乘积锥上的多齐性传播、衍射与散射

散射、多项齐次性及从过去到未来光无穷远处的二次波方程渐进行为

Scattering, Polyhomogeneity and Asymptotics for Quasilinear Wave Equations From Past to Future Null Infinity

本文提出了一种构造二次波方程半全局散射解的方法，证明了多项齐次性传播，并扩展了爱因斯坦真空方程的散射理论。

锥的逆几何衍射问题

Inverse Geometric Diffraction by a Cone

本文研究三维空间中从衍射系数等数据恢复严格凸锥形障碍物的逆问题，通过阿达马参数式等技术建立了对应几何衍射理论的严格逆理论。

On resonances generated by conic diffraction

We describe the resonances closest to the real axis generated by diffraction of waves among cone points on a manifold with Euclidean ends. These resonances lie asymptotically evenly spaced along a curve of the form $$\frac{\Im λ}{\log \left |\Re λ\right |}= -ν;$$ here $ν=(n-1)/2 L_0$ where $n$ is the dimension and $L_0$ is the length of the longest geodesic connecting two cone points. Moreover there are asymptotically no resonances below this curve and above the curve $$ \frac{\Im λ}{\log \left |\Re λ\right |}= -Λ$$ for a fixed $Λ>ν.$

Strichartz estimates for the wave equation on flat cones

We consider the solution operator for the wave equation on the flat Euclidean cone over the circle of radius $ρ> 0$, the manifold $\mathbb{R}_+ \times \mathbb{R} / 2 πρ\mathbb{Z}$ equipped with the metric $\g(r,θ) = dr^2 + r^2 dθ^2$. Using explicit representations of the solution operator in regions related to flat wave propagation and diffraction by the cone point, we prove dispersive estimates and hence scale invariant Strichartz estimates for the wave equation on flat cones. We then show that this yields corresponding inequalities on wedge domains, polygons, and Euclidean surfaces with coni…

锥点处的半经典传播

Semiclassical propagation through cone points

该研究提出高频波锥点衍射的通用框架，证明非对称首项算子下锥点半经典正则性的几乎精确损失及严格衍射测地线的正则性改善，应用于高能预解估计等，采用半经典锥微积分完成证明。

Diffraction of Elastic Waves by Edges

We investigate the diffraction of singularities of solutions to the linear elastic equation on manifolds with edge singularities. Such manifolds are modeled on the product of a smooth manifold and a cone over a compact fiber. For the fundamental solution, the initial pole generates a pressure wave (p-wave), and a secondary, slower shear wave (s wave). If the initial pole is appropriately situated near the edge, we show that when a p-wave strikes the edge, the diffracted p-waves and s-waves (i.e. loosely speaking, do not correspond to limits of p-rays which just miss the edge) are weaker in a S…