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了解领域论文（摘要翻译）

1. 各向异性 Bourgain–Brezis 不等式

Anisotropic Bourgain–Brezis inequalities

作者: Dmitriy Stolyarov

类别: math.CA, math.AP

发布日期: 2026-08-21

arXiv ID: 2608.21135v1

链接: http://arxiv.org/abs/2608.21135v1

摘要翻译

中文标题: 各向异性 Bourgain–Brezis 不等式

摘要翻译: 我们将 $p=1$ 时的 Hardy–Littlewood–Sobolev 不等式推广至各向异性情形。主定理的若干推论给出了各向异性 Bourgain–Brezis 不等式的多个实例。

2. 通过 Nash–Moser 迭代在黎曼流形上对 Filtration 方程建立微分 Harnack 估计

Differential Harnack Estimates for the Filtration Equations on Riemannian Manifolds via Nash–Moser Iteration

作者: Jian-Hua Hao, Yu-Zhao Wang

类别: math.DG, math.AP

发布日期: 2026-08-21

arXiv ID: 2608.21213v1

链接: http://arxiv.org/abs/2608.21213v1

摘要翻译

中文标题: 通过 Nash–Moser 迭代在黎曼流形上对 Filtration 方程建立微分 Harnack 估计

摘要翻译: 我们证明了在完全黎曼流形上，Filtration 方程 $u_t=ΔF(u)$ 的光滑正解在一致抛物范围（uniformly parabolic ranges）内的局部微分 Harnack 估计。Bochner–判别式论证给出了一个强制性正部不等式（coercive positive-part inequality），该不等式通过 Nash–Moser 迭代闭合。我们导出了 Harnack 和 Liouville 推论，恢复了多孔介质方程（porous medium equation）和快速扩散方程（fast diffusion equation）的 Aronson-Bénilan 估计，并描述了一类真正非幂次的 filtration 律。

3. 弱空间约束下非因子化动力学 Fokker-Planck 方程的 Lyapunov 估计

Lyapunov estimates for non-factorised kinetic Fokker-Planck equations under weak spatial confinements

作者: Émeric Bouin

类别: math.AP

发布日期: 2026-08-21

arXiv ID: 2608.21111v1

链接: http://arxiv.org/abs/2608.21111v1

摘要翻译

中文标题: 弱空间约束下非因子化动力学 Fokker-Planck 方程的 Lyapunov 估计

摘要翻译: 本文针对具有弱外部约束的动力学 Fokker-Planck 方程构造了弱 Lyapunov 泛函。该问题在现有文献中一直是一个广泛开放的问题。除了揭示模型中有趣的结构外，推导这些 Foster-Lyapunov 权重使我们能够利用 Harris 型论证来证明这些方程稳态的存在性，以及以显式拉伸指数速率向平衡收敛。

4. KP 与修正 KP 方程在 $\mathbb T^2$ 和 $\mathbb R^2$ 上的非唯一奇异解

Non-unique singular solutions for KP and modified KP equations on $\mathbb T^2$ and $\mathbb R^2$

作者: Alexandru F. Radu

类别: math.AP

发布日期: 2026-08-21

arXiv ID: 2608.20972v1

链接: http://arxiv.org/abs/2608.20972v1

摘要翻译

中文标题: KP 与修正 KP 方程在 $\mathbb T^2$ 和 $\mathbb R^2$ 上的非唯一奇异解

摘要翻译: 我们在 $\mathbb T^2$ 和 $\mathbb R^2$ 上构造了无穷多个具有零初值和紧时间支撑的三阶及五阶 KP-I、KP-II 及其修正对应方程的弱奇异解。它们的非线性项是截断无关的绝对收敛傅里叶卷积。二次型解属于 $C_tL^p$（对 $p<2$）以及 $C_t(H^{-σ,0}\cap H^{-σ})$（对 $σ>0$）。修正型解属于 $C_tL^p$（对 $p<3$）。一个三次型族属于 $C_tH^α$（对 $α<1/3$）；另一个具有抛物线型傅里叶支撑，且属于 $C_tH^{s,0}$（对 $s<1/2$）以及 $C_tH^α$（对 $α<1/4$）。指数 $1/3$ 和 $1/2$ 在 $L^3$ 乘积阈值处是最佳的。对于 $\mathbb R^2$ 上的二次型五阶 KP 方程，非唯一性范围几乎是最佳的。我们还构造了周期定常 KP-I 和 KP-II 解，并证明 $L^2$ 是奇异定常 KP-I 解与光滑性之间的最佳阈值。

