{+}^d)$ and $\varphi = (\varphi_1, \varphi_2, \ldots, \varphi_d) \in C^1(\mathbb R_{+}^d, \mathbb{R}^{d})$, define the ${\mathcal L}^{\alpha}$-gradient and ${\mathcal L}^{\alpha}$-divergence operators that are associated with $\mathcal{L}^{\alpha}$:

which also gives

Naturally, we denote by ${\mathcal B}{\mathcal V}_{\mathcal L^{\alpha}}(\Omega)$ the set of all functions possessing Laguerre bounded variation ($\mathcal{L}^{\alpha}$-BV in short) on $\Omega$. Based on the results of ^[2], we investigate some related topics for the Laguerre setting, and the plan of the notes is given as follows. Section 2.1 collects some basic facts and notations used later, the lower semicontinuity (Lemma 2.1), the completeness (Lemma 2.2), the structure theorem (Theorem 2.3) and approximation via $C_{c}^{\infty}$-functions (Theorem 2.4). Unlike Theorem 2 in [9, Section 5.2.2], we must utilize the mean value theorem for multivariate functions and the intrinsic nature of the Laguerre variation. Section 2.2 is focused on the perimeter $P_{ \mathcal{L}^{\alpha}}(\cdot, \Omega)$ induced by ${\mathcal B}{\mathcal V}_{\mathcal L^{\alpha}}(\Omega)$, as shown in equation (2.6) below.

Remember that the classical perimeter of $E\subseteq \mathbb{R}^d$ is defined as

here let $\mathcal{F}(\mathbb R^d)$ be the set that contains all functions

satisfying

As we all know that

is an inherent property of $P(E)$ at the elementary level.

In Lemma 2.10, we proved that (1.1) is valid for the Laguerre perimeter $P_{ \mathcal{L}^{\alpha}}(\cdot)$. In Section 2.3, a coarea formula for $\mathcal{L}^{\alpha}$-BV functions is derived. In Theorem 2.12, we conclude that the isoperimetric inequality

shares equivalence with the Sobolev type inequality

as an application. We point out that, in the proof of (1.2), the inequality $|\nabla f(x)|\lesssim| \nabla_{\mathcal{L}^{\alpha}} f(x)|$ on $\Omega_{1}$ holds true. With this in mind, we consider the subset

of $\Omega$ which is a reasonable substitute of $\Omega$ and whose figure is given as follows:

Our motivation comes not only from the fact that these objects are interesting on their own, but also from the possibility of their potential applications in further research concerning the Laguerre operator. Consequently, our aim in Section 3 is to examine the Laguerre mean curvature of a set that has a finite Laguerre perimeter. It is interesting to note that the sets of finite perimeter introduced by E. De Giorgi for the Laplace operator $\Delta$ have found applications in classical problems of the calculus of variations, such as the Plateau problem and the isoperimetric problem, see ^[10,11,12]. Barozzi, Gonzalez, and Tamanini ^[13] demonstrated that for any finite classical perimeter set $E$ within $\mathbb R^{d}$, its mean curvature is included in $L^1(\mathbb R^d)$. One might naturally wonder whether $P_{ \mathcal{L}^{\alpha}}(E, \Omega)$, $\alpha\in (-1, \infty)^{d}$ holds similarly as ^[13]. Note that it is necessary to use identity (1.1) in the proof of the main theorem of ^[13]. In Theorem 3.1, we generalize the result of ^[13] to ${P}_{ \mathcal{L}^{\alpha}}(\cdot, \Omega_1)$ and show that if a set $E$ is a subset of $\Omega_1$ such that ${P}_{ \mathcal{L}^{\alpha}}(E, \Omega_1) < \infty$, then the mean curvature of $E$ is in $L^{1}(\Omega_1, d\mu_{ \alpha})$.

Throughout this paper by $C$ we always denote a positive constant that may vary at each occurrence; $A\approx B$ means that $\frac{1}{C}A\le B\le CA$ and the notation $X\lesssim Y$ is used to indicate that $X\le CY$ with a positive constant $C$ independent of significant quantities. Similarly, one writes $X \gtrsim Y$ for $X\ge CY$.

