Nonlinear Analysis-Theory Methods & Applications最新文献

英文中文

Optimal planar immersions of prescribed winding number and Arnold invariants 规定圈数和阿诺德不变量的最优平面浸入

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-10 DOI: 10.1016/j.na.2025.113942

Anna Lagemann, Heiko von der Mosel

Vladimir Arnold defined three invariants for generic planar immersions, i.e. planar curves whose self-intersections are all transverse double points. We use a variational approach to study these invariants by investigating a suitably truncated knot energy, the tangent-point energy. We prove existence of energy minimizers for each truncation parameter

δ > 0

in a class of immersions with prescribed winding number and Arnold invariants, and establish Gamma convergence of the truncated tangent-point energies to a limiting renormalized tangent-point energy as

δ \to 0

. Moreover, we show that any sequence of minimizers subconverges in

C^{1}

, and the corresponding limit curve has the same topological invariants, self-intersects exclusively at right angles, and minimizes the renormalized tangent-point energy among all curves with right self-intersection angles. In addition, the limit curve is an almost-minimizer for all of the original truncated tangent-point energies as long as the truncation parameter

δ

is sufficiently small. Therefore, this limit curve serves as an “optimal” curve in the class of generic planar immersions with prescribed winding number and Arnold invariants.

Vladimir Arnold定义了一般平面浸入式的三个不变量，即自交均为横向双点的平面曲线。我们使用变分的方法来研究这些不变量，通过研究一个适当截断的结能量，切点能量。在给定圈数和Arnold不变量的浸入式中，证明了每一个截断参数δ>；0的能量极小值的存在性，并建立了截断的切点能量的伽玛收敛到一个极限重归一化切点能量为δ→0。此外，我们还证明了任何最小值序列在C1中都是子收敛的，并且相应的极限曲线具有相同的拓扑不变量，在直角处完全自交，并且在所有自交角为直角的曲线中极小化了的切点能量。此外，只要截断参数δ足够小，对于所有原始截断的切点能量，极限曲线几乎是最小的。因此，该极限曲线可作为具有规定圈数和阿诺德不变量的一般平面浸没的一类“最优”曲线。

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引用次数: 0

Γ-convergence of higher-order phase transition models Γ-convergence的高阶相变模型

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-09 DOI: 10.1016/j.na.2025.113971

Denis Brazke , Gianna Götzmann , Hans Knüpfer

We investigate the asymptotic behavior as

ɛ \to 0

of singularly perturbed phase transition models of order

n \geq 2

, given by

G_{ɛ}^{λ, n} [u] ≔ \int_{I} \frac{1}{ɛ} W (u) - λ ɛ^{2 n - 3} {(u^{(n - 1)})}^{2} + ɛ^{2 n - 1} {(u^{(n)})}^{2} d x, u \in W^{n, 2} (I),

where

λ > 0

is fixed,

I \subset R

is an open bounded interval, and

W \in C^{0} (R)

is a suitable double-well potential. We find that there exists a positive critical parameter depending on

W

and

n

, such that the

Γ

-limit of

G_{ɛ}^{λ, n}

with respect to the

L^{1}

-topology is given by a sharp interface functional in the subcritical regime. The cornerstone for the corresponding compactness property is a novel nonlinear interpolation inequality involving higher-order derivatives, which is based on Gagliardo–Nirenberg type inequalities.

研究了n≥2阶奇异摄动相变模型的渐近性，其中λ λ，n[u]是∫I1 ^ W(u)−λ ^ 2n−3(u(n−1))2+ ^ 2n−1(u(n))2dx,u∈Wn,2(I)，其中λ ^ gt；0是固定的，I∧R是一个开有界区间，W∈C0(R)是一个合适的双阱势。我们发现存在一个依赖于W和n的正临界参数，使得G λ，n关于l1拓扑的Γ-limit由亚临界区中的锐界面泛函给出。该紧性的基础是基于Gagliardo-Nirenberg型不等式的一种涉及高阶导数的非线性插值不等式。

引用次数: 0

Geometric analysis on weighted manifolds under lower 0-weighted Ricci curvature bounds 下0权Ricci曲率界下加权流形的几何分析

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-09 DOI: 10.1016/j.na.2025.113965

Yasuaki Fujitani , Yohei Sakurai

We develop geometric analysis on weighted Riemannian manifolds under lower 0-weighted Ricci curvature bounds. Under such curvature bounds, we prove a first non-zero Steklov eigenvalue estimate of Wang–Xia type on compact weighted manifolds with boundary, and a first non-zero eigenvalue estimate of Choi–Wang type on closed weighted minimal hypersurfaces. We also produce an ABP estimate and a Sobolev inequality of Brendle type.