5. 缓慢均匀旋转降低气体恒星稳定性对应的临界绝热指数

Slow uniform rotation reduces the critical adiabatic exponent for the stability of gaseous stars

作者: Yucong Wang

类别: math.AP

发布日期: 2026-08-21

arXiv ID: 2608.20764v1

链接: http://arxiv.org/abs/2608.20764v1

摘要翻译

中文标题: 缓慢均匀旋转降低气体恒星稳定性对应的临界绝热指数

摘要翻译: 我们证明了，在轴对称扰动下，任意足够慢的均匀旋转均能稳定超大质量恒星，且该稳定性对于质量临界值左侧邻域内的每一个绝热指数均成立。线性分析中的关键新要素是自相似族所继承自质量临界标度的一个结构性性质：与之相关的自相似方向成为修正后线性化算子的约化核方向。

6. 通过经典 Schottky 统一化对小振幅 KP 有限间隙解的参数化及单间隙与双间隙解的持续性

A Parameterization of Small Amplitude KP Finite Gap Solutions via Classical Schottky Uniformization and Persistence of One and Two Gap Solutions

作者: Haoxuan Liu, Zuhong You, Xiaoping Yuan

类别: math.AP

发布日期: 2026-08-21

arXiv ID: 2608.20721v1

链接: http://arxiv.org/abs/2608.20721v1

摘要翻译

中文标题: 通过经典 Schottky 统一化对小振幅 KP 有限间隙解的参数化及单间隙与双间隙解的持续性

摘要翻译: 我们利用经典 Schottky 统一化方法，发展了一种针对具有预设空间波数向量的 KP 方程小振幅有限间隙解的参数化。该参数化适用于 Lyapunov-Schmidt 约化。基于此参数化，我们证明了在 Hamilton 扰动下，KP-I 和 KP-II 的周期单间隙解以及 KP-I 的双周期双间隙解的持续性。

7. Schrödinger Ornstein–Uhlenbeck 流：色散、限制性与非线性动力学

The Schrödinger Ornstein–Uhlenbeck flow: dispersion, restriction and nonlinear dynamics

作者: Nicola Garofalo

类别: math.AP

发布日期: 2026-08-20

arXiv ID: 2608.20577v1

链接: http://arxiv.org/abs/2608.20577v1

摘要翻译

中文标题: Schrödinger Ornstein–Uhlenbeck 流：色散、限制性与非线性动力学

摘要翻译: 我们研究由各向同性 Ornstein–Uhlenbeck 算子

$$ \mathscr L=Δ-\langle x,\nabla\rangle $$

在 $L^2(\mathbb R^d,dγ)$ 中生成的 Schrödinger 演化，其中 $dγ$ 是不变高斯测度。通过结合自由 Schrödinger 方程与谐振子之间的经典透镜对应以及高斯共轭，我们识别出隐藏在 Ornstein–Uhlenbeck 演化中的自由 Schrödinger 结构。 主要推论是一个时空型的动态限制性定理。在对时空变量进行显式变换之后，伴随的 Ornstein–Uhlenbeck 延拓算子变为限制在经典 Schrödinger 抛物面上的普通时空 Fourier 变换。在 Ornstein–Uhlenbeck 一侧，这对应于一个显式的时变高斯时空测度。因此，一种既不由 $\mathscr L$ 的符号所暗示，也不由 Ornstein–Uhlenbeck 方程的缩放对称性所选择的限制性几何，从自由 Schrödinger 表示中动态地涌现出来。 同样的对应关系识别了齐次和非齐次 Ornstein–Uhlenbeck Strichartz 估计中的加权高斯空间，并将质量临界非线性 Schrödinger 方程转化为

$$ i\partial_tu+\mathscr Lu = μe^{-|x|^2/d}|u|^{4/d}u. $$

我们还获得了酉 Ornstein–Uhlenbeck 群的全局描述，包括其在焦散点之外的振荡表示及其在焦散时刻的精确取值，连同相关的尖锐加权色散估计和 Hardy 型不确定性原理。

