生成时间: 2025年10月05日 02:48:40

重点关注论文数量: 0

了解领域论文数量: 6

不相关论文数量: 21

了解领域论文（摘要翻译）

1. Blow up dynamics for the 3D energy-critical Nonlinear Schrödinger

equation

三维能量临界非线性薛定谔方程的爆破动力学

作者: Tobias Schmid

类别: math.AP

发布日期: 2023-08-03

ArXiv ID: 2308.01883v3

链接: http://arxiv.org/abs/2308.01883v3

摘要翻译

中文标题: 三维能量临界非线性薛定谔方程的爆破动力学

摘要翻译: 我们构造了在空间维数 $ d = 3 $ 下，能量临界聚焦非线性薛定谔方程（NLS）的一类双参数连续的II型爆破产生解。这些解在有限时间内坍缩为一个单一的能量泡，具体形式为 $ u(t,x) = e^{i \alpha(t)}\lambda(t)^{\frac{1}{2}}W(\lambda(t) x) + \eta(t, x )$，其中 $ t \in[0, T) $，$ x \in \mathbb{R}^3 $。这里 $ W(x) = \big( 1 + \frac{|x|^2}{3}\big)^{-\frac{1}{2}} $ 是基态解，$ \lambda(t) = (T-t)^{- \frac{1}{2} - \nu} $，对应适当的 $ \nu > 0 $，$ \alpha(t) = \alpha_0 \log(T - t) $，且 $ T= T(\nu, \alpha_0) > 0 $。此外，当 $ t \to T^- $ 时，有 $ \|\eta(t) - \eta_T\|_{\dot{H}^1 \cap \dot{H}^2} = o(1) $，其中 $ \eta_T \in \dot{H}^{1} \cap \dot{H}^2 $。

2. Linear damping estimates for periodic roll wave solutions of the

inviscid Saint-Venant equations and related systems of hyperbolic balance laws

无黏圣维南方程及相关的双曲平衡律系统的周期滚波解的线性阻尼估计

作者: L. Miguel Rodrigues, Kevin Zumbrun

类别: math.AP, 35B35, 35L67, 15A18, 15A42, 35Q35, 35P15, 15A60, 15A63

发布日期: 2025-04-17

ArXiv ID: 2504.12804v3

链接: http://arxiv.org/abs/2504.12804v3

摘要翻译

中文标题: 无黏圣维南方程及相关的双曲平衡律系统的周期滚波解的线性阻尼估计

摘要翻译: 本文在作者先前关于黏性情形下光滑解所得结果的基础上，进行了显著推广，针对无黏圣维南方程及相关双曲平衡律系统的周期滚波解，建立了线性阻尼估计。此类阻尼估计由 $H^s$ 能量估计构成，能够实现高阶导数范数对低阶导数范数的指数型支配关系（exponential slaving），在双曲或部分抛物型系统中行波解的非线性稳定性分析中起着关键作用——既可用于获得高频预解估计，也可用于闭合非线性迭代过程，而在该过程中现有的线性化稳定性估计通常会出现正则性损失。本文证明了对于规模 $n \leq 6$ 的系统，存在一个类李雅普诺夫（Lyapunov-type）定理：只要满足严格的高频谱稳定性条件，则此类能量估计必然存在；而对于维度为7及以上的情形，高频谱稳定性与本文所发展的这类能量估计的存在性之间可能存在一般性的差距。其中的一个关键工具是一个依赖于维度的线性代数引理，其形式类似于常微分方程中的李雅普诺夫引理，据我们所知，该引理在以往文献中尚未出现。

