Achilles Tertikas, Kyril Tintarev
We show existence of minimizers for the Hardy-Sobolev-Maz'ya inequality in $R^{m+n}\setminus\R^n$ for $m=1$ and $n>2$ or for $m>2$ and $n>0$.
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In this paper we establish several Hardy and Hardy-Sobolev type inequalities with homogeneous weights on the first orthant $\displaystyle \mathbb{R}_{*}^n:=\{(x_1, \ldots, x_n):x_1>0, \ldots, x_n>0 \}$. We then use some of them to produce Hardy type inequalities with remainder terms. Furthermore, we obtain some interpolation inequalities and Maz'ya type inequalities with remainder terms with the help of Maz'ya inequality and Sobolev inequality of Cabré and Ros-Orton on the upper half space $\mathbb{R}_{+}^n:=\{x=(x_1, \ldots, x_n)|\, x_n>0 \}$.
Weighted fractional Hardy-Sobolev and Hardy-Sobolev-Maz'ya inequalities with singularities on flat submanifold
本文研究了带权重的分数Hardy-Sobolev和Hardy-Sobolev-Maz'ya不等式，探讨了奇点在平坦子流形上的应用，并证明了相关不等式在特定条件下的不成立性。
In this article we will study the existence, multiplicity and Morse index of sign changing solutions for the Hardy-Sobolev-Maz'ya(HSM) equation in bounded domain and involving critical growth. We obtain infinitely many sign changing solutions for HSM equation. We also establish an estimate on the Morse index for the sign changing solutions.
Best constant and extremal functions for a class Hardy-Sobolev-Maz'ya inequalities
本研究通过推导一类p-Laplace方程的积分恒等式、分类其正有限能量柱对称解，结合已有文献结果，得到了相关Hardy-Sobolev-Maz'ya不等式的最佳常数与极值函数。
In this work we establish trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants, for weakly mean convex domains. We accomplish this by obtaining a new weighted Hardy type estimate which is of independent inerest. We then produce Hardy-Sobolev-Maz'ya inequalities for the spectral half Laplacian. This covers a critical case left open in \cite{FMT1}.
We prove several Sobolev inequalities, which are then used to establish a fractional Hardy-Sobolev- Maz'ya inequality on the upper halfspace.
In this article we first establish a complete characterization of Hardy's inequalities in $\mathbb{R}^n$ involving distances to different codimension subspaces. In particular the corresponding potentials have strong interior singularities. We then provide necessary and sufficient conditions for the validity of Hardy-Sobolev-Maz'ya inequalities with optimal Sobolev terms.
We study linear and non-linear equations related to the fractional Hardy--Sobolev inequality. We prove nondegeneracy of ground state solutions to the basic equation and investigate existence and qualitative properties, including symmetry of solutions to some perturbed equations.
In this article we establish new improvements of the optimal Hardy inequality in the half space. We first add all possible linear combinations of Hardy type terms thus revealing the structure of this type of inequalities and obtaining best constants. We then add the critical Sobolev term and obtain necessary and sufficient conditions for the validity of Hardy-Sobolev-Maz'ya type inequalities.
By using, among other things, the Fourier analysis techniques on hyperbolic and symmetric spaces, we establish the Hardy-Sobolev-Maz'ya inequalities for higher order derivatives on half spaces. The proof relies on a Hardy-Littlewood-Sobolev inequality on hyperbolic spaces which is of its independent interest. We also give an alternative proof of Benguria, Frank and Loss' work concerning the sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half space. Finally, we show the sharp constant in the Hardy-Sobolev-Maz'ya inequality for bi-Laplacian in the upper half…
In this article, we derive the existence of positive solutions of a semi-linear, non-local elliptic PDE, involving a singular perturbation of the fractional laplacian, coming from the fractional Hardy-Sobolev-Maz'ya inequality, derived in this paper. We also derive symmetry properties and a precise asymptotic behaviour of solutions.
In this paper, we prove Hardy-Leray inequality for three-dimensional solenoidal (i.e., divergence-free) fields with the best constant. To derive the best constant, we impose the axisymmetric condition only on the swirl components. This partially complements the former work by O. Costin and V. Maz'ya \cite{Costin-Mazya} on the sharp Hardy-Leray inequality for axisymmetric divergence-free fields.