2.   $\mathcal{L}^{\alpha}$-BV functions

2.1.   Fundamentals of $\mathcal{L}^{\alpha}$-BV Space

This section presents the $\mathcal{L}^{\alpha}$-BV space, which is defined as the set of all functions that exhibit Laguerre bounded variation and investigates its properties. The Laguerre variation ($\mathcal{L}^{\alpha}$-variation in short) of $f \in {L^1}(\Omega, d\mu_{\alpha})$ is defined by

where ${\mathcal F}(\Omega)$ denotes the class of all functions

satisfying

We say that an function $f \in {L^1}(\Omega, d\mu_{\alpha})$ has the $\mathcal{L}^{\alpha}$-bounded variation on $\Omega$ if

and denote by ${\mathcal B}{\mathcal V}_{\mathcal L^{\alpha}}(\Omega)$ the class of all such functions, and it is a Banach space with the norm

Definition 2.1. Suppose $\Omega$ is an open set in $\mathbb{R}_{+}^{d}$. Let $1 \le p \le \infty$. The Sobolev space $W_{ \mathcal{L}^{\alpha}}^{k, p}(\Omega)$ associated with $\mathcal{L}^{\alpha}$ is defined as the set of all functions $f \in {L^p}(\Omega, d\mu_{ \alpha})$ such that

The norm of $f\in W_{ \mathcal{L}^{\alpha}}^{k, p}(\Omega)$ is given by

The upcoming results will gather certain properties of the space $\mathcal{BV}_{ \mathcal{L}^{\alpha}}(\Omega)$. We omit the details of their proofs, since we can use the similar arguments as ^[2] to prove them.

Lemma 2.1. $({\rm{i}})$ Suppose $f \in W_{ \mathcal{L}^{\alpha}}^{1, 1} (\Omega)$, then

which implies $W_{ \mathcal{L}^{\alpha}}^{1, 1} (\Omega)\subseteq \mathcal{BV}_{ \mathcal{L}^{\alpha}}(\Omega).$

$({\rm{ii}})$ (Lower semicontinuity). Suppose $f_k \in {\mathcal B}{\mathcal V}_{\mathcal L^{\alpha}}(\Omega), \ k \in \mathbb N$ and $f_k \to f$ in $L_{loc}^1(\Omega, d\mu_\alpha)$, then

Lemma 2.2. The space $\big({\mathcal B}{\mathcal V}_{\mathcal L^{\alpha}}(\Omega), \left \|\cdot \right\|_{ {\mathcal B}{\mathcal V}_{\mathcal L^{\alpha}}(\Omega)}\big)$ is a Banach space.

The Hahn-Banach theorem and the Riesz representation theorem can be used to prove the structure theorem for $\mathcal{L}^{\alpha}$-BV functions, as presented in the following lemma.

Lemma 2.3. (Structure theorem for $\mathcal{BV}_{ \mathcal{L}^{\alpha}}$ functions). Let $f\in {\mathcal B}{\mathcal V}_{\mathcal L^{\alpha}}(\Omega)$. Then there exists a Radon measure $\mu_{ \mathcal{L}^{\alpha}}$ on $\Omega$ such that

for every $\varphi \in C_{c}^{\infty} (\Omega, \mathbb{R}^{d})$ and

where $|\mu_{ \mathcal{L}^{\alpha}}|$ represents the total variation of the measure $\mu_{ \mathcal{L}^{\alpha}}$.

We can obtain an approximation result for the $\mathcal{L}^{\alpha}$-variation in the following theorem.