3. The centered maximal operator removes the non-concave Cantor part from

the gradient

中心极大算子消除了梯度中的非凹Cantor部分

作者: Panu Lahti, Julian Weigt

类别: math.CA, math.AP, 42B25, 26B30

发布日期: 2025-10-02

ArXiv ID: 2510.01936v1

链接: http://arxiv.org/abs/2510.01936v1

摘要翻译

中文标题: 中心极大算子消除了梯度中的非凹Cantor部分

摘要翻译: 我们研究定义在 $\mathbb{R}^d$（其中 $d \in \mathbb{N}$）上的有界变差函数 $f$ 的中心Hardy–Littlewood极大函数 $M f$ 的正则性。特别地，我们证明：在 $|D^c f|$-几乎处处使得 $f$ 具有非凹爆破的点 $x$ 处，均有 $M f(x) > f^*(x)$。进一步，由此我们推导出：若 $f$ 的变差测度不含跳跃部分，且其Cantor部分具有非凹爆破，则 $M f$ 的有界变差正则性可提升至Sobolev正则性。

4. On sharp Strichartz estimate for hyperbolic Schrödinger equation on

$\mathbb{T}^3$

关于三维环面上双曲薛定谔方程的Sharp Strichartz估计

作者: Baoping Liu, Xu Zheng

类别: math.AP, math.CA, 35Q55

发布日期: 2025-10-02

ArXiv ID: 2510.01886v1

链接: http://arxiv.org/abs/2510.01886v1

摘要翻译

中文标题: 关于三维环面上双曲薛定谔方程的Sharp Strichartz估计

摘要翻译: 本文通过结合几何测度论中的相交几何方法，证明了双曲薛定谔方程在 $\mathbb{T}^3$ 上的Sharp Strichartz估计。作为应用，我们得到了非线性双曲薛定谔方程的最优局部适定性结果。

5. Strichartz and dispersive estimates for quantum bouncing ball model:

exponential sums and Van der Corput methods in 1d semi-classical Schrödinger equations

量子弹跳球模型的Strichartz估计与色散估计：一维半经典薛定谔方程中的指数和与Van der Corput方法

作者: Oana Ivanovici

类别: math.AP, math.NT

发布日期: 2025-10-02

ArXiv ID: 2510.01779v1

链接: http://arxiv.org/abs/2510.01779v1

摘要翻译

中文标题: 量子弹跳球模型的Strichartz估计与色散估计：一维半经典薛定谔方程中的指数和与Van der Corput方法

摘要翻译: 本文研究定义在半直线上、具有线性势和Dirichlet边界条件的一维半经典薛定谔方程。我们的主要目标是为该模型建立改进的色散估计与Strichartz估计，以刻画解的空间-时间行为。我们利用Van der Corput型导数判别法，证明了更为精细的Strichartz估计，优于此前已知的结果——以往的Strichartz估计存在1/4阶的损失。此外，在假设某些指数和具有精确上界的条件下，我们的结果表明，这些损失可进一步降至对任意$\epsilon>0$均成立的$1/6 + \epsilon$阶，而这将是最佳可能的结果。我们还推测，类似的Strichartz估计应在高维情形下的Friedlander模型区域中同样成立。

6. On dispersive decay for the generalized Korteweg–de Vries equation

关于广义Korteweg–de Vries方程的色散衰减性研究

作者: Matthew Kowalski, Minjie Shan

类别: math.AP, 35Q53

发布日期: 2025-10-02

ArXiv ID: 2510.01728v1

链接: http://arxiv.org/abs/2510.01728v1

摘要翻译

中文标题: 关于广义Korteweg–de Vries方程的色散衰减性研究

摘要翻译: 我们证明了广义Korteweg–de Vries（gKdV）方程解的时间逐点色散估计。特别地，对于质量临界模型的解，我们仅假设初值数据属于 $\dot{H}^{\frac{1}{4}} \cap \dot{H}^{-\frac{1}{12}}$，并证明了解在 $L^\infty$ 范数下的衰减速率为 $|t|^{-\frac{1}{3}}$。为实现这一目标，我们建立了gKdV方程解的负正则性的保持性质，并将Lorentz–Strichartz估计推广至混合范数情形。

不相关论文（基本信息）

1. Uniqueness in the Plateau problem for calibrated currents

校准流的Plateau问题解的唯一性

作者: Bryan Dimler, Chen-Kuan Lee

类别: math.DG, math.AP

发布日期: 2025-10-02

ArXiv ID: 2510.02299v1

链接: http://arxiv.org/abs/2510.02299v1

摘要: We show that every compactly supported calibrated integral current with connected $C^{3,\alpha}$ boundary is the unique solution to the oriented Plateau problem for its boundary data. This is proved as a consequence of the boundary regularity theory for area-minimizing currents and classical unique continuation principles adapted to the minimal surface system.