给出了下0权Ricci曲率界下加权黎曼流形的几何分析。在这样的曲率边界下，证明了紧加权流形上Wang-Xia型的第一个非零Steklov特征值估计，以及封闭加权极小超曲面上Choi-Wang型的第一个非零特征值估计。我们也得到了一个ABP估计和一个Brendle型的Sobolev不等式。

引用次数: 0

A priori estimates for anti-symmetric solutions to a fractional Laplacian equation in a bounded domain 有界域上分数阶拉普拉斯方程反对称解的先验估计

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-08 DOI: 10.1016/j.na.2025.113970

Chenkai Liu , Shaodong Wang , Ran Zhuo

In this paper, we obtain a priori estimates for the set of anti-symmetric solutions to a fractional Laplacian equation in a bounded domain using a blowing-up and rescaling argument. In order to establish a contradiction to possible blow-ups, we apply a certain variation of the moving planes method in order to prove a monotonicity result for the limit equation after rescaling.

在有界区域上，利用放大和重标尺论证，给出了分数阶拉普拉斯方程的反对称解集的先验估计。为了证明极限方程在重新标度后的单调性，我们对运动平面法进行了一定的变换。

引用次数: 0

Particle approximation of nonlocal interaction energies 非局部相互作用能量的粒子近似

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-07 DOI: 10.1016/j.na.2025.113974

Davide Carazzato , Aldo Pratelli , Ihsan Topaloglu

We consider Riesz-type nonlocal energies with general interaction kernels and their discretizations related to particle systems. We prove that the discretized energies

Γ

-converge in the weak-

*

topology to the Riesz functional defined over the space of probability measures. We also address the minimization problem for the discretized energies, and prove the existence of minimal configurations of particles in a very general and natural setting.

我们考虑具有一般相互作用核的riesz型非局部能量及其与粒子系统相关的离散化。我们证明了弱- *拓扑中的离散能量Γ-converge在概率测度空间上定义的Riesz泛函。我们还讨论了离散能量的最小化问题，并证明了在非常一般和自然的情况下粒子的最小构型的存在性。

引用次数: 0

The Dirichlet problem on lower dimensional boundaries: Schauder estimates via perforated domains 低维边界上的Dirichlet问题：通过穿孔区域的Schauder估计

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-07 DOI: 10.1016/j.na.2025.113973

Gabriele Fioravanti

In this paper, we investigate the Dirichlet problem on lower dimensional manifolds for a class of weighted elliptic equations with coefficients that are singular on such sets. Specifically, we study the problem

(\begin{matrix} - div ({| y |}^{a} A (x, y) \nabla u) = {| y |}^{a} f + div ({| y |}^{a} F), \\ u = ψ, on Σ_{0}, \end{matrix})

where

(x, y) \in R^{d - n} \times R^{n}

2 \leq n \leq d

a + n \in (0, 2)

, and

Σ_{0} = {| y | = 0}

is the lower dimensional manifold where the equation loses uniform ellipticity.

Our primary objective is to establish

C^{0, α}

and

C^{1, α}

regularity estimates up to

Σ_{0}

, under suitable assumptions on the coefficients and the data. Our approach combines perforated domain approximations, Liouville-type theorems and a blow-up argument.

本文研究了一类系数为奇异的加权椭圆型方程在低维流形上的Dirichlet问题。具体来说，我们研究了−div(|y|aA(x,y)∇u)=|y|af+div(|y| af),u=ψ,onΣ0，其中(x,y)∈Rd - n×Rn, 2≤n≤d, a+n∈(0,2)，Σ0={|y|=0}是方程失去一致椭圆性的低维流形。我们的主要目标是在对系数和数据的适当假设下，建立到Σ0的C0，α和C1，α正则性估计。我们的方法结合了穿孔域近似、liouville型定理和一个放大论证。

引用次数: 0

A Mean Field Game system and a related deterministic optimal control problem 平均场博弈系统及相关的确定性最优控制问题

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-07 DOI: 10.1016/j.na.2025.113977

Ştefana-Lucia Aniţa

This paper concerns a Mean Field Game (MFG) system related to a Nash type equilibrium for dynamical games associated to large populations. One shows that the MFG system may be viewed as the Euler–Lagrange system for an optimal control problem related to a Fokker–Planck equation with control in the drift. One derives the existence of a weak solution to the MFG system and under more restrictive assumptions one proves some uniqueness results.

本文研究了与大种群动态博弈纳什均衡相关的平均场博弈（MFG）系统。一个证明了MFG系统可以被看作是一个最优控制问题的欧拉-拉格朗日系统，该最优控制问题与漂移中的控制Fokker-Planck方程有关。导出了MFG系统弱解的存在性，并在更严格的假设条件下证明了一些唯一性结果。

引用次数: 0

Formation of delta shock waves in the limit of Riemann solutions to the Aw–Rascle traffic model with a damping term 带阻尼项的Aw-Rascle交通模型黎曼解极限下三角洲激波的形成

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-07 DOI: 10.1016/j.na.2025.113969

Jie Cheng , Tianrui Bai , Fangqi Chen

In this paper, we consider the Riemann problem of the Aw–Rascle traffic model with a damping term and the formation of delta shock waves in the limit of the Riemann solutions as

