Nonlinear Analysis-Theory Methods & Applications最新文献

英文中文

The hydrostatic approximation of the Boussinesq equations with rotation in a thin domain 薄域中带有旋转的布森斯克方程的静力学近似值

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-11-02 DOI: 10.1016/j.na.2024.113688

Xueke Pu , Wenli Zhou

In this paper, the global existence of strong solutions to the primitive equations with only horizontal viscosity and diffusivity is established under the assumption of initial data

(v_{0}, T_{0}) \in H^{1}

with additional regularity

\partial_{z} v_{0} \in L^{4}

. Moreover, we prove that the scaled Boussinesq equations with rotation strongly converge to the primitive equations with only horizontal viscosity and diffusivity, with the convergence rate

O (λ^{min {2, β - 2, γ - 2} / 2}) (2 < β, γ < \infty)

, in the cases of initial data

(v_{0}, T_{0}) \in H^{1}

with

\partial_{z} v_{0} \in L^{4}

and initial data

(v_{0}, T_{0}) \in H^{2}

, respectively, as the aspect ratio

λ

goes to zero.

本文在初始数据（v0,T0）∈H1 和附加正则性∂zv0∈L4 的假设下，建立了只有水平粘性和扩散性的原始方程的强解的全局存在性。此外，我们证明了带旋转的缩放布森斯克方程强烈收敛于只有水平粘性和扩散性的原始方程，收敛速率为 O(λmin{2,β-2,γ-2}/2)(2<；β,γ<∞），分别适用于初始数据（v0,T0）∈H1 且∂zv0∈L4 和初始数据（v0,T0）∈H2 的情况，当纵横比 λ 变为零时。

引用次数: 0

Partial gradient regularity for parabolic systems with degenerate diffusion and Hölder continuous coefficients 具有退化扩散和赫尔德连续系数的抛物线系统的部分梯度正则性

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-10-31 DOI: 10.1016/j.na.2024.113691

Fabian Bäuerlein

We consider vector valued weak solutions

u : Ω_{T} \to R^{N}

with

N \in N

of degenerate or singular parabolic systems of type

\partial_{t} u - div a (z, u, D u) = 0 in Ω_{T} = Ω \times (0, T),

where

Ω

denotes an open set in

R^{n}

for

n \geq 1

and

T > 0

a finite time. Assuming that the vector field

a

is not of Uhlenbeck-type structure, satisfies

p

-growth assumptions and

(z, u) \mapsto a (z, u, ξ)

is Hölder continuous for every

ξ \in R^{N n}

, we show that the gradient

D u

is partially Hölder continuous, provided the vector field degenerates like that of the

p

-Laplacian for small gradients.

我们考虑矢量值弱解 u:ΩT→RN，N∈N 的∂tu-diva(z,u,Du)=0inΩT=Ω×(0,T)类型的退化或奇异抛物线系统，其中Ω表示 Rn 中的开集，n≥1，T>0 为有限时间。假定向量场 a 不是乌伦贝克型结构，满足 p 生长假设，且 (z,u)↦a(z,u,ξ) 对于每个 ξ∈RNn 都是霍尔德连续的，我们证明梯度 Du 部分是霍尔德连续的，条件是向量场像 p-Laplacian 的梯度一样退化为小梯度。

引用次数: 0

Gap results and existence of free boundary CMC surfaces in rotational domains 旋转域中的间隙结果和自由边界 CMC 表面的存在

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-10-30 DOI: 10.1016/j.na.2024.113681

Allan Freitas , Márcio S. Santos , Joyce S. Sindeaux

In this paper, we work with the existence and uniqueness of free boundary constant mean curvature surfaces in rotational domains. These are domains whose boundary is generated by a rotation of a graph. We classify the free boundary CMC surfaces as topological disks or annulus under some conditions on the function that generates the graph and a gap condition on the umbilicity tensor. Also, we construct some examples of free boundary CMC surfaces in the rotational ellipsoid that, in particular, satisfy our gap condition.

本文研究旋转域中自由边界恒均值曲率曲面的存在性和唯一性。这些域的边界由图形的旋转产生。根据生成图形的函数的一些条件和脐张量的间隙条件，我们将自由边界 CMC 曲面归类为拓扑盘或环面。此外，我们还构建了一些旋转椭球体中的自由边界 CMC 曲面实例，这些曲面尤其满足我们的间隙条件。

引用次数: 0

Positive and nodal limiting profiles for a semilinear elliptic equation with a shrinking region of attraction 具有收缩吸引力区域的半线性椭圆方程的正极限和节点极限剖面

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-10-22 DOI: 10.1016/j.na.2024.113680

Mónica Clapp , Víctor Hernández-Santamaría , Alberto Saldaña

We study the existence and concentration of positive and nodal solutions to a Schrödinger equation in the presence of a shrinking self-focusing core of arbitrary shape. Via a suitable rescaling, the concentration gives rise to a limiting profile that solves a nonautonomous elliptic semilinear equation with a sharp sign change in the nonlinearity. We characterize the (radial or foliated Schwarz) symmetries and the (polynomial) decay of the least-energy positive and nodal limiting profiles.