2. Optimal Lieb-Thirring type inequalities for Schrödinger and Jacobi

operators with complex potentials

关于具有复势的薛定谔算子与雅可比算子的最优Lieb-Thirring型不等式

作者: Sabine Bögli, Sukrid Petpradittha

类别: math.SP, math-ph, math.AP, math.MP, 47B36, 34L40, 47A10, 47A75

发布日期: 2025-10-02

ArXiv ID: 2510.02288v1

链接: http://arxiv.org/abs/2510.02288v1

摘要: We prove optimal Lieb-Thirring type inequalities for Schr"odinger and Jacobi operators with complex potentials. Our results bound eigenvalue power sums (Riesz means) by the $L^p$ norm of the potential, where in contrast to the self-adjoint case, each term needs to be weighted by a function of the ratio of the distance of the eigenvalue to the essential spectrum and the distance to the endpoint(s) thereof. Our Lieb-Thirring type bounds only hold for integrable weight functions. To prove optimality, we establish divergence estimates for non-integrable weight functions. The divergence rates exhibit a logarithmic or even polynomial gain compared to semiclassical methods (Weyl asymptotics) for real potentials.

3. Transfer of Stability from the Classical to the Fractional Anisotropic

Calderón Problem

从经典各向异性Calderón问题到分数阶各向异性Calderón问题的稳定性传递

作者: Hendrik Baers, Angkana Rüland

类别: math.AP

发布日期: 2025-10-02

ArXiv ID: 2510.02242v1

链接: http://arxiv.org/abs/2510.02242v1

摘要: We discuss two spectral fractional anisotropic Calder'on problems with source-to-solution measurements and their quantitative relation to the classical Calder'on problem. Firstly, we consider the anistropic fractional Calder'on problem from [FGKU25]. In this setting, we quantify the relation between the local and nonlocal Calder'on problems which had been deduced in [R25] and provide an associated stability estimate. As a consequence, any stability result which holds on the level of the local problem with source-to-solution data has a direct nonlocal analogue (up to a logarithmic loss). Secondly, we introduce and discuss the fractional Calder'on problem with source-to-solution measurements for the spectral fractional Dirichlet Laplacian on open, bounded, connected, Lipschitz sets on $\mathbb{R}^n$. Also in this context, we provide a qualitative and quantitative transfer of uniqueness from the local to the nonlocal setting. As a consequence, we infer the first stability results for the principal part for a fractional Calder'on type problem for which no reduction of Liouville type is known. Our arguments rely on quantitative unique continuation arguments. As a result of independent interest, we also prove a quantitative relation between source-to-solution and Dirichlet-to-Neumann measurements for the classical Calder'on problem.

4. On Lieb-Thirring inequalities for multidimensional Schrödinger

operators with complex potentials

关于具有复势的多维薛定谔算子的Lieb-Thirring不等式研究

作者: Sabine Bögli, Sukrid Petpradittha, František Štampach

类别: math.SP, math-ph, math.AP, math.MP, 35P15, 47A10, 47A75, 81Q12

发布日期: 2025-10-02

ArXiv ID: 2510.02192v1

链接: http://arxiv.org/abs/2510.02192v1

摘要: We solve the open problem by Demuth, Hansmann, and Katriel announced in [Integr. Equ. Oper. Theory 75 (2013), 1-5] by a counter-example construction. The problem concerns a possible generalisation of the Lieb-Thirring inequality for Schr"odinger operators in to the case of complex-valued potentials. A counter-example has already been found for the one-dimensional case by the first and third authors in [J. Spectr. Theory 11 (2021), 1391-1413]. Here we generalise the counter-example to higher dimensions.