8. 通过外推方法获得退化的 Sobolev 不等式与 Poincaré 不等式

Degenerate Sobolev and Poincaré inequalities via extrapolation

作者: David Cruz-Uribe, Feyza Elif Dal, Scott Rodney

类别: math.AP

发布日期: 2026-08-20

arXiv ID: 2608.20498v1

链接: http://arxiv.org/abs/2608.20498v1

摘要翻译

中文标题: 通过外推方法获得退化的 Sobolev 不等式与 Poincaré 不等式

摘要翻译: 本文利用源自 Rubio de Francia 外推理论的证明技巧，建立了矩阵加权 Sobolev 不等式与 Poincaré 不等式。给定权重 $w,\,v$ 以及定义在 $\mathbb{R}^n$ 的一个连通开子集 $Ω$ 上的对称非负定矩阵值函数 $Q$，且满足如下下椭圆性条件

$$ w(x)^p \leq |\sqrt{Q(x)}ξ|^p,\quad ξ\in \mathbb{R}^n, $$

我们给出了权重 $w,v$ 的 Lebesgue 可积性条件，从而保证存在 $τ\geq 1$，使得如下形式的 Sobolev 不等式与 Poincaré 不等式成立：

$$ \bigg(\int_Ω|u|^{τp} \,vdx\bigg)^{\frac{1}{τp}} \leq C(v,w) \bigg(\int_Ω|\sqrt{Q}\nabla u|^p\,dx\bigg)^{\frac{1}{ p}},\textrm{ and} $$

$$ \bigg(\int_Ω|u-\langle u\rangle_{Ω,v}|^{τp} \,v dx\bigg)^\frac{1}{τp} \leq C(v,w)\bigg(\int_Ω|\sqrt{Q}\nabla u|^p \, dx\bigg)^{\frac{1}{p}} $$

其中 $u$ 为光滑函数。我们在 John 域、Heisenberg 群和 CR 流形等多个具体例子中探讨了这些结果及其相关结论。

不相关论文（基本信息）

1. $s$ 接近零时 $oldsymbol{\mathbb{R}^3}$ 中稳定的 $s$-极小锥是平的

Stable $s$-minimal cones in $\mathbb{R}^3$ are flat for $s$ close to zero

作者: Raphael Tsiamis

类别: math.DG, math.AP

发布日期: 2026-08-21

arXiv ID: 2608.21340v1

链接: http://arxiv.org/abs/2608.21340v1

摘要: We prove that stable nonlocal minimal cones in three dimensions are flat, for $s$ sufficiently close to zero. Our approach develops new structural properties of stable $s$-minimal surfaces in every dimension as $s \downarrow 0$, notably a compactness theory and pinching theorems near a half-space.

2. 一个逆自由边界问题

An inverse free boundary problem

作者: Cătălin I. Cârstea, Matti Lassas, Jinpeng Lu, Lauri Oksanen, Ziyao Zhao

类别: math.AP

发布日期: 2026-08-21

arXiv ID: 2608.21153v1

链接: http://arxiv.org/abs/2608.21153v1

摘要: We study inverse problems for the elliptic and parabolic obstacle problems from boundary measurements. For the classical elliptic obstacle problem with strictly superharmonic obstacle function, we show that the Dirichlet-to-Neumann map admits a one-sided linearization at every boundary datum lying strictly above the obstacle. The linearized map is the Dirichlet-to-Neumann map for a rough Dirichlet problem on the a priori unknown non-contact set. We show that these linearized Cauchy data uniquely determine the non-contact set up to set of Sobolev $2$-capacity zero and consequently determine the obstacle. Our result applies to the inverse problem for a parabolic obstacle problem where both the coefficient and the obstacle function are time-independent by reducing to the elliptic inverse problem.

