论文标题
半线性PDE溶液中类似扇形样域的对称性和单调性结果
Symmetry and monotonicity results for solutions of semilinear PDEs in sector-like domains
论文作者
论文摘要
在此手稿中，我们考虑在类似扇区的域中具有凸非线性的半线性PDE。使用圆柱坐标$（r，θ，z）$，我们研究了$θ$衍生品在边界上消失的衍生物的形状。我们证明，与$θ$相对于$θ$，任何具有MORSE索引的解决方案都必须独立于$θ$或严格单调。在平面结构域的特殊情况下，结果构成圆形扇区和环形，也可以扩展到矩形域。还考虑了较高维度中的相应问题，以及对无界域的扩展。证明基于旋转平面参数：引入方便的歧管，以避免在开口大于$π$的情况下与反射图像重叠的域。
In this manuscript we consider semilinear PDEs, with a convex nonlinearity, in a sector-like domain. Using cylindrical coordinates $(r, θ, z)$, we investigate the shape of solutions whose derivative in $θ$ vanishes at the boundary. We prove that any solution with Morse index less than two must be either independent of $θ$ or strictly monotone with respect to $θ$. In the special case of a planar domain, the result holds in a circular sector as well as in an annular, and it can also be extended to a rectangular domain. The corresponding problem in higher dimensions is also considered, as well as an extension to unbounded domains. The proof is based on a rotating-plane argument: a convenient manifold is introduced in order to avoid overlapping the domain with its reflected image in the case when its opening is larger than $π$.