We prove a Hardy-Sobolev-Maz'ya inequality for arbitrary domains Ω\subset\R^N with a constant depending only on the dimension N\geq 3. In particular, for convex domains this settles a conjecture by Filippas, Maz'ya and Tertikas. As an application we derive Hardy-Lieb-Thirring inequalities for eigenvalues of Schrödinger operators on domains.
In this paper, we investigate the following elliptic problem involving double critical Hardy-Sobolev-Maz'ya terms: $$ \left\{\begin{array}{ll} -Δu = μ\frac{|u|^{2^*(t)-2}u}{|y|^t} + \frac{|u|^{2^*(s)-2}u}{|y|^s} + a(x) u, & {\rm in}\ Ω,\\ \quad u = 0, \,\, &{\rm on}\ \partial Ω, \end{array} \right. $$ where $μ\geq0$, $a(x)>0$, $2^*(t)=\frac{2(N-t)}{N-2}$, $2^*(s) = \frac{2(N-s)}{N-2}$, $0\leq t<s<2$, $x = (y,z)\in \mathbb{R}^k\times \mathbb{R}^{N-k}$, $2\leq k<N$, $(0,z^*) \in \barΩ$ and $Ω$ is an bounded domain in $\mathbb{R}^N$. Applying an abstract theorem in \cite{sz}, we prove that if $N>…
It is shown that the sharp constant in the Hardy-Sobolev-Maz'ya inequality on the three dimensional upper half space is given by the Sobolev constant. This is achieved by a duality argument relating the problem to a Hardy-Littlewood-Sobolev type inequality whose sharp constant is determined as well.
The paper deals with natural generalizations of the Hardy-Sobolev-Maz'ya inequality and some related questions, such as the optimality and stability of such inequalities, the existence of minimizers of the associated variational problem, and the natural energy space associated with the given functional.
In this work we establish trace Hardy and trace Hardy-Sobolev-Maz'ya inequalities with best Hardy constants, for domains satisfying suitable geometric assumptions such as mean convexity or convexity. We then use them to produce fractional Hardy-Sobolev-Maz'ya inequalities with best Hardy constants for various fractional Laplacians. In the case where the domain is the half space our results cover the full range of the exponent $s \in (0,1)$ of the fractional Laplacians. We answer in particular an open problem raised by Frank and Seiringer \cite{FS}.
Though Adams and Hardy-Adams inequalities can be extended to general symmetric spaces of noncompact type fairly straightforwardly by following closely the systematic approach developed in our early works on real and complex hyperbolic spaces, higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities are more difficult to establish. The main purpose of this goal is to establish the Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane. A crucial part of our work is to establish appropriate factorization theorems on the…
Non-radial minimizers for Hardy--Sobolev inequalities in non-convex cones
本文研究非凸锥中由Hardy--Sobolev不等式生成的$p$-Laplacian Neumann问题，证明在球截面第一Neumann特征值满足一定条件时径向解不是极值函数，从而极小化子必为非径向，并由此得到多重解。
Let $ (M,g) $ be a smooth compact Riemannian manifold of dimension $ N \geq 3 $. Given $p_0 \in M$, $λ\in \mathcal{R}$ and $σ\in (0,2]$, we study existence and non existence of minimizers of the following quotient: \begin{equation}\label{Paper Equation} μ_{λ,σ}=\inf_{u \in H^1(M)\setminus \lbrace0\rbrace} \frac{\displaystyle\int_M |\nabla u|^2 dv_g -λ\int_M u^2 dv_g }{\biggl(\displaystyle\int_M ρ^{-σ} |u|^{2^*(σ)} dv_g\biggl)^{2/2^*(σ)}}, \end{equation} where $ρ(.):=dist(p_0,.)$ denoted the geodesic distance from $p \in M$ to $p_0$. In particular for $σ=2$, we provide sufficient and necessary …
Existence of a bi-radial sign-changing solution for Hardy-Sobolev-Mazya type equation
本文研究Hardy-Sobolev-Maz'ya型方程，通过将其提升到双曲球模型并构造不变子空间、应用集中紧性原理，证明了双径向变号解的存在性。
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $N\geq 3$ and we let $Σ$ to be a closed submanifold of dimension $1 \leq k \leq N-2. $ In this paper we study existence and non-existence of minimizers of Hardy inequality with weight function singular on $Σ$ within the framework of Brezis-Marcus-Shafrir. In particular we provide necessary and sufficient conditions for existence of minimizers.