Theorem 2.4. Let $\Omega_1$ be an open set defined in (1.3). Assume that $u\in {\mathcal B}{\mathcal V}_{\mathcal L^{\alpha}}(\Omega_1)$, then there exists a sequence $\{u_h\}_{h\in\mathbb{N}}\in {\mathcal B}{\mathcal V}_{\mathcal L^{\alpha}}(\Omega_1) \cap C ^\infty (\Omega_1)$ such that

and

Proof. The approach we take differs from the proof presented in [9, Section 5.2.2, Theorem 2] as we utilize the mean value theorem of multivariate functions and the intrinsic nature of the $\mathcal{L}^{\alpha}$-variation. Via the lower semicontinuity of ${\mathcal L}^{\alpha}$-BV functions, it suffices to demonstrate that for $\varepsilon > 0$ there exists a function $u_\varepsilon\in C^\infty(\Omega_1)$ such that

and

Fix $\varepsilon > 0$. If $m$ is a given positive integer, then construct a series of open sets,

where $\mathrm{dist}(x, \partial\Omega_1) = \inf\{|x-y|:y\in \partial \Omega_1\}$. Note that ${\Omega_{1, j}} \subset {\Omega_{1, j + 1}} \subset \Omega_1, \ j\in\mathbb{N}$ and $\mathop\cup\limits_{j = 0}^\infty{\Omega_{1, j}} = \Omega_1$. Since $| \nabla_{\mathcal{L}^{\alpha}} u|(\cdot)$ is a measure, then choose a value $m\in\mathbb{N}$ to be sufficiently large such that

Set ${U_0}: = \Omega_{1, 0}$ and $U_j: = \Omega_{1, j+1}\backslash {\overline\Omega}_{1, j-1}$ for $j\ge1$. Based on the standard outcomes from [9, Section 5.2.2, Theorem 2], our inference is that there exists a partition of unity connected to the covering $\{U_j\}_{j\in\mathbb N}$. Namely, there exist functions $\{f_j\}_{j\in\mathbb N} \in C_c^\infty(U_j)$ such that $0\le f_j \le 1, \ j\ge0$ and $\sum\limits_{j = 0}^\infty f_j = 1$ on $\Omega_1$. Thus we have the fact that

on $\Omega_1$. Given $\varepsilon > 0$ and $u \in {L^1}(\Omega_1, \mathbb{R})$, extended to zero out of $\Omega_1$, the regularization can be defined as

where $\eta\in C_c^\infty (\mathbb{R}_{+}^d)$ is a nonnegative radial function satisfying

and $\text{supp}\ \eta \subset B(0, 1)\cap \mathbb R_{+}^d$. Then for each $j$, there exists $0 < \varepsilon_j < \varepsilon$ so small such that

Construct

In some neighborhood of each point $x\in \Omega_1$, there are only finitely many nonzero terms in this sum, hence $v_\varepsilon \in C^\infty(\Omega_1)$ and $u = \sum\limits_{j = 0}^\infty {u{f_j}}$. Therefore, by a simple computation, we obtain

Consequently,

Now, assume $\varphi\in C_c^1(\Omega_1, \mathbb{R}^d)$ and $|\varphi|\le 1$. We decompose the integral as follows:

where

For the sake of research, let

As for $I$, we obtain

where in the last equality we have used the fact (2.2). In fact, when $\|\varphi\|_{L^\infty}\le 1$, then $|f_j(\eta_{\varepsilon_j} *\varphi)(x)|\le 1, \ j\in\mathbb{N}$, and each point in $\Omega$ is contained in at most three of the sets $\{U_j\}_{j = 0}^\infty$. Furthemore, (2.3) implies that $|I_2| < \varepsilon$.

On the other hand, we modify the integration order to obtain

Therefore, the estimation presented for term $I_2$ above indicates that

where

and

Furthermore,

where we applied the fact (2.1) in the final inequality. Note that $\psi(x_k) = \frac{\alpha_k+\frac{1}{2}-x_k}{\sqrt{x_k}}$, $\| \varphi\|_{L^\infty}\le 1$ and $\text{supp}\ \eta\subseteq B(0, 1)\cap \mathbb R_{+}^d$. Assuming $|x_k-y_k| < {\varepsilon_j} < |y_k|/{2}$, the mean value theorem of multivariate functions guarantees the existence of $\theta\in (0, 1)$ such that

Consequently, we obtain

where we have used the facts that

and in the third inequality we have used the fact that

Through taking the supremum over $\varphi$ and considering the arbitrariness of $\varepsilon > 0$, we prove the theorem.

