Stochastic homogenization and geometric singularities : a study on corners

在有尖角的领域中BV迹不等式最优常数的研究

Optimal constant for the trace inequality in $BV$ for domains with corners

研究确定了在有特定奇点的领域中，BV函数迹不等式最优常数的显式值。

Stochastic Homogenization for Reaction-Diffusion Equations

In the present paper we study stochastic homogenization for reaction-diffusion equations with stationary ergodic reactions. We first show that under suitable hypotheses, initially localized solutions to the PDE asymptotically become approximate characteristic functions of a ballistically expanding Wulff shape. The next crucial component is the proper definition of relevant front speeds and subsequent establishment of their existence. We achieve the latter by finding a new relation between the front speeds and the Wulff shape, provided the Wulff shape does not have corners. Once front speeds ar…

一维空间中存在角点奇异性时非线性演化方程在 t=0 附近的数值分辨率

Numerical Resolution near t = 0 of Nonlinear Evolution Equations in the Presence of Corner Singularities in Space Dimension 1

针对初边值数据不兼容导致时空角点奇异性影响数值精度的问题，提出了两种修复方法以恢复数值格式的精度，并应用于一维粘性Burgers方程和非线性反应扩散方程。

强奇异振荡势的散射、均匀化和界面效应

Scattering, homogenization and interface effects for oscillatory potentials with strong singularities

研究了具有快速周期振荡和强局域奇点的衰减势场的一维散射。特别是，我们考虑薛定谔方程\[ H_εψ:= (-\partial_x^2 + V_0(x) + q(x,x/ε)) ψ= k^2 ψ, \]的$k\in\RR$和$ε<< 1$。这里，$q(.,y+1)=q(.,y)$的均值为零，当$|x|$趋于无穷大时，$|V_0(x)+q(x,.)|$趋于零。$H_ε$的扭曲平面波是以下形式的解：$e^{\pm ikx}+u^s_\pm(x;k)$，当$|x|$走向无穷大时，$u^s_\pm$向外输出。我们推导出它们的$ε$小的渐近行为，从该渐近行为的散射量，如透射系数，$t^ε(k)$，遵循。令$t_0^{hom}(k)$表示与平均电势$V_0$相关联的均匀化透射系数。如果势是光滑的，那么经典的均匀化理论给出了例如畸变平面波、透射系数和反射系数的渐近展开式。我们的理论承认，$V_0$的奇异性或$q_ε$的不连续性是一个解必须满足界面条件（连续性或跳跃条件）的“界面”。为了满足这些条件，有必要引入界面校正器，这是高度振荡的$ε$。我们的主要结果的一个后果是，$t^ε(k)-t_0^{hom}(k)$，在均匀化的透射系数的误差是（i）$O(ε^2)$，如果$q_ε$是连续的和（ii）$O(ε)$，如果$q_ε$有不连续。此外，在不连续的情况下，校正器在$ε$中是高度振荡的，使得一阶校正器不…

关于角域弹性散射行为的研究

On elastic scattering behavior of corner domains

本文研究弹性波在角域中的散射，证明具有角或边缘奇异性的可穿透障碍物会散射所有满足条件的入射波，将标量结果推广到方程组，确立“角总是散射”原理。

具有奇性的区域中非局部极小曲面的边界连续性及Borthagaray、Li和Nochetto提出的一个问题

Boundary continuity of nonlocal minimal surfaces in domains with singularities and a problem posed by Borthagaray, Li, and Nochetto

本文证实了Borthagaray、Li和Nochetto关于非局部极小曲面粘附量与边界非局部平均曲率关系的猜想，证明了其在凸角点连续、凹角点不连续，并推广了边界连续性与奇性的判定条件。

带特征角的水族箱内波

Internal waves in aquariums with characteristic corners

本研究拓展了此前针对光滑边界区域的内波研究，对带角的有效二维水族箱内波长时间传播的奇异剖面做了微局部描述，发现除对应经典动力学吸引子的奇异性外，角点还会向外传播更温和的奇异性。

The distance function from the boundary of a domain with corners

We study the regularity of the distance function to the boundary of a domain in $\mathbb{R}^2$, with respect to some asymmetric norms. We allow the boundary of the domain to have corners. We obtain an explicit formula for the second derivative of these distance functions. Furthermore, we study a generalized notion of the ridge of a domain, which is the set of singularities of a distance function to the boundary of the domain. We completely characterize the ridge by using a generalized notion of curvature.

Optimal estimates of the field enhancement in presence of a bow-tie structure of perfectly conducting inclusions in two dimensions

This paper deals with the field enhancement, that is, the gradient blow-up, due to presence of a bow-tie structure of perfectly conducting inclusions in two dimensions. The bow-tie structure consists of two disjoint bounded domains which have corners with possibly different aperture angles. The domains are parts of cones near the vertices, and they are nearly touching to each other. We characterize the field enhancement using explicit functions and, as consequences, derive optimal estimates of the gradient in terms of the distance between two inclusions and aperture angles of the corners. The …

有限时间出现奇点的正则平面曲线流的数值研究

Numerical study of a flow of regular planar curves that develop singularities at finite time

本文数值研究了Goldstein-Petrich几何流，该流在Euler方程下描述平面涡斑演化，并再现了Perelman-Vega证明的有限时间角状奇点形成。

differential forms上wave equation的角衍射

Diffraction at corners for the wave equation on differential forms

在本文中，我们证明了奇性传播的波动方程的微分形式与自然（即相对或绝对）边界条件的Lorentzian流形上的角，其中特别包括麦克斯韦方程的制定。这些结果类似于作者以前对标量波动方程和Dirichlet或Neumann边界条件系统上的波动方程所得到的结果。因此，主要的新颖性是自然边界条件的存在下，有效地使问题的非标量，甚至"领导秩序“，在角落的余维至少2。

Numerical solution for the stress near a hole with corners in an infinite plate under biaxial loading

We consider the elastic stress near a hole with corners in an infinite plate under biaxial stress. The elasticity problem is formulated using complex Goursat functions, resulting in a set of singular integro-differential equations on the boundary. The resulting boundary integral equations are solved numerically using a Chebyshev collocation method which is augmented by a fractional power term, derived by asymptotic analysis of the corner region, to resolve stress singularities at corners of the hole. We apply our numerical method to the test case of the hole formed by two partially-overlapping…

凸多边形上Stokes问题的非常弱解

Very weak solutions of the Stokes problem in a convex polygon

本文从研究空腔流中的角奇异性出发，证明了当$Ω$是光滑区域或凸多边形凸区域时，$Ω,$中Stokes问题解的存在唯一性.我们还建立了迹定理，证明了在$L^2(\partialΩ)^2.$中$u$的迹可以是任意的，并将结果推广到线性发展Stokes问题.