5. Sharp remainder terms and stability of weighted Hardy-Poincaré and

Heisenberg-Pauli-Weyl inequalities related to the Baouendi-Grushin operator

与Baouendi-Grushin算子相关的加权Hardy-Poincaré和Heisenberg-Pauli-Weyl不等式的精确余项及其稳定性

作者: Yerkin Shaimerdenov, Nurgissa Yessirkegenov, Amir Zhangirbayev

类别: math.AP, 26D10, 35J70

发布日期: 2025-08-22

ArXiv ID: 2508.16380v2

链接: http://arxiv.org/abs/2508.16380v2

摘要: In this paper, we obtain sharp remainder terms for the Hardy-Poincar'e inequalities with general non-radial weights in the setting of Baouendi-Grushin vector fields (see Theorem 2.5). It is worth emphasizing that all of our results are new both in the Baouendi-Grushin and standard Euclidean settings. The method employed allows us to not only unify, but also improve the results of Kombe and Yener [KY18] for any $1<p<\infty$ while holding true for complex-valued functions and providing explicit constants (Corollary 2.7). As a result, we are able to obtain sharp remainder terms to many known weighted Hardy-type inequalities (see Section 3.1). Aside from weighted Hardy-type inequalities, we also recover a sharp remainder formula for the $L^{p}$-Poincar'e inequality (Corollary 3.5). In the special case of radial weights, we are naturally able to introduce the notion of Baouendi-Grushin $p$-Bessel pairs (see Definition 2.9). Finally, we apply the technique to establish the sharp remainder term of the Heisenberg-Pauli-Weyl inequality in $L^{p}$ for $1<p<\infty$ (Corollary 3.13), which includes the sharp constant. This makes it possible to obtain the $L^{p}$-analogue for $2\leq p < n$ (Theorem 3.17) of a stability result by Cazacu, Flynn, Lam and Lu [CFLL24].

6. Equivalence between solvability of the Dirichlet and Regularity problem

under an $L^1$ Carleson condition on $\partial_t A$

在 $\partial_t A$ 满足 $L^1$ Carleson 条件下 Dirichlet 问题与正则性问题可解性的等价性

作者: Martin Ulmer

类别: math.AP, 35J25, 35J15

发布日期: 2025-09-12

ArXiv ID: 2509.10328v2

链接: http://arxiv.org/abs/2509.10328v2

摘要: We study an elliptic operator $L:=\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that solvability of the Regularity problem in $\dot{W}^{1,p}$ implies solvability of the adjoint Dirichlet problem in $L^{p'}$. Previously, Shen (2007) established a partial reverse result. In our work, we show that if we assume an $L^1$-Carleson condition on only $|\partial_t A|$ the full reverse direction holds. As a result, we obtain equivalence between solvability of the Dirichlet problem $(D)^*_{p'}$ and the Regularity problem $(R)_p$ under this condition. As a further consequence, we can extend the class of operators for which the $L^p$ Regularity problem is solvable by operators satisfying the mixed $L^1-L^\infty$ condition. Additionally in the case of the upper half plane, this class includes operators satisfying this $L^1$-Carleson condition on $|\partial_t A|$.