Let $p \in (1,\infty)$ and $Ω\subset \mathbb{R}^N$ be a domain. Let $ A: =(a_{ij}) \in L^{\infty}_{\text{loc}}(Ω; \mathbb{R}^{N\times N})$ be a symmetric and locally uniformly positive definite matrix. Set $|ξ|_A^2:= \displaystyle \sum_{i,j=1}^N a_{ij}(x) ξ_i ξ_j$, $ξ\in \mathbb{R}^N$, and let $V$ be a given potential in a certain local Morrey space. We assume that the energy functional $$Q_{p,A,V}(ϕ):=\displaystyle \int_Ω [|\nabla ϕ|_A^p + V|ϕ|^p] {\rm dx} $$ is nonnegative in $W^{1,p}(Ω)\cap C_c(Ω)$. We introduce a generalized notion of $Q_{p,A,V}$-capacity and characterize the space of all …
We establish some qualitative properties of minimizers in the fractional Hardy--Sobolev inequalities of arbitrary order.
This paper continues the program initiated in the works by the authors [60], [61] and [62] and by the authors with Li [51] and [52] to establish higher order Poincaré-Sobolev, Hardy-Sobolev-Maz'ya, Adams and Hardy-Adams inequalities on real hyperbolic spaces using the method of Helgason-Fourier analysis on the hyperbolic spaces. The aim of this paper is to establish such inequalities on the Siegel domains and complex hyperbolic spaces. Firstly, we prove a factorization theorem for the operators on the complex hyperbolic space which is closely related to Geller' operator, as well as the CR inva…
We obtain existence, multiplicity, and bifurcation results for the Brezis-Nirenberg problem for the fractional $p$\nobreakdash-Laplacian operator, involving critical Hardy-Sobolev exponents. Our results are mainly extend results in the literature for $α=0$. In the absence of an explicit formula for a minimizer in the fractional Hardy-Sobolev inequality $α\not= 0$, we get around this difficulty by working with certain asymptotic estimates for minimizers recently obtained in \cite{MM}.
The paper studies the existence of minimizers for Rayleigh quotients $μ_Ω=\inf\frac{\int_Ω|\nabla u|^2}{\int_ΩV{|u|^2}} $, where $Ω$ is a domain in $\mathbb{R}^N$, and $V$ is a nonzero nonnegative function that may have singularities on $\partialΩ$. As a model for our results one can take $Ω$ to be a Lipschitz cone and $V$ to be the Hardy potential $V(x)=\frac{1}{|x|^2} $.
Using the Fourier analysis techniques on hyperbolic spaces and Green's function estimates, we confirm in this paper the conjecture given by the same authors in [43]. Namely, we prove that the sharp constant in the $\frac{n-1}{2}$-th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension $n$ coincides with the best $\frac{n-1}{2}$-th order Sobolev constant when $n$ is odd and $n\geq9$ (See Theorem 1.6). We will also establish a lower bound of the coefficient of the Hardy term for the $k-$th order Hardy-Sobolev-Maz'ya inequality in upper half space in the remaining cases of d…
We consider the $L^p$ Hardy inequality involving the distance to the boundary of a domain in the $n$-dimensional Euclidean space with nonempty compact boundary. We extend the validity of known existence and non-existence results, as well as the appropriate tight decay estimates for the corresponding minimizers, from the case of domains of class $C^2$ to the case of domains of class $C^{1,γ}$ with $γ\in (0,1]$. We consider both bounded and exterior domains. The upper and lower estimates for the minimizers in the case of exterior domains and the corresponding related non-existence result seem to…
Let $Ø$ be a smooth bounded domain in $\R^N$ with $N\ge 1$. In this paper we study the Hardy-Poincaré inequality with weight function singular at the boundary of $Ø$. In particular we provide sufficient and necessary conditions on the existence of minimizers.