我们研究了在任意形状的收缩自聚焦核心存在的情况下，薛定谔方程正解和节点解的存在与集中。通过适当的重定标，这种集中会产生一个极限轮廓，该轮廓可以求解一个非自主椭圆半线性方程，其非线性符号会发生急剧变化。我们描述了（径向或叶状施瓦茨）对称性以及最小能量正向和节点极限剖面的（多项式）衰减。

引用次数: 0

Precise asymptotics near a generic S1×R3 singularity of mean curvature flow 平均曲率流一般 S1×R3 奇点附近的精确渐近线

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-10-16 DOI: 10.1016/j.na.2024.113679

Zhou Gang , Shengwen Wang

In the present paper we study a type of generic singularity of mean curvature flow modelled on the bubble-sheet

S^{1} \times R^{3}

, and we derive an asymptotic profile for a neighbourhood of singularity.

本文以气泡片 S1×R3 为模型，研究了平均曲率流的一种通用奇点，并推导出奇点邻域的渐近曲线。

引用次数: 0

Thermostatted kinetic theory in measure spaces: Well-posedness 度量空间中的恒温动力学理论：摆平性

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-10-09 DOI: 10.1016/j.na.2024.113666

Carlo Bianca , Nicolas Saintier

This paper is devoted to the generalization of the thermostatted kinetic theory within the framework of probability measures. Specifically well-posedness of the Cauchy problem related to a thermostatted kinetic equation for measure-valued functions is established. The external force applied to the system is assumed to be Lipschitz, in contrast to previous work where external forces are generally constant. Existence is obtained by employing an Euler-like approximation scheme which is shown to converge assuming the initial condition has moment of order greater than 2. Uniqueness is proved assuming the gain operator is Lipschitz w.r.t a (new) Monge–Kantorovich–Wasserstein distance

W_{2 -}

, intermediate between the classical

W_{2}

and

W_{r}

r < 2

, distances. The assumptions on the gain operator are quite general covering

n

-ary interaction, and apply in particular to the Kac equation.

本文致力于在概率度量框架内对恒温动力学理论进行概括。具体而言，本文建立了与量值函数恒温动力学方程相关的考奇问题的良好拟合。假定施加在系统上的外力为 Lipschitz，这与之前的研究不同，之前的研究通常认为外力是恒定的。假设增益算子在（新的）Monge-Kantorovich-Wasserstein 距离 W2--介于经典的 W2 和 Wr, r<2 距离之间是 Lipschitz，则证明了唯一性。关于增益算子的假设非常普遍，涵盖了 n-ary 相互作用，尤其适用于 Kac 方程。

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引用次数: 0

Positive solutions for a Kirchhoff problem of Brezis–Nirenberg type in dimension four 四维布雷齐斯-尼伦堡型基尔霍夫问题的正解

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-10-09 DOI: 10.1016/j.na.2024.113675

Giovanni Anello, Luca Vilasi

We consider a Kirchhoff problem of Brezis–Nirenberg type in a smooth bounded domain of

R^{4}

with Dirichlet boundary conditions. Our approach, novel in this framework and based upon approximation arguments, allows us to cope with the interaction between the higher order Kirchhoff term and the critical nonlinearity, typical of the dimension four. We derive several existence results of positive solutions, complementing and improving earlier results in the literature. In particular, we provide explicit bounds of the parameters

b

and

λ

coupled, respectively, with the higher order Kirchhoff term and the subcritical nonlinearity, for which the existence of solutions occurs.

我们考虑的是 R4 平滑有界域中的布雷齐斯-尼伦堡型基尔霍夫问题，其边界条件为狄利克特。我们的方法在这个框架中是新颖的，基于近似论证，使我们能够处理高阶基尔霍夫项与临界非线性之间的相互作用，这是典型的四维问题。我们推导出了几个正解的存在性结果，补充并改进了文献中的早期结果。特别是，我们提供了分别与高阶基尔霍夫项和次临界非线性耦合的参数 b 和 λ 的明确边界，对于这两个参数，解的存在是必然的。

引用次数: 0

Minimization of Dirichlet energy of j−degree mappings between annuli 环面间 j 度映射的 Dirichlet 能量最小化

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-10-09 DOI: 10.1016/j.na.2024.113671

David Kalaj

Let

A

and

A_{*}

be circular annuli in the complex plane, and consider the Dirichlet energy integral of

j

-degree mappings between

A

and

A_{*}

. We aim to minimize this energy integral. The minimizer is a

j

-degree harmonic mapping between the annuli

A

and

A_{*}

, provided it exists. If such a harmonic mapping does not exist, then the minimizer is still a

j

-degree mapping which is harmonic in

A^{'} \subset A

, and it is a squeezing mapping in its complementary annulus

A^{''} = A ∖ A^{'}

. This result is an extension of a certain result by Astala et al. (2010).

