Nonlinear Analysis-Theory Methods & Applications最新文献

英文中文

Gromov–Hausdorff stability of tori under Ricci and integral scalar curvature bounds 里奇和积分标量曲率约束下的环的格罗莫夫-豪斯多夫稳定性

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-08-12 DOI: 10.1016/j.na.2024.113629

Shouhei Honda , Christian Ketterer , Ilaria Mondello , Raquel Perales , Chiara Rigoni

We establish a nonlinear analogue of a splitting map into a Euclidean space, as a harmonic map into a flat torus. We prove that the existence of such a map implies Gromov–Hausdorff closeness to a flat torus in any dimension. Furthermore, Gromov–Hausdorff closeness to a flat torus and an integral bound on $r_{M} (x)$ , the smallest eigenvalue of the Ricci tensor ${ric}_{x}$ in $x$ , imply the existence of a harmonic splitting map. Combining these results with Stern’s inequality, we provide a new Gromov–Hausdorff stability theorem for flat 3-tori. The main tools we employ include the harmonic map heat flow, Ricci flow, and both Ricci limits and RCD theories.

我们建立了欧几里得空间分裂映射的非线性类比，即平面环的谐波映射。我们证明了这种映射的存在意味着在任何维度上都与平环面的格罗莫夫-豪斯多夫接近。此外，Gromov-Hausdorff 与平坦环面的接近性和 rM(x) 的积分约束（即 x 中里奇张量 ricx 的最小特征值）意味着谐波分裂映射的存在。将这些结果与斯特恩不等式相结合，我们为平面 3 蝶形提供了一个新的格罗莫夫-豪斯多夫稳定性定理。我们使用的主要工具包括谐波图热流、利玛窦流以及利玛窦极限和 RCD 理论。

引用次数: 0

Properties of fractional p-Laplace equations with sign-changing potential 符号变化势分数 p 拉普拉斯方程的性质

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-08-09 DOI: 10.1016/j.na.2024.113628

Yubo Duan , Yawei Wei

In this paper, we consider the nonlinear equation involving the fractional p-Laplacian with sign-changing potential. This model draws inspiration from De Giorgi Conjecture. There are two main results in this paper. Firstly, we obtain that the solution is radially symmetric within the bounded domain, by applying the moving plane method. Secondly, by exploiting the idea of the sliding method, we construct the appropriate auxiliary functions to prove that the solution is monotone increasing in some direction in the unbounded domain. The different properties of the solution in bounded and unbounded domains are mainly attributed to the inherent non-locality of the fractional p-Laplacian.

在本文中，我们考虑了涉及符号变化势的分数 p-拉普拉奇的非线性方程。这一模型的灵感来自 De Giorgi 猜想。本文有两个主要结果。首先，我们通过应用移动平面法，得到了解在有界域内是径向对称的。其次，利用滑动法的思想，我们构造了适当的辅助函数，证明解在无界域中的某个方向上是单调递增的。有界域和无界域解的不同性质主要归因于分数 p-Laplacian 固有的非位置性。

引用次数: 0

Convergence of Multiplier Operators on Compact Manifolds 紧凑流形上乘法算子的收敛性

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-08-09 DOI: 10.1016/j.na.2024.113632