3. 四维 Brezis-Nirenberg 问题中变号多泡解

Sign-changing multi-bubble solutions for the Brezis-Nirenberg problem in four dimensions

作者: Angela Pistoia, Giuseppe Mario Rago, Giusi Vaira

类别: math.AP

发布日期: 2026-08-21

arXiv ID: 2608.21239v1

链接: http://arxiv.org/abs/2608.21239v1

摘要: We construct families of sign-changing solutions for the four-dimensional Brezis–Nirenberg problem

$$ -Δu=u^3+\varepsilon u\quad\text{in }Ω,\qquad u=0\quad\text{on }\partialΩ, $$

as $\varepsilon\to0^+$. A Lyapunov–Schmidt reduction shows that the location and relative scales of the bubbles are governed by a signed Green–Robin interaction matrix. We formulate an abstract existence criterion in terms of a simple positive eigenvalue admitting a positive eigenvector and a stable critical set. We then apply it to a positive–negative pair in a general domain and to several symmetric multi-peak configurations, including alternating regular polygons, orthogonal polygons, one central peak surrounded by peaks of the opposite sign, and aligned three-, four-, and five-peak patterns. For the two-peak solution we also prove that it has exactly two nodal domains and, under a natural balance condition and connectedness of the boundary, that the closure of its nodal set meets the boundary.

4. 图标量曲率方程的 Robin 与 Neumann 问题

Robin and Neumann problems for the graph scalar curvature equation

作者: Guohuan Qiu

类别: math.AP

发布日期: 2026-08-21

arXiv ID: 2608.21085v1

链接: http://arxiv.org/abs/2608.21085v1

摘要: We study Robin and Neumann problems for the scalar curvature equation of admissible graphs over bounded uniformly convex domains in three dimensions. Under a small-volume assumption, we prove existence and uniqueness for the Robin problem and obtain a classical Neumann solution as the Robin parameter tends to zero. The volume threshold is optimal among conditions depending only on the volume. The main step is a boundary second-derivative estimate uniform in the Robin parameter; known interior and global-to-boundary curvature estimates then give the global bound.

5. 来自可测集与 $C^{1,1}$ 区域上热可观性的边界谱不等式

Boundary Spectral Inequalities from Measurable Sets and Heat Observability on $C^{1,1}$ Domains

作者: Lu Chen, Haolin Liu, Hongyu Liu

类别: math.AP

发布日期: 2026-08-21

arXiv ID: 2608.21039v1

链接: http://arxiv.org/abs/2608.21039v1

摘要: In this paper, we consider the boundary control problem. For bounded connected $C^{1,1}$ domains, we establish quantitative boundary spectral inequalities for both Dirichlet and Neumann eigenfunctions. Moreover, on the basis of the spectral inequality, we study the boundary control of the solution of the heat equation and get the following observability inequality $\|u(T)\|_{L^2(Ω)}^2 \leq C\int_{\mathcal J}|\partial_νu(x,t)|^2 \, dσ\, dt,$ where the observation domain $\mathcal J\subset\partialΩ\times(0,T)$ is measurable with $|\mathcal J|>0$. The proof of the observability inequality combines quantitative continuation from positive-measure boundary sets with low-frequency spectral concentration. This result shows that boundary observability of solutions to the heat equation can also be achieved in $C^{1,1}$ domains.

6. 几何多调和热流的集中紧致性

Concentration-compactness for the geometric polyharmonic heat flow

作者: James McCoy, Scott Parkins, Glen Wheeler, Valentina Wheeler

类别: math.AP

发布日期: 2026-08-21

arXiv ID: 2608.20950v1

链接: http://arxiv.org/abs/2608.20950v1

摘要: We develop a concentration-compactness theory for geometric evolution equations of arbitrarily high order, using the geometric polyharmonic heat flow of closed immersed surfaces in $\R^3$ as the model case. The flow is the $(2p+2)$-order normal evolution

$$ \partial_t f=(-1)^{p+1}Δ^p H\,ν,\qquad p\geq1, $$

which includes the surface diffusion flow when $p=1$. We prove localised energy estimates with sharp cut-off bookkeeping, interior estimates, a lifespan/concentration alternative, tracefree-curvature $\varepsilon$-regularity estimates, and a gap theorem for stationary solutions. These tools are then combined with a blowup argument, the preservation of signed enclosed volume, and the monotonicity of area to rule out singularities below a small tracefree-curvature threshold. Consequently, for connected initial immersions satisfying $\|A^o\|_2^2<\varepsilon$, where $\varepsilon>0$ depends only on the order of the flow, the solution exists for all time and converges exponentially in $C^\infty$ to a round sphere with the preserved enclosed volume.