Remark 2.5. By computation, we conclude that the function $u\in {\mathcal B}{\mathcal V}_{\mathcal L^{\alpha}}(\Omega)$ satisfies (2.5) in Theorem 2.4 when $d\ge 3$, at this time, Theorem 2.4 is valid for any open set $\Omega\subseteq \mathbb{R}^{d}$.

Additionally, the max-min property of the $\mathcal{L}^{\alpha}$-variation can be observed from Lemma 2.1 and Theorem 2.4.

Theorem 2.6. Let $\Omega_1$ be an open set defined in (1.3). Suppose $u, v\in L^1(\Omega_1, d\mu_\alpha)$, then

Proof. One may assume, without any loss of generality,

By Theorem 2.4, we take two functions

such that

Since

Via Lemma 2.1, it follows that

2.2.   Basic properties of Laguerre perimeter

This subsection presents a new type of perimeter: the Laguerre perimeter ($\mathcal{L}^{\alpha}$-perimeter in short). Moreover, we establish the related results for it.

We define the $\mathcal{L}^{\alpha}$-perimeter of $E\subset \Omega$ as follows:

where $\mathcal F(\Omega)$ is defined in Section 2.1. Specifically, we will also use the notation

We immediately deduce Lemma 2.1 by replacing $f$ with $1_{E}$.

Corollary 2.7. (Lower semicontinuity of $P_{ \mathcal{L}^{\alpha}}$). Assume $1_{E_k}\to 1_E$ in $L_{loc}^1(\Omega, d\mu_\alpha)$, where $E$ and $E_k$, $k\in\mathbb N,$ are subsets of $\Omega$, then

Additionally, utilizing Theorem 2.6 and selecting $u = 1_E$ and $v = 1_F$ for every compact subsets $E, F$ in $\Omega_1$, we can promptly acquire the subsequent corollary. According to Xiao and Zhang's result in [14, Section 1.1 (ⅲ)], the equality condition of (2.7) is also provided by us.

Corollary 2.8. For all compact subsets $E, F$ within $\Omega_1$, we get

where $\Omega_1$ is an open set defined in (1.3). Especially, if $P_{ \mathcal{L}^{\alpha}}(E\setminus (E\cap F), \Omega_1) \cdot P_{ \mathcal{L}^{\alpha}}(F\setminus (F\cap E), \Omega_1) = 0$, the equality of (2.7) holds true.

Proof. Given that (2.7) is true, we only need to demonstrate that its opposite inequality is also valid, provided that the above condition is satisfied. It is evident that the condition $P_{ \mathcal{L}^{\alpha}}(E\setminus (E\cap F), \Omega_1)\cdot P_{ \mathcal{L}^{\alpha}}(F\setminus (E\cap F), \Omega_1) = 0$ leads to $P_{ \mathcal{L}^{\alpha}}(E\setminus (E\cap F), \Omega_1) = 0$ or $P_{ \mathcal{L}^{\alpha}}(F\setminus (E\cap F), \Omega_1) = 0$. Suppose $P_{ \mathcal{L}^{\alpha}}(E\setminus (E\cap F), \Omega_1) = 0$. By (2.7), we have

Via (2.6) and $E\cup F = F\cup (E\setminus (E\cap F))$, we have

Combining (2.8) with (2.9) deduces that

the desired result can be obtained from it. Another similar case can be proven as well, but the details are omitted.

We will now demonstrate that sets with finite $\mathcal{L}^{\alpha}$-perimeter satisfy the Gauss-Green formula.

Theorem 2.9. (Gauss-Green formula). Let $E\subseteq\Omega$ be subset with finite $\mathcal{L}^{\alpha}$-perimeter. Then we have

where the outward normal to $E$ is represented by the unit vector $\vec{n}(x)$ and $\mathrm{\widetilde{div}}_{ \mathcal{L}^{\alpha}}(\cdot)$ is defined in (2.4).

