小角度极限下带拐角的平面波导

严格凸角散射

Strictly convex corners scatter

我们证明了角小于$π$的平面上的势不存在非散射能。这扩展了Blasten，Päivärinta和Sylvester考虑矩形角的早期结果。在三维空间中，我们证明了一个类似的结果，任何潜在的圆锥角的开口角是外面的一个可数子集的$(0,π)$。

Does elastic stress modify the equilibrium corner angle?

We consider the influence of elasticity and anisotropic surface energy on the energy-minimizing shape of a two-dimensional void under biaxial loading. In particular, we consider void shapes with corners for which the strain energy density is singular at the corner. The elasticity problem is formulated as a boundary integral equation using complex potentials. By incorporating the asymptotic behavior of the singular elastic fields at corners of the void, we develop a numerical spectral method for determining the stress for a class of arbitrary void shapes and corner angles. We minimize the total…

带角点区域上二维Euler流的涡度梯度增长

The growth of the vorticity gradient for the two-dimensional Euler flows on domains with corners

本文研究了带角点的非光滑区域上的二维Euler方程。结果表明，当角点$θ$严格小于$π/2$时，角点涡量的Lipschitz估计最多为单指数增长性，且上界是尖锐的. %，接近停滞点。对于角$π/2 < θ<2π$，$θ\neq π$较大的角，我们构造了一个瞬时失去连续性的涡量的例子。对于$θ\le π/2$情况，涡度在区域内保持连续。因此，我们确定的阈值的角度的涡度保持连续性。对于边界角$θ=π/2$，还表明涡度Lipschitz常数的增长率可以是双指数的，这与Kiselev-Sverak的结果（Annals of Math.，2014年）。

Uniqueness of the 2D Euler equation on a corner domain with non-constant vorticity around the corner

We consider the 2D incompressible Euler equation on a corner domain $Ω$ with angle $νπ$ with $\frac{1}{2}<ν<1$. We prove that if the initial vorticity $ω_0 \in L^{1}(Ω)\cap L^{\infty}(Ω)$ and if $ω_0$ is non-negative and supported on one side of the angle bisector of the domain, then the weak solutions are unique. This is the first result which proves uniqueness when the velocity is far from Lipschitz and the initial vorticity is nontrivial around the boundary.

V形波导中的分数阶拉普拉斯算子

Fractional Laplacian in V-shaped waveguide

本文研究了V形波导中受限分数阶Dirichlet拉普拉斯算子的谱性质，证明了任意连接角下离散谱的存在性及其对角度的单调依赖。

“角点处有界毛细曲面径向极限的存在性和行为”一文的推广

A Generalization of "Existence and Behavior of the Radial Limits of a Bounded Capillary Surface at a Corner"

存在原理（即定理1）的“在角处有界毛细表面的径向极限的存在性和行为”（Pacific J.Math.Vol.176，No.1（1996），165-194）延伸到接触角$γ$不远离$0$和$π$的情况（并且取决于在具有在${\cal O}=(0,0)$处的凸角的有界域$Ω\in {\bf R}^{2}$中的位置）。将“侧扇”尺寸的下界（即上述文献中的定理2）推广到凸角和非凸角的这种接触角的情况。

Robin特征值问题中的非共振角

Non-resonant angles in a Robin eigenvalue problem

该研究确定了Robin特征值问题中非共振角的完整范围，解决了Dauge与Lafranche提出的经数值支持的猜想，明确了临界角的表征方式。

Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons

We obtain asymptotic formulae for the Steklov eigenvalues and eigenfunctions of curvilinear polygons in terms of their side lengths and angles. These formulae are quite precise: the errors tend to zero as the spectral parameter tends to infinity. The Steklov problem on planar domains with corners is closely linked to the classical sloshing and sloping beach problems in hydrodynamics; as we show it is also related to quantum graphs. Somewhat surprisingly, the arithmetic properties of the angles of a curvilinear polygon have a significant effect on the boundary behaviour of the Steklov eigenfunc…

某些带两个突出角物体的亚音速无旋无粘绕流

Subsonic irrotational inviscid flow around certain bodies with two protruding corners

我们证明了不存在的非平凡的均匀亚音速无粘无旋绕流的几类固体有两个突出的角落，特别是垂直和倾斜的平板，水平板是唯一的情况下，存在的解决方案。这一方面填补了经典的机构与一个突出的角落和最近的工作机构与三个或更多的突出角落的结果之间的差距。因此，即使在零粘性和滑移边界条件下，固体也能产生涡量，在这个意义上说，至少有一个旋转解，但没有无旋解。我们的观察补充了普遍接受的解释涡产生的基础上普朗特的粘性边界层理论。

非光滑域上弹性薄板的多项式稳定性

Polynomial Stability of an Elastic Thin Plate on Non-Smooth Domain

研究非光滑域上弹性板的动力边界条件下的多项式稳定化，通过变分框架处理边界奇点引起的解正则性损失，证明小角域保持多项式衰减率，大角域通过反馈控制恢复稳定性。

Eigenvalue Asymptotics in a Twisted Waveguide

We consider a twisted quantum wave guide, and are interested in the spectral analysis of the associated Dirichlet Laplacian H. We show that if the derivative of rotation angle decays slowly enough at infinity, then there is an infinite sequence of discrete eigenvalues lying below the infimum of the essential spectrum of H, and obtain the main asymptotic term of this sequence.

Weighted elliptic estimates for a mixed boundary system related to the Dirichlet-Neumann operator on a corner domain

Based on the $H^2$ existence of the solution, we investigate weighted estimates for a mixed boundary elliptic system in a two-dimensional corner domain, when the contact angle $\om\in(0,π/2)$. This system is closely related to the Dirichlet-Neumann operator in the water-waves problem, and the weight we choose is decided by singularities of the mixed boundary system. Meanwhile, we also prove similar weighted estimates with a different weight for the Dirichlet boundary problem as well as the Neumann boundary problem when $\om\in(0,π)$.

Uniqueness of Positive Vorticity Solutions to the 2D Euler Equations on Singular Domains

We show that particle trajectories for positive vorticity solutions to the 2D Euler equations on fairly general bounded simply connected domains cannot reach the boundary in finite time. This includes domains with possibly nowhere $C^1$ boundaries and having corners with arbitrary angles, and can fail without the sign hypothesis when the domain has large angle corners. Hence positive vorticity solutions on such domains are Lagrangian, and we also obtain their uniqueness if the vorticity is initially constant near the boundary.

具有锐角接触角的可压缩毛细重力水波的局部适定性

Local Well-Posedness for Compressible Capillary-Gravity Water Waves with Acute Contact Angles

本文研究具有锐角接触角的可压缩毛细重力水波的局部适定性，通过建立几何结构、奇异性分析与先验能量估计，得到了首个含接触点的可压缩欧拉方程自由边值问题的局部适定性结果。

具有弯曲角的区域的热迹

The heat trace for domains with curved corners

研究了具有弯曲角的平面多边形的热迹，发现其在阶$ t^{1/2} $处首次出现角和曲率的相互作用，并证明了Dirichlet角因子的符号法则，表明其符号仅由角是否为凸或凹决定。