Fractional Hardy--Maz'ya inequality on a half-space
本文证明了半空间上的分数阶Maz'ya不等式，给出了最佳常数，并推广到Sobolev--Bregman形式。
Finsler $p$-Laplace equation with a potential: Maz'ya-type characterization and attainments of the Hardy constant
本文研究带位势的Finsler $p$-Laplace方程，通过建立Bregman距离双边估计得到Hardy权重的Maz'ya型刻画，并给出Hardy常数可达的充分条件。
In this paper, we show the existence and non-existence of minimizers of the following minimization problems which include an open problem mentioned by Horiuchi and Kumlin in 2012: \begin{align*} G_a := \inf_{u \in W_0^{1,N}(Ω) \setminus \{ 0\} } \dfrac{\int_Ω |\nabla u |^{N} \,dx}{\left( \int_Ω |u|^{q} f_{a,β}(x) dx \right)^{\frac{N}{q}}}, \,\text{where} \,\,f_{a, β}(x):=|x|^{-N}\left( \log \frac{aR}{|x|} \right)^{-β}. \end{align*} First, we give an answer to the open problem when $Ω=B_R(0)$. Next, we investigate the minimization problems on general bounded domains. In this case, the results d…
Existence and symmetry of extremals for the high order Hardy-Sobolev-Maz'ya inequalities
本文研究了高阶Hardy-Sobolev-Maz'ya不等式极值函数的存在性及对称性，通过等价版本的Poincaré-Sobolev不等式和超曲面分析方法，解决了临界情况下的Brezis-Nirenberg方程正对称解的存在性问题。
Let $Ω$ be a smooth bounded domain in $\mahbb R^N$ with $N\ge 3$ and let $Σ_k$ be a closed smooth submanifold of $δΩ$ of dimension $1\le k\le N-2$. In this paper we study the weighted Hardy inequality with weight function singular on $Σ_k$. In particular we provide sufficient and necessary conditions for existence of minimizers.
The Hardy-Sobolev trace inequality can be obtained via Harmonic extensions on the half-space of the Stein and Weiss weighted Hardy-Littlewood-Sobolev inequality. In this paper we consider a bounded domain and study the influence of the boundary mean curvature in the Hardy-Sobolev trace inequality on the underlying domain. We prove existence of minimizers when the mean curvature is negative at the singular point of the Hardy potential.
An inequality of Hardy type is established for quadratic forms involving Dirac operator and a weight $r^{-b}$ for functions in $\R^n$. The exact Hardy constant $c_b=c_b(n)$ is found and generalized minimizers are given. The constant $c_b$ vanishes on a countable set of $b$, which extends the known case $n=2$, $b=0$ which corresponds to the trivial Hardy inequality in $\R^2$. Analogous inequalities are proved in the case $c_b=0$ under constraints and, with error terms, for a bounded domain.
Sharp gradient stability for a class of Hardy-Sobolev-Maz'ya inequalities
本文针对具有部分（更强）奇异权和非径向极值函数的一类Hardy-Sobolev-Maz'ya不等式，证明了尖锐梯度稳定性，这是首个关于非径向极值函数的稳定性结果。
As our main result we prove a variant of the fractional Hardy-Sobolev-Maz'ya inequality for half spaces. This result contains a complete answer to a recent open question by Musina and Nazarov. In the proof we apply a new version of the fractional Hardy-Sobolev inequality that we establish also for more general unbounded John domains than half spaces.
In this paper we obtain an inequality on the unit disc $B$ in the plane, which improves the classical Moser-Trudinger inequality and the classical Hardy inequality at the same time. Namely, there exists a constant $C_0>0$ such that \[ \int_B e^{\frac {4πu^2}{H(u)}} dx \le C_0 < \infty, \quad \forall\; u\in C^\infty_0(B),\] where $$H(u) := \int_B |\n u|^2 dx - \int_B \frac {u^2}{(1-|x|^2)^2} dx.$$ This inequality is a two dimensional analog of the Hardy-Sobolev-Maz'ya inequality in higher dimensions, which was recently intensively studied. We also prove that the supremum is achieved in a suitab…
Let $Ω$ be an open set in a metric measure space $X$. Our main result gives an equivalence between the validity of a weighted Hardy-Sobolev inequality in $Ω$ and quasiadditivity of a weighted capacity with respect to Whitney covers of $Ω$. Important ingredients in the proof include the use of a discrete convolution as a capacity test function and a Maz'ya type characterization of weighted Hardy-Sobolev inequalities.