Proof. By calculating, we have

where we have used the classical Gauss-Green formula and the following facts regarding the derivatives of $\omega(x)$:

for $1\le i\le d$. This completes the proof.

Lemma 2.10. If a set $E$ is in $\Omega$ and has finite $\mathcal{L}^{\alpha}$-perimeter, then

Proof. For any $\varphi\in \mathcal {F}(\mathbb R_{+}^{d})$, since ${P}_{ \mathcal{L}^{\alpha}}(E, \Omega) < \infty$, then

Via the extended Gauss-Green formula (Theorem 2.9) and taking into consideration the fact that $\varphi$ has a compact support, we obtain

where the unit exterior normal vector to $E$ at $x$ is denoted by $\vec{n}(x)$. The arbitrary nature of $\varphi$ results in the attainment of

through the use of supremum.

2.3.   Coarea formula of $\mathcal{L}^{\alpha}$-BV functions and the Sobolev inequality

Below we prove the coarea formula and the Sobolev inequality for $\mathcal{L}^{\alpha}$-perimeter.

Theorem 2.11. Let $\Omega_1$ be an open set defined in (1.3). If $f\in {\mathcal B}{\mathcal V}_{\mathcal L^{\alpha}}(\Omega_1)$, then

where $E_t = \{x\in\Omega_1: f(x) > t\}$ for $t\in\mathbb{R}$.

Proof. At first, assume

It is straightforward to prove that for $i = 1, 2, \ldots, d$,

and

where the proof of [9, Section 5.5, Theorem 1] displays the latter. Therefore,

Therefore, we conclude that for all $\varphi\in \mathcal{F}(\Omega_1)$,

Furthermore,

Secondly, it can be assumed without any loss of generality that we simply need to confirm that

holds for $f\in\mathcal{BV}_{ \mathcal{L}^{\alpha}}(\Omega_1)\bigcap C^\infty(\Omega_1)$. The idea of [15, Proposition 4.2] can be referenced in this proof. Denote by

Obviously,

Define the following function $g_h$ as

where $t\in \mathbb{R}$. Set the sequence ${v_h}(x): = g_h(f(x))$. At this time, ${v_h} \to {1_{{E_t}}}$ in $L^1(\Omega_1, d\mu_{ \alpha})$. In fact,

As $h\rightarrow \infty$, $\{x\in\Omega_1: t < f(x)\le t+1/h\}\rightarrow \emptyset$, we then obtain

By utilizing Theorem 2.4 and taking the limit as $h \rightarrow \infty$ we can derive

Integrating (2.11) reaches

Ultimately, through approximation and using the lower semicontinuity of the $\mathcal{L}^{\alpha}$-perimeter, we can deduce that (2.10) is valid for every $f\in\mathcal {BV}_{ \mathcal{L}^{\alpha}}(\Omega_1)$.

We can eventually establish the Sobolev inequality and the isoperimetric inequality for $\mathcal{L}^{\alpha}$-BV functions. Since the domain $\Omega_{1}$ is a reasonable substitute of $\Omega$, we can obtain the isoperimetric inequality and the Sobolev inequality for $f\in\mathcal{BV}_{ \mathcal{L}^{\alpha}}(\Omega_{1})$, where $\Omega_{1}$ is given in (1.3).

Theorem 2.12.

$({\rm{i}})$ (Sobolev inequality). Let $\Omega_1$ be an open set defined in (1.3). Then for all $f\in {\mathcal{BV}}_{ \mathcal{L}^{\alpha}}(\Omega_1)$, we have

$({\rm{ii}})$ (Isoperimetric inequality). Suppose that $E$ is a bounded set having finite $\mathcal{L}^{\alpha}$-perimeter in $\Omega_1$. Then

$({\rm{iii}})$ The two statements mentioned above are equivalent.

