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Annular type surfaces with fixed boundary and with prescribed, almost constant mean curvature 具有固定边界和规定的几乎恒定的平均曲率的环形曲面
Pub Date : 2024-01-25 DOI: 10.1007/s00030-023-00915-2
Paolo Caldiroli, Gabriele Cora, Alessandro Iacopetti

We prove existence and nonexistence results for annular type parametric surfaces with prescribed, almost constant mean curvature, characterized as normal graphs of compact portions of unduloids or nodoids in ({mathbb {R}}^{3}), and whose boundary consists of two coaxial circles of the same radius.

我们证明了具有规定的、几乎恒定的平均曲率的环形参量曲面的存在性和不存在性结果,这些曲面的特征是在({mathbb {R}}^{3}) 中的UNDULOID或NODOID的紧凑部分的法线图形,其边界由两个半径相同的同轴圆组成。
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引用次数: 0
On radial positive normalized solutions of the Nonlinear Schrödinger equation in an annulus 论环面非线性薛定谔方程的径向正归一化解
Pub Date : 2024-01-22 DOI: 10.1007/s00030-023-00917-0
Jian Liang, Linjie Song

We are interested in the following semilinear elliptic problem:

$$begin{aligned} {left{ begin{array}{ll} -Delta u + lambda u = u^{p-1}, x in T, u > 0, u = 0 text {on} partial T, int _{T}u^{2} , dx= c end{array}right. } end{aligned}$$

where (T = {x in mathbb {R}^{N}: 1< |x| < 2}) is an annulus in (mathbb {R}^{N}), (N ge 2), (p > 1) is Sobolev-subcritical, searching for conditions (about c, N and p) for the existence of positive radial solutions. We analyze the asymptotic behavior of c as (lambda rightarrow +infty ) and (lambda rightarrow -lambda _1) to get the existence, non-existence and multiplicity of normalized solutions. Additionally, based on the properties of these solutions, we extend the results obtained in Pierotti et al. in Calc Var Partial Differ Equ 56:1–27, 2017. In contrast of the earlier results, a positive radial solution with arbitrarily large mass can be obtained when (N ge 3) or if (N = 2) and (p < 6). Our paper also includes the demonstration of orbital stability/instability results.

我们对以下半线性椭圆问题感兴趣: $$begin{aligned} {left{ begin{array}{ll} -Delta u + lambda u = u^{p-1}, x in T, u > 0, u = 0 text {on} Partial T, int _{T}u^{2}dx= cend{array}right.}end{aligned}$ 其中(T = x in R}^{N}: 1< |x| < 2} )是在(mathbb {R}^{N} )中的一个环面,(N ge 2 ),(p > 1 )是索博勒夫次临界,为正径向解的存在寻找条件(关于 c、N 和 p)。我们分析了 c 作为 (lambda rightarrow +infty ) 和 (lambda rightarrow -lambda _1)的渐近行为,从而得到归一化解的存在、不存在和多重性。此外,基于这些解的性质,我们扩展了 Pierotti 等人在 Calc Var Partial Differ Equ 56:1-27, 2017 中得到的结果。与之前的结果不同,当 (N ge 3) 或 (N = 2) and (p < 6) 时,可以得到具有任意大质量的正径向解。我们的论文还包括轨道稳定性/不稳定性结果的证明。
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引用次数: 0
Optimization of the Dirichlet problem for gradient differential inclusions 梯度微分夹杂的德里赫特问题优化
Pub Date : 2024-01-20 DOI: 10.1007/s00030-023-00904-5
Elimhan N. Mahmudov, Dilara Mastaliyeva

The paper is devoted to optimization of the gradient differential inclusions (DFIs) on a rectangular area. The discretization method is the main method for solving the proposed boundary value problem. For the transition from discrete to continuous, a specially proven equivalence theorem is provided. To optimize the posed continuous gradient DFIs, a passage to the limit is required in the discrete-approximate problem. Necessary and sufficient conditions of optimality for such problems are derived in the Euler–Lagrange form. The results obtained in terms of the divergence operation of the Euler–Lagrange adjoint inclusion are extended to the multidimensional case. Such results are based on locally adjoint mappings, being related coderivative concept of Mordukhovich.