假设 A 和 A∗ 是复平面上的圆形环面，并考虑 A 和 A∗ 之间 j 阶映射的 Dirichlet 能量积分。我们的目标是最小化这个能量积分。最小值是环面 A 和 A∗ 之间的 j 度谐波映射，前提是它存在。如果不存在这样的调和映射，那么最小化映射仍然是一个在 A′⊂A 中调和的 j 度映射，并且是其互补环面 A′′=A∖A′ 中的挤压映射。这一结果是对阿斯塔拉等人（2010）的某个结果的扩展。

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引用次数: 0

Asymmetric affine Poincaré–Sobolev–Wirtinger inequalities on BV(Ω) and characterization of extremizers in one-dimension BV(Ω) 上的非对称仿射 Poincaré-Sobolev-Wirtinger 不等式和一维极值的表征

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-10-03 DOI: 10.1016/j.na.2024.113673

Raul Fernandes Horta, Marcos Montenegro

The present work deals with sharp asymmetric Poincaré–Sobolev–Wirtinger inequalities involving the Zhang’s energy on the space of bounded variation functions

B V (Ω)

for any bounded domain

Ω \subset R^{n}

in any dimension

n \geq 1

. We establish the existence of a curve of optimal constants along with several of its properties such as attainability, symmetry, monotonicity, positivity, continuity and also asymptotic ones. Moreover, for

n = 1

, our approach allows to exhibit its precise shape and to characterize all extremizers.

本研究涉及任何维数 n≥1 的有界域 Ω⊂Rn 的有界变化函数 BV(Ω)空间上涉及张氏能的尖锐非对称 Poincaré-Sobolev-Wirtinger 不等式。我们确定了最优常数曲线的存在性及其若干性质，如可达性、对称性、单调性、正向性、连续性和渐近性。此外，对于 n=1，我们的方法可以展示其精确形状并描述所有极值。

引用次数: 0

The asymptotic behavior of constant sign and nodal solutions of (p,q)-Laplacian problems as p goes to 1 当 p 变为 1 时，(p,q)-拉普拉斯问题的恒定符号和节点解的渐近行为

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2024-10-01 DOI: 10.1016/j.na.2024.113677

Giovany M. Figueiredo , Marcos T.O. Pimenta , Patrick Winkert

In this paper we study the asymptotic behavior of solutions to the

(p, q)

-equation

- Δ_{p} u - Δ_{q} u = f (x, u) in Ω, u = 0 on \partial Ω,

p \to 1^{+}

, where

N \geq 2

1 < p < q < 1^{*} ≔ N / (N - 1)

and

f

is a Carathéodory function that grows superlinearly and subcritically. Based on a Nehari manifold treatment, we are able to prove that the

(1, q)

-Laplace problem given by

- div (\frac{\nabla u}{| \nabla u |}) - Δ_{q} u = f (x, u) in Ω, u = 0 on \partial Ω,

has at least two constant sign solutions and one sign-changing solution, whereby the sign-changing solution has least energy among all sign-changing solutions. Furthermore, the solutions belong to the usual Sobolev space

W_{0}^{1, q} (Ω)

which is in contrast with the case of 1-Laplacian problems, where the solutions just belong to the space

BV (Ω)

of all functions of bounded variation. As far as we know this is the first work dealing with

(1, q)

-Laplace problems even in the direction of constant sign solutions.

本文研究了(p,q)方程 -Δpu-Δqu=f(x,u)inΩ,u=0on∂Ω,as p→1+ 的解的渐近行为，其中 N≥2, 1<p<q<1∗≔N/(N-1) 且 f 是超线性亚临界增长的 Carathéodory 函数。基于对 Nehari 流形的处理，我们能够证明由 -div∇u|∇u|-Δqu=f(x,u)inΩ,u=0on∂Ω 所给出的 (1,q)-Laplace 问题至少有两个恒符号解和一个符号变化解，其中符号变化解在所有符号变化解中能量最小。此外，解属于通常的 Sobolev 空间 W01,q(Ω)，这与 1 拉普拉斯问题的情况不同，后者的解只是属于所有有界变化函数的空间 BV(Ω)。据我们所知，这是第一部处理 (1,q) - 拉普拉斯问题的著作，甚至是在常数符号解的方向上。

引用次数: 0

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