7. Refined asymptotics for the Cauchy problem for the fast $p$-Laplace

evolution equation

快速 $p$-Laplace 演化方程柯西问题的精细渐近分析

作者: Matteo Bonforte, Iwona Chlebicka, Nikita Simonov

类别: math.AP

发布日期: 2024-05-08

ArXiv ID: 2405.05405v3

链接: http://arxiv.org/abs/2405.05405v3

摘要: Our focus is on the fast diffusion equation driven by the $p$-Laplacian operator, that is $\partial_t u=\Delta_p u$ with $1<p<2$, posed in the whole space $\mathbb{R}^N$, $N\geq 2$. The nonnegative solutions are expected to converge in time toward a stationary profile. While such convergence had been previously established for $p$ close to $2$, no quantitative rates were known, and the asymptotic behaviour remained poorly understood across the full fast diffusion range. In fact, the long time behaviour of solutions to the $p$-Laplace Cauchy problem drastically change in different subranges of the $p$. Some of them are analysed here for the first time. In this work, we provide the convergence rates for nonnegative, integrable solutions in the so-called good fast diffusion range, $p_c=\tfrac{2N}{N+1} <p<2$, where mass is conserved. We prove that solutions converge to a self-similar profile with matching mass, with explicit rates measured in relative error. Our constructive proof is based on a new entropy method that remains effective even when the entropy is not displacement convex – where optimal transport techniques fail. In the very fast diffusion range $1<p<p_c$, we give the first asymptotic analysis near the extinction time. We uncover new critical exponents – especially in high dimensions – that give rise to markedly different qualitative behaviour depending on the value of $p$. We also establish convergence rates for the gradients of radial solutions in the good fast diffusion range, again measured in relative error. Finally, we analyze the structural properties required for the entropy method to apply, thereby opening a broader investigation into the basin of attraction of Barenblatt-type profiles, particularly in the singular case of $p$ close to $1$.

8. Low regularity Sobolev well-posedness for Vlasov–Poisson

Vlasov–Poisson方程的低正则性Sobolev适定性

作者: In-Jee Jeong, Sangwook Tae

类别: math.AP

发布日期: 2025-10-02

ArXiv ID: 2510.02112v1

链接: http://arxiv.org/abs/2510.02112v1

摘要: We consider the Vlasov–Poisson equation on $\mathbb{R}^n \times \mathbb{R}^n$ with $n \ge 3$. We prove local well-posedness in $H^{s}(\mathbb{R}^n \times \mathbb{R}^n)$ with $s> n/2-1/4$, for initial distribution $f_{0} \in H^{s}(\mathbb{R}^n \times \mathbb{R}^n)$ having compact support in $v$. In particular, data not belonging to $L^p(\mathbb{R}^n \times \mathbb{R}^n)$ for large $p$ are allowed.

9. Numerical Approximation of the Critical Value of Eikonal Hamilton-Jacobi

Equations on Networks

网络上Eikonal Hamilton-Jacobi方程临界值的数值逼近

作者: Valentina Coscetti, Marco Pozza

类别: math.NA, cs.NA, math.AP, 35R02, 65M15, 49L25, 35B40

发布日期: 2025-02-28

ArXiv ID: 2502.20993v2

链接: http://arxiv.org/abs/2502.20993v2

摘要: The critical value of an eikonal equation is the unique value of a parameter for which the equation admits solutions and is deeply related to the effective Hamiltonian of a corresponding homogenization problem. We study approximation strategies for the critical value of eikonal equations posed on networks. They are based on the large time behavior of corresponding time-dependent Hamilton-Jacobi equations. We provide error estimates and some numerical tests, showing the performance and the convergence properties of the proposed algorithms.

10. Perturbation theory for the parabolic Regularity and Neumann problem

抛物型正则性与诺伊曼问题的摄动理论

作者: Martin Ulmer

类别: math.AP, 35K10, 35K20

发布日期: 2024-08-22

ArXiv ID: 2408.12529v2

链接: http://arxiv.org/abs/2408.12529v2

摘要: We show small and large Carleson perturbation results for the parabolic Regularity boundary value problem with boundary data in $\dot{L}_{1,1/2}^p$ and small Carelson perturbation results for the Neumann problem with boundary data in $L^p$. The operator we consider is $L:=\partial_t -\mathrm{div}(A\nabla\cdot)$ and the domains are parabolic cylinders $\Omega=\mathcal{O}\times\mathbb{R}$, where $\mathcal{O}$ is a Lipschitz domain.