7. 径向对称函数的高阶导数

Higher-order derivatives of radially symmetric functions

作者: Zdeněk Mihula, Jan Vybíral

类别: math.FA, math.AP

发布日期: 2026-08-21

arXiv ID: 2608.20941v1

链接: http://arxiv.org/abs/2608.20941v1

摘要: We prove a surprisingly simple pointwise formula for the Frobenius norm of the tensor of $n$-th order partial derivatives of a radially symmetric function $f(x)=g(r(x))$. Using the iterations of the differential operator $\mathcal{D} g(r)=g'(r)/r$, we avoid technical difficulties usually caused by higher-order radial derivatives. As a consequence, we obtain a complete characterization of the subspace of radially symmetric functions in both inhomogeneous and homogeneous Sobolev spaces of arbitrarily high order and all integrability parameters $p\in [1,\infty).$ Furthermore, the pointwise nature of our approach allows us to obtain similar results also for Sobolev-type spaces built upon more general Banach lattices.

8. 图形 $σ_2$-曲率方程的严格凸性与精确幂次凹性

Strict Convexity and Sharp Power Concavity for a Graphical $σ_2$-Curvature Equation

作者: Shuning Xu

类别: math.AP

发布日期: 2026-08-21

arXiv ID: 2608.20907v1

链接: http://arxiv.org/abs/2608.20907v1

摘要: We prove strict convexity of the square-root transformation $v=-\sqrt{-u}$ for admissible solutions of a graphical $σ_2$-curvature Dirichlet problem on smooth uniformly convex domains. A key ingredient is a constant-rank theorem for $D^2v$, proved by a direct Ma-Xu type argument in dimension three and by the Bian-Guan microscopic convexity principle together with inverse-convexity methods in arbitrary dimensions. Combined with boundary strict convexity and a domain-deformation argument, the constant-rank theorem yields $D^2v>0$ throughout the domain.

9. 集中现象与质量消失猜想

Concentration phenomena and the Vanishing Mass Conjecture

作者: Luca Gennaioli, Filip Rindler

类别: math.AP

发布日期: 2026-08-21

arXiv ID: 2608.20899v1

链接: http://arxiv.org/abs/2608.20899v1

摘要: Concentrating (that is, non-equi-integrable) sequences of functions satisfying linear PDE constraints arise in a wide variety of problems in PDE, the calculus of variations, and geometric measure theory. In 2003, Bouchitté conjectured that such concentrations can always be represented as superpositions of simple'' concentrations, a claim he termed the Vanishing Mass Conjecture. We fully resolve this conjecture for all first-order constant-coefficient linear differential operators, proving that the value-distribution (Young) measure of any such sequence admits a Choquet-type decomposition into probability measures whose barycenters lie in the associated Tartar wave cone. In fact, we prove a significantly stronger statement than originally conjectured, namely that the constituent measures are themselves generated by concentrating sequences satisfying the same PDE constraint. This provides a complete structural description of concentrating sequences and yields several applications, including a new proof of a theorem of De Philippis and the second author on the singular polar of PDE-constrained measures, two types of compensated compactness results, a general lower bound related to the Optimal Light Structures Conjecture in shape optimization, and a surprising result on the support cardinality of extremal concentration Young measures. We also place our results in the context of Morrey's Conjecture, showing that its analogue for pure concentrations is false. Our proofs introduce several new techniques, most notably convexity arguments involving barrier functions’’ and a careful smoothing procedure via the heat flow that replaces classical Fourier methods.

