A note on the index of cone differential operators

A note on the extreme points of the cone of quasiconvex quadratic forms with orthotropic symmetry

We study the extreme points of the cone of quasiconvex quadratic forms with linear elastic orthotropic symmetry. We prove that if the determinant of the acoustic matrix of the associated forth order tensor of the quadratic form is an extremal polynomial, then the quadratic form is an extreme point of the cone in the same symmetry class. The extremality of polynomials and quadratic forms here is understood in classical convex analysis sense.

A note on the cone restriction conjecture in the cylindrically symmetric case

In this note, we present two arguments showing that the classical \textit{linear adjoint cone restriction conjecture} holds for the class of functions supported on the cone and invariant under the spatial rotation in all dimensions. The first is based on a dyadic restriction estimate, while the second follows from a strengthening version of the Hausdorff-Young inequality and the Hölder inequality in the Lorentz spaces.

Characterization of isoperimetric sets inside almost-convex cones

In this note we characterize isoperimetric regions inside almost-convex cones. More precisely, as in the case of convex cones, we show that isoperimetric sets are given by intersecting the cone with a ball centered at the origin.

关于多边形域中拉普拉斯算子平移定理的注记（扩展版）

A note on the shift theorem for the Laplacian in polygonal domains (extended version)

本文针对有限平面锥形区域（及多边形）上泊松方程，在三种边界条件下给出平移定理，其适用范围依赖锥角，右端点用贝索夫空间描述。

球面扇区和锥中半线性椭圆方程的对称性破缺与不稳定性

Symmetry breaking and instability for semilinear elliptic equations in spherical sectors and cones

本文研究了锥内球面扇区中半线性椭圆方程，通过分析径向正解的Morse指数及其对单位球面区域的依赖性，证明了Gidas-Ni-Nirenberg型径向对称性结果在一般非凸锥中不成立，并发现无界锥中标准bubbles解在非凸锥中会变得不稳定。

A note on the Morse index of 2k-ended phase transitions in R^2

We show that the Morse index of every 2k-ended solution of the Allen-Cahn equation in R^2 is >= k-1. This bound is expected to be sharp.