We consider the existence and the non-existence of a minimizer of the following minimization problems associated with an improved Hardy-Sobolev type inequality introduced by Ioku. $$ I_a := \inf_{u \in W_0^{1,p}(B_R ) \setminus \{ 0\} } \frac{\int_{B_R} |\nabla u |^{p} \,dx}{\left( \int_{B_R} |u|^{p^*(s)} V_a(x) \,dx \right)^{\frac{p}{p^*(s)}}}, \,\,\text{where}\,\, V_a (x) =\frac{1}{|x|^s \left( 1- a \,\left( \frac{|x|}{R} \right)^{\frac{N-p}{p-1}} \right)^β} \ge \frac{1}{|x|^s}. $$ Only for radial functions, the minimization problem $I_a$ is equivalent to it associated with the classical Har…
Nonlinear elliptic systems involving Hardy-Sobolev Criticalities
本文研究含Hardy势和Hardy-Sobolev临界耦合项的非线性椭圆系统，通过变分法在Nehari流形上寻找基态和束缚态，并确定解存在的参数范围。
Existence of solutions for a system with general Hardy--Sobolev singular criticalities
本文研究由奇异临界 Hardy--Sobolev 项耦合的系统，其奇异阶数独立且范围宽，借助变分技巧证明了正有界态和基态的存在性。
On fractional critical problems with multi-polar Hardy potentials
研究分数方程在双重临界性下的正解存在性，包含多极Hardy型势和Sobolev临界非线性，克服非局部算子和单极方程无显式基态的困难，通过扩展形式和渐进行为估计，利用集中-紧致性论证，证明极小值的存在由质量大小和极间距离决定。
In this paper, we present the geometric Hardy inequality for the sub-Laplacian in the half-spaces on the stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space on the Heisenberg group with a sharp constant \begin{equation*} \int_{\mathbb{H}^+} |\nabla_{H}u|^p dξ\geq \left(\frac{p-1}{p}\right)^p \int_{\mathbb{H}^+} \frac{\mathcal{W}(ξ)^p}{dist(ξ,\partial \mathbb{H}^+)^p} |u|^p dξ, \,\, p>1, \end{equation*} which solves the conjecture in the paper \cite{Larson}. Also, we obtain a version of the Hardy-Sobolev inequality in a half-space on the Heise…
Global higher integrability and Hardy inequalities for double-phase functionals under a capacity density condition
本文在容量密度条件下证明双相泛函的全局高阶可积性，引入新变分容量，建立Hardy不等式及等价性，并给出反例说明Maz'ya型不等式不成立。
We prove a new inequality which improves on the classical Hardy inequality in the sense that a nonlinear integral quantity with super-quadratic growth, which is computed with respect to an inverse square weight, is controlled by the energy. This inequality differs from standard logarithmic Sobolev inequalities in the sense that the measure is neither Lebesgue's measure nor a probability measure. All terms are scale invariant. After an Emden-Fowler transformation, the inequality can be rewritten as an optimal inequality of logarithmic Sobolev type on the cylinder. Explicit expressions of the sh…
Explicit Formulas of Fractional GJMS operators on hyperbolic spaces and sharp fractional Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities
本文通过散射理论给出双曲空间上分数阶GJMS算子的显式公式，并建立相关尖锐分数阶Poincaré-Sobolev与Hardy-Sobolev-Maz'ya不等式，证明其常数与欧氏空间最佳Sobolev常数一致。
We present the best constant and the existence of extremal functions for an Improved Hardy-Sobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in $\mathbb{R}^N$. We also discuss the connection of the related functional spaces and as a result we obtain some Caffarelli - Kohn - Nirenberg inequalities. Our starting point is the existence of a minimizer for the Bliss' inequality and the indirect dependence of the Hardy inequality at the origin.