10. Lotka–Volterra 竞争-扩散系统中双稳态行波前传播方向

Propagation Direction of Bistable Traveling Fronts in the Lotka–Volterra Competition–Diffusion System

作者: Shizhao Ma, Dongyuan Xiao, Maolin Zhou

类别: math.AP

发布日期: 2026-08-21

arXiv ID: 2608.20795v1

链接: http://arxiv.org/abs/2608.20795v1

摘要: We study the propagation direction of bistable traveling fronts in the two-species Lotka–Volterra competition–diffusion system under strong competition. A complete characterization of the sign of the wave speed has remained a long-standing unsolved problem. We establish the first global necessary and sufficient criterion for zero wave speed by combining a Maxwell-type identity with a phase-plane rigidity argument. This criterion yields a unique zero-speed threshold, identifies its value in the symmetric case and its limiting values in two extreme regimes, and consequently determines the propagation direction throughout the strong-competition parameter region.

11. 在 $n-2$ 个反射对称性下的对数 Brunn–Minkowski 不等式

Logarithmic Brunn–Minkowski Inequality under $n-2$ Reflection Symmetries

作者: XiaoRui Lu

类别: math.MG, math.AP, math.DG

发布日期: 2026-08-21

arXiv ID: 2608.20730v1

链接: http://arxiv.org/abs/2608.20730v1

摘要: We prove the log-Brunn-Minkowski inequality for origin-symmetric convex bodies with symmetries to n-2 orthogonal hyperplanes, and discuss the equality case and the uniqueness of the related logarithmic Minkowski problem. We also generalize the inequality from the Lebesgue measure to certain log-concave measure.

12. 各向同性-向列相问题的各向异性弹性渐近极限

Asymptotic limit for the isotropic-nematic problem with anisotropic elasticity

作者: Fanghua Lin, Wei Wang, Zhifei Zhang

类别: math.AP

发布日期: 2026-08-21

arXiv ID: 2608.20701v1

链接: http://arxiv.org/abs/2608.20701v1

摘要: We consider the isotropic-nematic interface problem for liquid crystals based on the Landau-de Gennes model. We show that when the anisotropic elastic constant $L_2$ is negative, then the surface tension strength of the isotropic-nematic interface is proportional to $\sqrt{L_1+\frac{2}{3}{L}_2}$, and the homeotropic alignment is more favored near the interface. This result gives a rigorous confirmation for de Gennes’ formal derivation in \cite{deGen1971} on the isotropic-nematic tension strength and the anchoring alignment condition in the case of $L_2<0$.

13. 逆问题中低傅里叶模态的改进稳定性

Improved stability of low Fourier modes in inverse problems for potentials

作者: Tony Liimatainen, William Trad, Mikko Salo

类别: math.AP

发布日期: 2026-08-21

arXiv ID: 2608.20673v1

链接: http://arxiv.org/abs/2608.20673v1

摘要: We prove Lipschitz, sub-Hölder and Hölder stability estimates for recovering the low Fourier modes of an unknown potential from the Dirichlet-to-Neumann (DN) map. We study three different cases, depending on the regularity of the difference $q_1-q_2$. First, we consider

$$ (-Δ- λ^2 + q)u=0 \quad \text{in} \quad Ω\subset \mathbb{R}^n. $$

We show that the difference $q_1-q_2$, assumed to be $M$-bandlimited, can be recovered in a Lipschitz stable way from the difference of the corresponding DN maps. This holds whenever $λ$ is sufficiently large relative to $M^{n/2}$. The proof involves real geometrical optics solutions. Secondly, we consider

$$ (-Δ+ q)u=0 \quad \text{in} \quad (-π,π)^n. $$

We show that the low Fourier coefficients of the difference $q_1-q_2$, assumed to be real-analytic and periodic, can be recovered in a sub-Hölder stable way from the difference of the corresponding DN maps. The number of recoverable Fourier modes grows as the DN maps become closer. Finally, we consider the case where the Fourier coefficients of the difference $q_1-q_2$ decay at a super-exponential rate $e^{-c|k|^{n/2}}$. We prove that the low Fourier modes can be recovered with Hölder stability, with the number of recoverable modes tending to infinity as the DN maps become closer. In all cases the $L^\infty$ potentials themselves do not need to satisfy additional assumptions or belong to a finite dimensional space. The constants in the stability estimates are uniform in the number of recovered Fourier modes.