11. Geometry of wave damping on the torus

环面上波阻尼的几何性质

作者: Kiril Datchev, Perry Kleinhenz, Antoine Prouff

类别: math.AP, math-ph, math.MP

发布日期: 2025-09-05

ArXiv ID: 2509.05239v2

链接: http://arxiv.org/abs/2509.05239v2

摘要: Energy decay rates of damped waves on the torus depend on the behavior of the damping near the undamped region and on the geometry of the damped set. In this paper we refine these geometric considerations, by introducing the concept of order of a glancing undamped point, and estimating decay rates in terms of this order. The proof is based on generalizing an averaging argument due to Sun. We also show that damping sets which attain these improvements are generic among polygons and smooth curves.

12. Perturbations of Minkowski spacetime with regular conformal

compactification

具有正则共形紧致化的闵可夫斯基时空的扰动

作者: Andrea Nützi

类别: gr-qc, math.AP

发布日期: 2025-10-02

ArXiv ID: 2510.01964v1

链接: http://arxiv.org/abs/2510.01964v1

摘要: We construct perturbations of Minkowski spacetime in general relativity, when given initial data that decays inverse polynomially to initial data of a Kerr spacetime towards spacelike infinity. We show that the perturbations admit a regular conformal compactification at null and timelike infinity, where the degree of regularity increases linearly with the rate of decay of the initial data to Kerr initial data. In particular, the compactification is smooth if the initial data decays rapidly to Kerr initial data. This generalizes results of Friedrich, who constructed spacetimes with a smooth conformal compactification in the case when the initial data is identical to Kerr initial data on the complement of a compact set. Our results rely on a novel formulation of the Einstein equations about Minkowski spacetime introduced by the author, that allows one to formulate the dynamic problem as a quasilinear, symmetric hyperbolic PDE that is regular at null infinity and with null infinity being at a fixed locus. It is not regular at spacelike infinity, due to the asymptotics of Kerr. Thus the main technical task is the construction of solutions near spacelike infinity, using tailored energy estimates. To accomplish this, we organize the equations according to homogeneity with respect to scaling about spacelike infinity, which identifies terms that are leading, respectively lower order, near spacelike infinity, with contributions from Kerr being lower order.

13. Mixed Christoffel-Minkowski problems for bodies of revolution

旋转体的混合Christoffel-Minkowski问题

作者: Leo Brauner, Georg C. Hofstätter, Oscar Ortega-Moreno

类别: math.MG, math.AP, math.DG, 52A39, 52A20, 52B45

发布日期: 2025-08-13

ArXiv ID: 2508.09794v3

链接: http://arxiv.org/abs/2508.09794v3

摘要: The mixed Christoffel-Minkowski problem asks for necessary and sufficient conditions for a Borel measure on the Euclidean unit sphere to be the mixed area measure of some convex bodies, one of which, appearing multiple times, is free and the rest are fixed. In the case where all bodies involved are symmetric around a common axis, we provide a complete solution to this problem, without assuming any regularity. In particular, we refine Firey’s classification of area measures of figures of revolution. In our argument, we introduce an easy way to transform mixed area measures and mixed volumes involving axially symmetric bodies, and we significantly improve Firey’s estimate on the local behavior of area measures. As a secondary result, we obtain a family of Hadwiger type theorems for convex valuations that are invariant under rotations around an axis.

14. Subwavelength resonances in two-dimensional elastic media with high

contrast

高对比度二维弹性介质中的亚波长共振

作者: Yuanchun Ren, Yixian Gao

类别: math.AP

发布日期: 2025-10-02

ArXiv ID: 2510.01911v1

链接: http://arxiv.org/abs/2510.01911v1

摘要: This paper employs layer potential techniques to investigate wave scattering in two-dimensional elastic media exhibiting high contrasts in both Lam'{e} parameters and density. Our contributions are fourfold. First, we construct an invertible operator based on the kernel spaces of boundary integral operators, which enables the characterization of resonant frequencies through an orthogonality condition. Second, we use asymptotic analysis to derive the equation governing the leading-order terms of these resonant frequencies. Third, we analyze the scattered field in the interior domain for incident frequencies across different regimes and characterize the longitudinal and transverse far-field patterns in the exterior domain. Finally, we examine the subwavelength bandgap in the phononic crystal with a dilute structure.