γ \to 1

. By introducing a new variable and employing generalized characteristic analysis methods, we construct solutions to the Riemann problem of the inhomogeneous Aw–Rascle traffic model. Specially, for the case

0 < u_{-} < u_{+}

, we prove the existence of a critical value

{\bar{γ}}_{0}

for

γ

such that when

0 < γ < {\bar{γ}}_{0}

, the Riemann solutions contain no vacuum states; otherwise, a vacuum state emerges. Furthermore, we demonstrate that as

γ \to 1

, the limit of the Riemann solutions with vacuum states aligns with the Riemann solutions to the inhomogeneous transport model under the same initial conditions, while the limit of solutions with shock waves converges to a curved delta shock solution. Notably, the weights supported on the delta shock solution differ from the Riemann solutions to the inhomogeneous transport model due to the influence of the damping term.

本文考虑了带阻尼项的Aw-Rascle交通模型的黎曼问题，以及黎曼解极限为γ→1时δ激波的形成。通过引入一个新的变量，利用广义特征分析方法，构造了非齐次交通模型的Riemann问题的解。特别地，对于0<；u−<；u+的情况，我们证明了γ的一个临界值γ¯0的存在性，使得当0<；γ<；γ¯0时，黎曼解不包含真空态；否则，出现真空状态。进一步证明，当γ→1时，具有真空态的黎曼解的极限与非均匀输运模型的黎曼解在相同初始条件下对准，而具有激波的黎曼解的极限收敛于弯曲的δ激波解。值得注意的是，由于阻尼项的影响，delta激波解所支持的权重与非均匀输运模型的黎曼解不同。

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引用次数: 0

Global existence for a Leibenson type equation with reaction on Riemannian manifolds 黎曼流形上带反应的Leibenson型方程的整体存在性

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-03 DOI: 10.1016/j.na.2025.113967

Giulia Meglioli , Francescantonio Oliva , Francesco Petitta

We show a global existence result for a doubly nonlinear porous medium type equation of the form

u_{t} = Δ_{p} u^{m} + u^{q}

on a complete and non-compact Riemannian manifold

M

of infinite volume. Here, for

1 < p < N

, we assume

m (p - 1) \geq 1

m > 1

and

q > m (p - 1)

. In particular, under the assumptions that

M

supports the Sobolev inequality, we prove that a solution for such a problem exists globally in time provided

q > m (p - 1) + \frac{p}{N}

and the initial datum is small enough; namely, we establish an explicit bound on the

L^{\infty}

norm of the solution at all positive times, in terms of the

L^{1}

norm of the data. Under the additional assumption that a Poincaré-type inequality also holds in

M

, we can establish the same result in the larger interval, i.e.

q > m (p - 1)

. This result has no Euclidean counterpart, as it differs entirely from the case of a bounded Euclidean domain due to the fact that

M

is non-compact and has infinite measure.

在无限体积的完全非紧黎曼流形M上，给出了形式为ut=Δpum+uq的双非线性多孔介质型方程的整体存在性结果。这里，对于1<；p<；N，我们假设m(p−1)≥1,m>；1和q>；m（p−1）。特别地，在M支持Sobolev不等式的假设下，我们证明了在q>； M (p−1)+pN且初始基准足够小的情况下，该问题的解在时间上全局存在；也就是说，我们根据数据的L1范数，在所有正时刻的解的L∞范数上建立一个显式的界。在附加的假设下，一个poincar型不等式在M中也成立，我们可以在更大的区间，即q>； M （p−1）中建立同样的结果。这个结果没有欧几里得对应物，因为它完全不同于有界欧几里得定义域的情况，因为M是非紧致的并且具有无限的度量。

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引用次数: 0

Improvement of the parabolic regularization method and applications to dispersive models 抛物正则化方法的改进及其在色散模型中的应用

IF 1.3 2区数学 Q1 MATHEMATICS

Nonlinear Analysis-Theory Methods & Applications

Pub Date : 2025-10-03 DOI: 10.1016/j.na.2025.113964

Alysson Cunha

We prove that the Benjamin–Ono equation is globally well-posed in

H^{s} (R)

for

s > \frac{1}{2}

. Our approach does not rely on the global gauge transformation introduced by Tao (Tao, 2004). Instead, we employ a modified version of the standard parabolic regularization method. In particular, this technique also enables us to establish global well-posedness, in the same Sobolev space, for the dispersion-generalized Benjamin–Ono (DGBO) equation.

我们证明了对于s>；12， Benjamin-Ono方程在Hs(R)上是全局适定的。我们的方法不依赖于Tao （Tao, 2004）引入的全局规范转换。相反，我们采用标准抛物线正则化方法的改进版本。特别地，该技术还使我们能够在相同的Sobolev空间中为色散广义Benjamin-Ono （DGBO）方程建立全局适定性。

引用次数: 0

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Nonlinear Analysis-Theory Methods & Applications

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