Xianghong Chen , Dashan Fan , Ziyao Liu

<div>We study a family of multiplier operators <math><mrow><msub><mrow><mi>T</mi></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>γ</mi></mrow></msub><mfenced><mrow><mi>t</mi><mi>⋅</mi></mrow></mfenced></mrow></msub><mfenced><mrow><mi>f</mi></mrow></mfenced></mrow></math> on compact manifolds <math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math>, which is an analogue of the spherical average <math><mrow><msubsup><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>γ</mi></mrow></msubsup><mfenced><mrow><mi>f</mi></mrow></mfenced></mrow></math> on <math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math>. We establish the almost everywhere convergence of <math><mrow><msub><mrow><mi>T</mi></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mi>γ</mi></mrow></msub><mfenced><mrow><mi>t</mi><mi>⋅</mi></mrow></mfenced></mrow></msub><mfenced><mrow><mi>f</mi></mrow></mfenced></mrow></math> as <math><mrow><mi>t</mi><mo>→</mo><mn>0</mn></mrow></math>. The result is an extension of a Stein’s theorem on <math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math>. Let <math><msubsup><mrow><mover><mrow><mi>S</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>t</mi></mrow><mrow><mi>γ</mi></mrow></msubsup></math> be an analogue of <math><mrow><msubsup><mrow><mi>S</mi></mrow><mrow><mi>t</mi></mrow><mrow><mi>γ</mi></mrow></msubsup><mspace></mspace></mrow></math>on the <math><mrow><mi>n</mi><mo>−</mo></mrow></math>torus <math><msup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msup></math>. As a consequence, we obtain that <math><mrow><msub><mrow><mo>lim</mo></mrow><mrow><mi>t</mi><mo>→</mo><mn>0</mn></mrow></msub><msubsup><mrow><mover><mrow><mi>S</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>t</mi></mrow><mrow><mi>γ</mi></mrow></msubsup><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math> almost everywhere if <math><mrow><mi>f</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>+</mo><mi>γ</mi></mrow></mfrac></mrow></msup><msup><mrow><mrow><mo>(</mo><mi>L</mi><mi>o</mi><msup><mrow><mi>g</mi></mrow><mrow><mo>+</mo></mrow></msup><mi>L</mi><mo>)</mo></mrow></mrow><mrow><mi>θ</mi></mrow></msup><mrow><mo>(</mo><msup><mrow><mi>T</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></mrow></mrow></math> with <math><mrow><mi>θ</mi><mo>></mo><mfrac><mrow><mn>1</mn><mo>−</mo><mi>γ</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn><mo>+</mo><mi>γ</mi></mrow></mfrac></mrow></math>, <math><mrow><mo>−</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo><</mo><mi>γ</mi><mo><</mo><mn>1</mn></mrow></math></sp

我们研究了紧凑流形 Mn 上的乘法算子 Tmγt⋅f 族，它是 Rn 上球面平均 Stγf 的类似物。我们建立了 Tmγt⋅f 在 t→0 时的几乎无处收敛性。这一结果是斯坦因定理在 Rn 上的扩展。让 S˜tγ成为 n-Torus Tn 上 Stγ 的类似物。因此，如果 f∈Lnn-1+γ(Log+L)θ(Tn) 且 θ>1-γn-1+γ, -n-22<γ<；1，并且存在一个 f∈Lnn-1+γ（Log+L）θ（Tn），θ<1-γn-1+γ，0<γ<1，使得 limsupt→0S˜tγ(f)(x)=∞ 几乎无处不在。

引用次数: 0

Second derivative Lδ-estimates for a class of singular fully nonlinear elliptic equations 一类奇异全非线性椭圆方程的二次导数 Lδ 估计值

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-08-09 DOI: 10.1016/j.na.2024.113630

Sumiya Baasandorj , Sun-Sig Byun , Jehan Oh

We provide global a priori second derivative $L^{δ}$ -estimates for a class of singular fully nonlinear elliptic equations with right hand side terms of $L^{n}$ .

我们为一类右手项为 Ln 的奇异全非线性椭圆方程提供了全局先验二阶导数 Lδ 估计值。

引用次数: 0

Existence results for Cahn–Hilliard-type systems driven by nonlocal integrodifferential operators with singular kernels 具有奇异内核的非局部积分微分算子驱动的卡恩-希利亚德型系统的存在性结果

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-08-02 DOI: 10.1016/j.na.2024.113623

Elisa Davoli , Chiara Gavioli , Luca Lombardini

We introduce a fractional variant of the Cahn–Hilliard equation settled in a bounded domain and with a possibly singular potential. We first focus on the case of homogeneous Dirichlet boundary conditions, and show how to prove the existence and uniqueness of a weak solution. The proof relies on the variational method known as minimizing movements scheme, which fits naturally with the gradient-flow structure of the equation. The interest of the proposed method lies in its extreme generality and flexibility. In particular, relying on the variational structure of the equation, we prove the existence of a solution for a general class of integrodifferential operators, not necessarily linear or symmetric, which include fractional versions of the $q$ -Laplacian.