本文致力于优化矩形区域上的梯度微分夹杂(DFIs)。离散化方法是求解拟议边界值问题的主要方法。为实现从离散到连续的过渡,提供了一个专门证明的等价定理。为了优化所提出的连续梯度 DFI,离散近似问题中需要通过极限。以欧拉-拉格朗日形式推导出了此类问题最优化的必要条件和充分条件。根据欧拉-拉格朗日邻接包含的发散运算得到的结果被扩展到多维情况。这些结果以局部邻接映射为基础,与莫尔杜霍维奇的 coderivative 概念相关。
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引用次数: 0
Fujita-type results for the degenerate parabolic equations on the Heisenberg groups 海森堡群上退化抛物方程的富士达型结果
Pub Date : 2024-01-20 DOI: 10.1007/s00030-023-00907-2
Ahmad Z. Fino, Michael Ruzhansky, Berikbol T. Torebek

In this paper, we consider the Cauchy problem for the degenerate parabolic equations on the Heisenberg groups with power law non-linearities. We obtain Fujita-type critical exponents, which depend on the homogeneous dimension of the Heisenberg groups. The analysis includes the case of porous medium equations. Our proof approach is based on methods of nonlinear capacity estimates specifically adapted to the nature of the Heisenberg groups. We also use the Kaplan eigenfunctions method in combination with the Hopf-type lemma on the Heisenberg groups.

在本文中,我们考虑了海森堡群上具有幂律非线性的退化抛物方程的考奇问题。我们得到了富士达型临界指数,它取决于海森堡群的同次元维度。分析包括多孔介质方程的情况。我们的证明方法基于非线性容量估计方法,特别适应海森堡群的性质。我们还将卡普兰特征函数法与海森堡群上的霍普夫型 Lemma 结合使用。
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引用次数: 0
Boundedness of solutions to a chemotaxis–haptotaxis model with nonlocal terms 具有非局部项的趋化-aptotaxis模型解的有界性
Pub Date : 2024-01-06 DOI: 10.1007/s00030-023-00908-1
Guoqiang Ren

In this paper, we consider the chemotaxis–haptotaxis model of two different types (parabolic–elliptic, fully parabolic) with nonlocal terms under Neumann boundary conditions in a bounded domain with smooth boundary. We show that the system possesses a unique global classical solution in different cases. Our results generalize and improve partial previously known ones.

在本文中,我们考虑了在具有光滑边界的有界域中,在 Neumann 边界条件下两种不同类型(抛物-椭圆、全抛物)的带有非局部项的趋化-aptotaxis 模型。我们证明,该系统在不同情况下具有唯一的全局经典解。我们的结果概括并改进了之前已知的部分结果。
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引用次数: 0
On a characterization of the Rellich–Kondrachov theorem on groups and the Bloch spectral cell equation 关于群上的雷利奇-康德拉乔夫定理的特征和布洛赫谱单元方程
Pub Date : 2024-01-03 DOI: 10.1007/s00030-023-00905-4
Vernny Ccajma, Wladimir Neves, Jean Silva

This paper is concerned with the Rellich–Kondrachov Theorem on Groups. We establish some conditions which characterize in a precise manner important properties related to this theorem and the Sobolev spaces on groups involved on it. The main motivation to study the Rellich–Kondrachov Theorem on Groups comes from the Bloch spectral cell equation, which is an eigenvalue-eigenfunction problem associated with the assymptotic limit of the anisotropic Schrödinger equation.

本文关注的是关于群的 Rellich-Kondrachov 定理。我们建立了一些条件,这些条件以精确的方式描述了与该定理相关的重要性质以及与该定理相关的群上的索波列夫空间。研究群上的 Rellich-Kondrachov 定理的主要动机来自布洛赫谱单元方程,这是一个与各向异性薛定谔方程的渐近极限相关的特征值-特征函数问题。
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引用次数: 0
Liouville-type theorems for fractional Hardy–Hénon systems 分数 Hardy-Hénon 系统的 Liouville 型定理
Pub Date : 2023-12-28 DOI: 10.1007/s00030-023-00903-6
Kui Li, Meng Yu, Zhitao Zhang

In this paper, we study Liouville-type theorems for fractional Hardy–Hénon elliptic systems with weights. Because the weights are singular at zero, we firstly prove that classical solutions for systems in ({mathbb {R}}^N backslash {0}) are also distributional solutions in ({mathbb {R}}^N). Then we study the equivalence between the fractional Hardy–Hénon system and a proper integral system, and we obtain new Liouville-type theorems for supersolutions and solutions by the method of integral estimates and scaling spheres respectively.