15. Reconstruction in the Calderón problem on a fixed partition from

finite and partial boundary data

基于有限且部分边界数据的固定区域划分下Calderón问题的重构

作者: Henrik Garde

类别: math.AP, cs.NA, math.NA, 35R30, 35Q60, 35R05, 47H05

发布日期: 2025-07-25

ArXiv ID: 2507.19410v2

链接: http://arxiv.org/abs/2507.19410v2

摘要: This short note modifies a reconstruction method by the author (Comm.~PDE, 45(9):1118–1133, 2020), for reconstructing piecewise constant conductivities in the Calder'on problem (electrical impedance tomography). In the former paper, a layering assumption and the local Neumann-to-Dirichlet map were needed since the piecewise constant partition also was assumed unknown. Here I show how to modify the method in case the partition is known, for general piecewise constant conductivities and only a finite number of partial boundary measurements. Moreover, no lower/upper bounds on the unknown conductivity are needed.

16. A note on the recovery sequence in the double gradient model for phase

transitions

关于相变双梯度模型中恢复序列的一点注记

作者: Jakob Deutsch

类别: math.AP, Primary: 49J45, Secondary: 35Q74, 49Q20, 74N99

发布日期: 2025-10-02

ArXiv ID: 2510.01893v1

链接: http://arxiv.org/abs/2510.01893v1

摘要: We investigate the $\limsup$ inequality in the double gradient model for phase transitions governed by a Modica–Mortola functional with a double-well potential in two dimensions. Specifically, we consider energy functionals of the form

for maps $ u \in H^2(\Omega; \mathbb{R}^2) $, where $ W $ vanishes only at two wells. Assuming a bound on the optimal profile constant – namely the cell problem on the unit cube – in terms of the geodesic distance between the two wells, we characterise the limiting interfacial energy via periodic recovery sequences as $\varepsilon \to 0^+$.

17. Monotonicity and Liouville-type theorems for semilinear elliptic

problems in the half space

半空间中半线性椭圆问题的单调性与Liouville型定理

作者: Berardino Sciunzi, Domenico Vuono

类别: math.AP, 35B06, 35B51, 35J61

发布日期: 2025-10-02

ArXiv ID: 2510.01865v1

链接: http://arxiv.org/abs/2510.01865v1

摘要: We consider classical solutions to $-\Delta u = f(u)$ in half-spaces, under homogeneous Dirichlet boundary conditions. We prove that any positive solution is strictly monotone increasing in the direction orthogonal to the boundary, provided that it is directionally bounded on finite strips. As a corollary, we deduce a new Liouville-type theorem for the Lane-Emden equation.

18. Efficient manifold evolution algorithm using adaptive B-Spline

interpolation

基于自适应B样条插值的高效流形演化算法

作者: Muhammad Ammad, Leevan Ling

类别: math.NA, cs.NA, math.AP, math.DG

发布日期: 2025-10-02

ArXiv ID: 2510.01790v1

链接: http://arxiv.org/abs/2510.01790v1

摘要: This paper explores an efficient Lagrangian approach for evolving point cloud data on smooth manifolds. In this preliminary study, we focus on analyzing plane curves, and our ultimate goal is to provide an alternative to the conventional radial basis function (RBF) approach for manifolds in higher dimensions. In particular, we use the B-Spline as the basis function for all local interpolations. Just like RBF and other smooth basis functions, B-Splines enable the approximation of geometric features such as normal vectors and curvature. Once properly set up, the advantage of using B-Splines is that their coefficients carry geometric meanings. This allows the coefficients to be manipulated like points, facilitates rapid updates of the interpolant, and eliminates the need for frequent re-interpolation. Consequently, the removal and insertion of point cloud data become seamless processes, particularly advantageous in regions experiencing significant fluctuations in point density. The numerical results demonstrate the convergence of geometric quantities and the effectiveness of our approach. Finally, we show simulations of curvature flows whose speeds depend on the solutions of coupled reaction–diffusion systems for pattern formation.