In the second part of the paper, we adapt the argument in order to prove the existence of solutions in the case of regional fractional operators. As a byproduct, this yields an existence result in the interesting cases of homogeneous fractional Neumann boundary conditions or periodic boundary conditions.

我们介绍了卡恩-希利亚德方程的分数变体，该方程在有界域中解决，并可能具有奇异势。我们首先关注同质 Dirichlet 边界条件的情况，并展示如何证明弱解的存在性和唯一性。证明依赖于称为，的变分法，它与方程的梯度流结构自然吻合。所提方法的趣味在于其极强的通用性和灵活性。特别是，依靠方程的变分结构，我们证明了一般整微分算子（不一定是线性或对称算子）的解的存在性，这些整微分算子包括分数版的-拉普拉奇。

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引用次数: 0

A Talenti-type comparison theorem for the p-Laplacian on RCD(K,N) spaces and some applications RCD(K,N)空间上 p-拉普拉斯的塔伦蒂型比较定理及其一些应用

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-08-02 DOI: 10.1016/j.na.2024.113631

Wenjing Wu

In this paper, we prove a Talenti-type comparison theorem for the $p$ -Laplacian with Dirichlet boundary conditions on open subsets of a normalized $RCD (K, N)$ space with $K > 0$ and $N \in (1, \infty)$ . The obtained Talenti-type comparison theorem is sharp, rigid and stable with respect to measured Gromov–Hausdorff topology. As an application of such Talenti-type comparison, we establish a sharp and rigid reverse Hölder inequality for first eigenfunctions of the $p$ -Laplacian and a related quantitative stability result.

本文证明了在K>0和N∈(1,∞)的归一化RCD(K,N)空间的开放子集上具有迪里希特边界条件的p-拉普拉奇的塔伦蒂型比较定理。所得到的塔伦提型比较定理对于测量的格罗莫夫-豪斯多夫拓扑学来说是尖锐的、刚性的和稳定的。作为塔伦提式比较定理的一个应用，我们为 p-拉普拉奇的第一特征函数建立了一个尖锐、刚性的反向赫尔德不等式和一个相关的定量稳定性结果。

引用次数: 0

Concentration limit for non-local dissipative convection–diffusion kernels on the hyperbolic space 双曲空间上非局部耗散对流-扩散核的浓度极限

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-08-01 DOI: 10.1016/j.na.2024.113618

María del Mar González , Liviu I. Ignat , Dragoş Manea , Sergiu Moroianu

We study a non-local evolution equation on the hyperbolic space $H^{N}$ . We first consider a model for particle transport governed by a non-local interaction kernel defined on the tangent bundle and invariant under the geodesic flow. We study the relaxation limit of this model to a local transport problem, as the kernel gets concentrated near the origin of each tangent space. Under some regularity and integrability conditions on the kernel, we prove that the solution of the rescaled non-local problem converges to that of the local transport equation. Then, we construct a large class of interaction kernels that satisfy those conditions.

We also consider a non-local, non-linear convection–diffusion equation on $H^{N}$ governed by two kernels, one for each of the diffusion and convection parts, and we prove that the solution converges to the solution of a local problem as the kernels get concentrated. We prove and then use in this sense a compactness tool on manifolds inspired by the work of Bourgain–Brezis–Mironescu.

我们研究双曲空间上的非局部演化方程。我们首先考虑一个粒子输运模型，该模型受切线束上定义的非局部相互作用核支配，并且在大地流作用下不变。当核集中在每个切向空间的原点附近时，我们将研究该模型向局部输运问题的松弛极限。在内核的一些正则性和可整性条件下，我们证明了重标度非局部问题的解收敛于局部输运方程的解。然后，我们构建了一大类满足这些条件的相互作用核。

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引用次数: 0

Approximate boundary controllability for parabolic equations with inverse square infinite potential wells 具有反平方无限势阱的抛物方程的近似边界可控性

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-08-01 DOI: 10.1016/j.na.2024.113624