在本文中,我们研究了带权重的分数哈代-赫农椭圆系统的 Liouville 型定理。因为权重在零点是奇异的,所以我们首先证明在 ({mathbb {R}}^N backslash {0})中系统的经典解也是({mathbb {R}}^N) 中的分布解。然后,我们研究了分数哈代-赫农系统与适当积分系统之间的等价性,并通过积分估计和缩放球的方法分别得到了超解和解的新的刘维尔型定理。
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引用次数: 0
The continuous dependence of the viscous Boussinesq equations uniformly with respect to the viscosity 粘滞布森斯克方程与粘度的均匀连续相关性
Pub Date : 2023-12-27 DOI: 10.1007/s00030-023-00902-7
Rong Chen, Zhichun Yang, Shouming Zhou

This paper focuses on the inviscid limit of the incompressible Boussinesq equations in the same topology as the initial data, and proved that the continuous dependence of the viscous Boussinesq equations uniformly in some Besov spaces with respect to the viscosity. Our results extends the work of Guo et al. (J Funct Anal 276(9):2821–2830, 2019) on Navier–Stokes equations to Boussinesq equations with both stratified limit and earth’s rotation.

本文重点研究了与初始数据拓扑相同的不可压缩布森斯克方程的无粘性极限,证明了粘性布森斯克方程在某些贝索夫空间中关于粘性的均匀连续依赖性。我们的研究成果将 Guo 等人 (J Funct Anal 276(9):2821-2830, 2019) 关于 Navier-Stokes 方程的研究扩展到了具有分层极限和地球自转的 Boussinesq 方程。
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引用次数: 0
Evolutionary stable strategies and cubic vector fields 进化稳定策略和立方向量场
Pub Date : 2023-12-26 DOI: 10.1007/s00030-023-00894-4
Jefferson Bastos, Claudio Buzzi, Paulo Santana

The introduction of concepts of Game Theory and Ordinary Differential Equations into Biology gave birth to the field of Evolutionary Stable Strategies, with applications in Biology, Genetics, Politics, Economics and others. In special, the model composed by two players having two pure strategies each results in a planar cubic vector field with an invariant octothorpe. Therefore, in this paper we study such class of vector fields, suggesting the notion of genericity and providing the global phase portraits of the generic systems with a singularity at the central region of the octothorpe.

将博弈论和常微分方程的概念引入生物学后,诞生了 "进化稳定策略"(Evolutionary Stable Strategies)领域,并应用于生物学、遗传学、政治学、经济学等领域。在特殊情况下,由各拥有两种纯策略的两个玩家组成的模型会产生一个平面立方向量场,该向量场有一个不变的八矢量。因此,我们在本文中研究了这类向量场,提出了通用性概念,并提供了在八角中心区域具有奇点的通用系统的全局相位图。
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引用次数: 0
Existence and non-existence results for cooperative elliptic systems without variational structure 无变结构合作椭圆系统的存在与不存在结果
Pub Date : 2023-12-15 DOI: 10.1007/s00030-023-00896-2
John Villavert

We consider general cooperative elliptic systems possibly without variational structure and with differential operator resembling that from an Euler–Lagrange equation for a sharp Hardy–Sobolev inequality. Under suitable growth conditions on the source nonlinearities and geometric assumptions on the domain, we derive various existence and non-existence results and Liouville theorems. The results are obtained by incorporating and adapting various techniques, including variants of the method of moving planes enhanced by Kelvin and Emden–Fowler type transformations, as well as degree theoretic shooting methods.

我们考虑的一般合作椭圆系统可能没有变分结构,其微分算子类似于尖锐 Hardy-Sobolev 不等式的欧拉-拉格朗日方程。在适当的源非线性增长条件和域几何假设下,我们推导出各种存在和不存在结果以及Liouville定理。这些结果是通过结合和调整各种技术得出的,包括通过开尔文和埃姆登-福勒类型变换增强的移动平面方法的变体,以及度论射影方法。
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Nonlinear Differential Equations and Applications (NoDEA)
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