14. 具有壳层相互作用的 Landau-Dirac 算子：自伴性与聚簇

The Landau-Dirac operator with shell interactions: self-adjointness and clustering

作者: Badreddine Benhellal, Vincent Bruneau, Pablo Miranda

类别: math-ph, math.AP, math.SP

发布日期: 2026-08-21

arXiv ID: 2608.20665v1

链接: http://arxiv.org/abs/2608.20665v1

摘要: We consider the two-dimensional Dirac operator with constant magnetic field that is perturbed by a combination of electrostatic and Lorentz-scalar delta interactions with variable coefficients supported on a smooth closed curve. Self-adjointness is studied in the so called non critical and critical cases. In the non-critical case the essential spectrum is unchanged - it remains to be the set of the Landau-Dirac levels, the eigenvalues of infinite multiplicity of the unperturbed operator - while in the critical case an additional interval of essential spectrum emerges in the spectral gap containing zero. Our main result concerns the discrete spectrum in the non-critical case: using the pseudodifferential properties of the involved boundary integral operators, we show that the eigenvalues accumulate at each Landau-Dirac level at a rate governed by the logarithmic capacity of the curve. A novel and surprising phenomenon is the change in the side of the accumulation depending on the position relative to the critical value. As a byproduct, clusters of eigenvalues for a family of exterior boundary value problems are obtained via confining couplings; the infinite-mass boundary condition arises as a special case.

15. FitzHugh-Nagumo 方程的差分 Harnack 不等式

A Differential Harnack Inequality for the FitzHugh-Nagumo Equation

作者: Xiaodong Cao, Harry Fluck, Sophia Teng

类别: math.AP

发布日期: 2026-08-20

arXiv ID: 2608.20590v1

链接: http://arxiv.org/abs/2608.20590v1

摘要: In this paper we develop a Li-Yau-Hamilton (LYH) type differential Harnack estimate for positive solutions to the FitzHugh-Nagumo equation on $\mathbb{R}^n$. We then use our LYH-differential Harnack inequality to prove several properties about positive solutions to the equation, including a classical Harnack inequality and a lower bound for the speed of traveling wave solutions.

16. 每个维度下 Neumann–Poincaré 算子的一个精确等周不等式

A sharp isoperimetric inequality for the Neumann–Poincaré operator in every dimension

作者: Matthew J. Colbrook, Siavash Sadeghi

类别: math.AP, math.NA, math.SP

发布日期: 2026-08-20

arXiv ID: 2608.20466v1

链接: http://arxiv.org/abs/2608.20466v1

摘要: Let $Ω\subset\mathbb{R}^d$, $d\ge2$, be a bounded connected domain with boundary of class $C^{1,α}$, where $0<α<1$. For the adjoint Neumann–Poincaré operator $K^*_{\partialΩ}$, normalised so that its distinguished eigenvalue is $1/2$, let $λ_j^+(Ω)$ denote the upper min–max values on the mean-zero energy space. We prove $\sum_{j=1}^{d}λ_j^+(Ω)\ge \frac{d-2}{2}.$ It follows that $λ_1^+(Ω)\ge \frac{d-2}{2d},$ with equality if and only if $Ω$ is a ball. In dimension three this proves the $1/6$-conjecture of Miyanishi and Suzuki. The proof uses the coordinate boundary-charge densities induced by uniform applied fields. Their energy Gram matrix is the perfect-conductor polarization tensor $M_\infty$. Positivity of a $2d\times2d$ Gram matrix yields the endpoint Hashin–Shtrikman inequality $|Ω|\mathrm{Tr}(M_\infty^{-1})\le1,$ and bounds the trace of the compression of $K^*_{\partialΩ}$ to the applied-field space. Equality in the inverse-trace inequality makes the interior Newtonian potential quadratic, so a converse to Newton’s theorem identifies $Ω$ as an ellipsoid. Equality in the spectral estimate also makes the Hessian of this potential isotropic, which forces the ellipsoid to be a ball.