Arick Shao , Bruno Vergara

We consider heat operators on a bounded domain $Ω \subseteq R^{n}$ , with a critically singular potential diverging as the inverse square of the distance to $\partial Ω$ . Although null boundary controllability for such operators was recently proved in all dimensions in Enciso et al. (2023) , it crucially assumed (i) $Ω$ was convex, (ii) the control must be prescribed along all of $\partial Ω$ , and (iii) the strength of the singular potential must be restricted to a particular subrange. In this article, we prove instead a definitive approximate boundary control result for these operators, in that we (i) do not assume convexity of $Ω$ , (ii) allow for the control to be localized near any $x_{0} \in \partial Ω$ , and (iii) treat the full range of strength parameters for the singular potential. Moreover, we lower the regularity required for $\partial Ω$ and the lower-order coefficients. The key novelty is a local Carleman estimate near $x_{0}$ , with a carefully chosen weight that takes into account both the appropriate boundary conditions and the local geometry of $\partial Ω$ .

我们考虑有界域 Ω⊆Rn 上的热算子，其临界奇异势发散为到 ∂Ω 的距离的反平方。尽管最近 Enciso 等人 (2023) 在所有维度上证明了此类算子的空边界可控性，但其关键假设是：(i) Ω 是凸的；(ii) 控制必须沿∂Ω 的所有方向规定；(iii) 奇异势的强度必须限制在特定子范围内。在本文中，我们证明了这些算子的近似边界控制结果，我们(i) 不假设 Ω 的凸性，(ii) 允许控制在任意 x0∈∂Ω 附近局部化，(iii) 处理奇异势的全部强度参数。此外，我们降低了对∂Ω 和低阶系数的正则性要求。关键的新颖之处在于 x0 附近的局部卡勒曼估计，其权重经过精心选择，既考虑到了适当的边界条件，又考虑到了∂Ω 的局部几何形状。

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引用次数: 0

Regularizing effect for a class of Maxwell–Schrödinger systems 一类麦克斯韦-薛定谔系统的正则效应

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-08-01 DOI: 10.1016/j.na.2024.113625

Ayana Pinheiro de Castro Santana , Luís Henrique de Miranda

In this paper we prove existence and regularity of weak solutions for the following system $(\begin{matrix} - div (M (x) \nabla u) + g (x, u, v) = f in Ω \\ - div (M (x) \nabla v) = h (x, u, v) in Ω \\ u = v = 0 on \partial Ω, \end{matrix})$ where $Ω$ is an open bounded subset of $R^{N}$ , for $N > 2$ , $f \in L^{m} (Ω)$ , $M$ is a matrix with Lipschitz coefficients, $m > 1$ and $g$ , $h$ are two Carathéodory functions. We prove that under appropriate conditions on $g$ and $h$ , there exist solutions which escape the predicted regularity by the classical Stampacchia’s theory causing the so-called regularizing effect.

本文证明了以下系统弱解的存在性和正则性-div(M(x)∇u)+g(x,u,v)=finΩ-div(M(x)∇v)=h(x,u,v)inΩu=v=0on∂Ω，其中Ω是 RN 的开放有界子集，对于 N>；2，f∈Lm(Ω)，M 是具有 Lipschitz 系数的矩阵，m>1 和 g, h 是两个 Carathéodory 函数。我们证明，在 g 和 h 的适当条件下，存在一些解，它们摆脱了经典的斯坦帕奇亚理论所预测的正则性，从而产生了所谓的正则化效应。

引用次数: 0

The classification on a class of weakly weighted Einstein–Finsler metrics 关于一类弱加权爱因斯坦-芬斯勒度量的分类

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-07-30 DOI: 10.1016/j.na.2024.113621

Hongmei Zhu , Peijuan Rao

In this paper, we study generalized weighted Ricci curvatures, which include the $N$ -Ricci curvature and the projective Ricci curvature with totally different geometric meanings. We completely classify a class of weakly weighted Einstein-Finsler metrics.

本文研究广义加权里奇曲率，其中包括几何意义完全不同的-里奇曲率和投影里奇曲率。我们对一类弱加权爱因斯坦-芬斯勒度量进行了完整分类。

引用次数: 0

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Nonlinear Analysis-Theory Methods & Applications

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