19. Notes on Schauder estimates by scaling for elliptic PDEs in divergence

form

关于散度形式椭圆型偏微分方程Schauder估计的尺度分析笔记

作者: Stefano Vita

类别: math.AP, 35B65, 35B44, 35B45, 35B53, 35B30, 35B08

发布日期: 2025-10-02

ArXiv ID: 2510.01765v1

链接: http://arxiv.org/abs/2510.01765v1

摘要: These are the notes of a part of the PhD course Regularity for free boundary problems and for elliptic PDEs, held in Pavia in the spring of 2025. The aim is to provide a comprehensive and self-contained treatment of classical interior and local Schauder estimates for second-order linear elliptic PDEs in divergence form via scaling in the spirit of Simon’s work. The main techniques presented here are geometric in nature and were primarily developed in the study of geometric problems such as minimal surfaces. The adopted approach relies on compactness and blow-up arguments, combined with rigidity results (Liouville theorems), and shares many features with the one used in the study of free boundary problems, which was the main topic of the other part of the PhD course.

20. Nonlinear Forward-Backward Problems

非线性前向-后向问题

作者: Anne-Laure Dalibard, Frédéric Marbach, Jean Rax

类别: math.AP

发布日期: 2025-10-02

ArXiv ID: 2510.01732v1

链接: http://arxiv.org/abs/2510.01732v1

摘要: We prove the existence and uniqueness of strong solutions to the equation $u u_x - u_{yy} = f$ in the vicinity of the linear shear flow, subject to perturbations of the source term and lateral boundary conditions. Since the solutions we consider have opposite signs in the lower and upper half of the domain, this is a quasilinear forward-backward parabolic problem, which changes type across a critical curved line within the domain. In particular, lateral boundary conditions can be imposed only where the characteristics are inwards. There are several difficulties associated with this problem. First, the forward-backward geometry depends on the solution itself. This requires to be quite careful with the approximation procedure used to construct solutions. Second, and more importantly, the linearized equations solved at each step of the iterative scheme admit a finite number of singular solutions, of which we provide an explicit construction. This is similar to well-known phenomena in elliptic problems in nonsmooth domains. Hence, the solutions to the equation are regular if and only if the source terms satisfy a finite number of orthogonality conditions. A key difficulty of this work is to cope with these orthogonality conditions during the nonlinear fixed-point scheme. In particular, we are led to prove their stability with respect to the underlying base flow. To tackle this deceivingly simple problem, we develop a methodology which we believe to be both quite natural and adaptable to other situations in which one wishes to prove the existence of regular solutions to a nonlinear problem for suitable data despite the existence of singular solutions at the linear level. This paper is a shorter version of [3].

21. The weighted isoperimetric inequality and Sobolev inequality outside

convex sets

凸集外部的加权等周不等式与索博列夫不等式

作者: Lu Chen, Jiali Lan

类别: math.AP

发布日期: 2025-10-02

ArXiv ID: 2510.01647v1

链接: http://arxiv.org/abs/2510.01647v1

摘要: In this paper, we establish a weighted capillary isoperimetric inequality outside convex sets using the $\lambda_w$-ABP method. The weight function $w$ is assumed to be positive, even, and homogeneous of degree $\alpha$, such that $w^{1/\alpha}$ is concave on $\R^n$. Based on the weighted isoperimetric inequality, we develop a technique of capillary Schwarz symmetrization outside convex sets, and establish a weighted P'{o}lya-Szeg"{o} principle and a sharp weighted capillary Sobolev inequality outside convex domain. Our result can be seen as an extension of the weighted Sobolev inequality in the half-space established by Ciraolo-Figalli-Roncoroni in \cite{CFR}.