Proof. (ⅰ) Let

such that

Since $\Omega_{1} = \Omega\setminus \{x\in\mathbb{R}_{+}^d:\exists i\in 1, \cdots, d\ \mathrm{such}\ \mathrm{that}\ \sqrt{x_i} < 1\}$, then for any $i = 1, \cdots, d$, we obtain $\sqrt{x_i}\geq 1$. It is easy to see that

After applying Fatou's lemma and the weighted Gagliardo-Nirenberg-Sobolev inequality, we get

where the relation between the gradient $\nabla$ and the Laguerre gradient $\nabla_{\mathcal{L}^{\alpha}}$ has been applied in (2.14).

(ⅱ) By setting $f = 1_E$ in (2.12), it can be demonstrated that (2.13) is true.

(ⅲ) Apparently, the implication from (ⅰ) to (ⅱ) has been proved. The statement below demonstrates that (ⅱ) implies (ⅰ). Let $0\le f\in C^\infty_c(\Omega_1)$. Applying the coarea formula from Theorem 2.11 and (ⅱ), we obtain

where $E_t = \big\{x\in\Omega_1:\ f(x) > t\big\}$. Let

One can easily observe that

Moreover, we can verify that $\chi(t)$ increases monotonically on $(0, \infty)$ and for any positive $h$,

Then $\chi(t)$ can be considered a Lipschitz function locally and $\chi'(t)\le |E_t|^{\frac{d-1}{d}}$ for a.e. $t\in (0, \infty)$. Thus,

Finally, Theorem 2.4 establishes the validity of (2.12) for all $f\in {\mathcal{BV}}_{ \mathcal{L}^{\alpha}}(\Omega_1)$.

As a direct result of the proof of (ⅰ) in Theorem 2.12, we can get the following corollary.

Corollary 2.13. Let $1 < p < d$ and let $\Omega_1$ be an open set defined in (1.3). For any $f\in W_{ \mathcal{L}^{\alpha}}^{1, 1}(\Omega_{1})$ one has

Proof. For some $\gamma > 1$ to be fixed later, via the Lemma 2.1 (ⅰ) and Hölder inequality we obtain

Choosing

and noting

then we conclude that (2.15) holds true.

3.   Laguerre mean curvature

The main concern of this section is to determine if the mean curvature of every set with finite $\mathcal{L}^{\alpha}$-perimeter in $\Omega_1\subseteq\mathbb R_{+}^{d}$ belongs to $L^{1}(\Omega_1, d\mu_{ \alpha})$. To obtain comprehensive information on the classical case, kindly consult ^[13]. In order to prove Theorem 3.1, it is necessary to use the important result for the Laguerre perimeter in Corollary 2.8. Therefore, we assume that the dimension $d\ge 3$ via Remark 2.5.

For a given $u\in L^{1}(\Omega_1, d\mu_{ \alpha})$, the functional corresponding to the $\mathcal{L}^{\alpha}$-perimeter, known as Massari type, is given by

where an arbitrary set of finite $\mathcal{L}^{\alpha}$-perimeter in $\mathbb R_{+}^{d}$ is denoted by $E$.

Theorem 3.1. For every set $E\subset\mathbb R_{+}^{d}$ that has finite $\mathcal{L}^{\alpha}$-perimeter, a function $u$ belonging to $L^{1}(\mathbb R_{+}^{d}, d\mu_{ \alpha})$ exists such that

is satisfied for every set $F\subset\Omega_1$ with finite $\mathcal{L}^{\alpha}$-perimeter.

Proof. Initially, we must identify a function $u\in L^{1}(\Omega_1, d\mu_{ \alpha})$ for a specified set $E$ such that

is true for every $F$ with either $F\subset E$ or $E\subset F$, then Theorem 3.1 is demonstrated, indicating that (3.1) applies to every $F\subset \Omega_1$. By including the inequality (3.1) that pertains to the test sets $E\cap F$ and $E\cup F$, we have

After taking that

we can get

that is, (3.1) holds for arbitrary $F$. Moreover, if (3.1) is vaild for a set $F\subset E$, then it is also applicable to the set $F$ such that $E\subset F$, i.e. $\Omega_1\backslash F \subset\Omega_1\backslash E$,

where we have utilized lemma 2.10 along with the property that $u$ equals zero outside of the set $E$. Therefore, it is sufficient to prove that the integrability of $u$ on $E$ is established and that (3.1) is valid for every $F\subset E$.

