Symmetry and monotonicity results for solutions of semilinear PDEs in sector-like domains

TL;DR

This paper establishes symmetry and monotonicity properties of solutions to semilinear PDEs in sector-like domains, showing solutions with low Morse index are either symmetric or monotone, extending classical results to more complex geometries.

Contribution

It introduces a rotating-plane method to prove symmetry and monotonicity of solutions in sector-like domains, including unbounded and higher-dimensional cases.

Findings

Solutions with Morse index less than two are either symmetric or monotone in the angular variable.

The results extend to higher dimensions and unbounded domains.

The rotating-plane argument effectively handles complex domain geometries.

Abstract

In this manuscript we consider semilinear PDEs, with a convex nonlinearity, in a sector-like domain. Using cylindrical coordinates $(r, \theta, z)$, we investigate the shape of solutions whose derivative in $\theta$ vanishes at the boundary. We prove that any solution with Morse index less than two must be either independent of $\theta$ or strictly monotone with respect to $\theta$. 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 $\pi$.