Let $Ω$ be a smooth bounded domain of $\mathbb{R}^{N+1}$ of boundary $\partial Ω= Γ_1 \cup Γ_2$ and such that $\partial Ω\cap Γ_2$ is a neighborhood of $0$, $h \in \mathcal{C}^0(\partial Ω\cap Γ_2) $ and $s \in [0,1)$. We propose to study existence of positive solutions to the following Hardy-Sobolev trace problem with mixed boundaries conditions \begin{align} \begin{cases} Δu= 0& \qquad \textrm{ in } Ω\\\ u=0 & \qquad \textrm{ on } Γ_1 \\\ \frac{\partial u}{\partial ν}=h(x) u + \frac{u^{q(s)-1}}{d(x)^{s}} & \qquad \textrm{ on } Γ_2, \end{cases} \end{align} where $q(s):=\frac{2(N-s)}{N-1}$ is …
We consider a bounded domain $Ω$ of $\mathbb{R}^N$, $N\ge3$, $h$ and $b$ continuous functions on $Ω$. Let $Γ$ be a closed curve contained in $Ω$. We study existence of positive solutions $u \in H^1_0\left(Ω\right)$ to the perturbed Hardy-Sobolev equation: $$ -Δu+h u+bu^{1+δ}=ρ^{-σ}_Γu^{2^*_σ-1} \qquad \textrm{ in } Ω, $$ where $2^*_σ:=\frac{2(N-σ)}{N-2}$ is the critical Hardy-Sobolev exponent, $σ\in [0,2)$, $0< δ<\frac{4}{N-2}$ and $ρ_Γ$ is the distance function to $Γ$. We show that the existence of minimizers does not depend on the local geometry of $Γ$ nor on the potential $h$. For $N=3$, th…
Sharp Hardy-Leray inequality for solenoidal fields
本文计算了 $\mathbb{R}^N$ 上无源向量场Hardy-Leray不等式的最佳常数，在无对称性假设下推导并简化了该常数，并证明其在积分有限的函数空间内无法达到。
On existence of minimizers for weighted $L^p$-Hardy inequalities on $C^{1,γ}$-domains with compact boundary
研究\(C^{1,\gamma}\)区域上加权\(L^p\)-Hardy不等式的两个变分常数，通过分析谱隙证明极小值存在的充要条件，并给出相应极小值的紧衰减估计。
A relativistic Hardy-type inequality with minimisers
本文证明了Dirac算子的精确且可达的加权Hardy型不等式，构造了满足等式的显式极小子，扩展了相对论量子力学中Dirac算子的谱性质研究。
We study the a priori estimates,existence/nonexistence of radial sign changing solution, and the Palais-Smale characterisation of the problem $-\De_{\Bn}u - \la u = |u|^{p-1}u, u\in H^1(\Bn)$ in the hyperbolic space $\Bn$ where $1<p\leq\frac{N+2}{N-2}$. We will also prove the existence of sign changing solution to the Hardy-Sobolev-Mazya equation and the critical Grushin problem.
Optimal Hardy Inequality for Fractional Laplacians on the Lattice
本文研究了整数晶格上分数阶拉普拉斯算子的最优哈代不等式，通过分析哈代权的临界性及阈值特性，证明了不等式常数的最优性。
Given a three dimensional pseudo-Einstein CR manifold $(M,T^{1,0}M,θ)$, we study the existence of a contact structure conformal to $θ$ for which the logarithmic Hardy-Littlewood-Sobolev (LHLS) inequality holds. Our approach closely follows \cite{Ok1} in the Riemannian setting. For this purpose, we introduce the notion of Robin mass as the constant term appearing in the expansion of the Green's function of the $P'$-operator. We show that the LHLS inequality appears when we study the variation of the total mass under conformal change. Then we exhibit an Aubin type result guaranteeing the existen…
Sharp weighted fractional Hardy inequalities
本文研究了加权分数阶Hardy不等式，确定了最佳常数，并获得了分数阶Hardy-Sobolev-Maz'ya不等式的加权版本。
Bivariate Hardy-Sobolev Inequality and Its Sharp Stability
本文研究了双变量Hardy-Sobolev不等式，推导了最佳常数并给出了极小化子集，同时在特定条件下获得了非负函数的精确稳定性结果。
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