Step I. Let $h(\cdot)$ be a measurable function on $E$ such that $h > 0$ and $\int_{E}h(x) d\mu_\alpha(x) < \infty$, and let $\Lambda$ be a measure that is both positive and totally finite:

Since $\Lambda(F) = 0$ if and only if $\mu_{ \alpha}(F) = 0$ is clearly true. For $\lambda > 0$ and $F\subset E$, we will examine the following functional

A commonly recognized fact is that any minimizing sequence is compact within $L^{1}_{loc}(\Omega_1, d\mu_{ \alpha})$, and this functional is lower semi-continuous in regards to the same convergence. Thus, we can deduce that, for any positive value of $\lambda$, there is a solution $E_{\lambda}$ to the problem:

Select a strictly increasing sequence of positive numbers $\{\lambda_{i}\}$ that tend to $\infty$ and use $E_{i}\equiv E_{\lambda_{i}}$ to refer to the associated solutions, so that $\forall i\ge 1$,

Given $i < j$. Let $F = E_{i}\cap E_{j}$. It follows from (3.3) that

that is,

this suggests

A direct computation gives

Conversely, by choosing $F = E_{i}\cup E_{j}\subset E$ from (3.3), we can obtain $\mathscr{F}_{\lambda_{j}}(E_{j})\leq \mathscr{F}_{\lambda_{j}}(E_{i}\cup E_{j})$. Hence,

equivalently,

which implies that

Remember that $h$ is a positive number. The previous estimate, along with (3.2) and the condition $\lambda_{i} < \lambda_{j}$, suggests that

i.e., $E_{i}\subset E_{j}$ and the sequence of minimizers $\{E_{i}\}$ is monotonically increasing. Conversely, by letting $F = E$, we get

which infers that $E_{i}$ converges to $E$ in a monotonic manner and within $L^{1}_{loc}(\mathbb R_{+}^{d}, d\mu_{ \alpha})$. Using Lemma 2.1 (ⅱ), we have

which means

Step II. Define $\lambda_{0} = 0$ and $E_{0} = \emptyset$, and let

It is evident that $u$ is negative almost everywhere on $E$, and

In (3.3), taking $F = E_{i+1}$, we have

that is, for every $i\geq 0$,

For values of $N$ that are large enough, we have

Letting $N\rightarrow \infty$, (3.4) indicates that

Let's assume an additional condition that $0 < \lambda_{i+1}-\lambda_{i}\leq c$, $i\geq 0$, where $c$ is a constant that doesn't depend on $i$, we can say that for any $N > 0$,

which gives

Then

In conclusion, $u\in L^{1}(\mathbb R_{+}^{d}, d\mu_{ \alpha})$.

Step III. We contend that the inequality

is vaild for all $F\subset E$ and every $i\geq 1$.

If $i = 1$, then $E_{i-1} = E_{0} = \emptyset$. Substituting this into (3.5) yields

which coincides with (3.3) for $i = 1$.

Now we assume that (3.5) holds for a fixed $i\geq 1$ and every $F\subset E$. Take $F\cap E_{i}$ as a test set. Observe that $\{E_{j}\}$ is increasing. It is evidently clear to show that

Then

Conversely, $\mathscr{F}_{\lambda_{i+1}}$ is minimized by $E_{i+1}$. Hence,

and noticing that

it is possible for us to obtain

This deduces

Therefore, we obtain that

i.e., (3.5) is true for $i+1$. Last but not least,

which gives (3.3).

Acknowledgments

This work was supported by National Natural Science Foundation of China (No. 11671031, No. 12271042) and Beijing Natural Science Foundation of China (No. 1232023).

Conflict of interest

The authors declare there is no conflict of interest.