Symmetry properties for positive solutions of mixed boundary value problems in a sub-spherical sector

TL;DR

This paper studies symmetry properties of positive solutions to semilinear elliptic equations with mixed boundary conditions in symmetric domains, introducing new maximum principles and analyzing derivatives to establish symmetry and monotonicity results.

Contribution

It develops a maximum principle for mixed-boundary problems, addresses non-vanishing derivatives along Neumann boundaries, and combines these with the moving plane method for symmetry analysis.

Findings

Established symmetry and monotonicity for solutions in sub-spherical sectors.

Developed a maximum principle applicable to small or narrow domains.

Identified signs of directional derivatives to facilitate symmetry proofs.

Abstract

In this paper, we investigate the symmetry properties of positive solutions $u$ to a semilinear elliptic equation under mixed Dirichlet-Neumann boundary conditions in symmetric domains. First, we establish a maximum principle tailored to mixed-boundary problems in domains of either small volume or narrow width, thereby enabling the application of the moving plane method. Secondly, in contrast to the purely Dirichlet case, a key challenge is to establish the non-vanishing of the tangential derivative of $u$ along the Neumann boundary. To address this, we employ local analysis techniques of angular derivatives, as introduced by Hartman and Wintner [Amer. J. Math., 1953]. Thirdly, we identify the signs of directional derivatives of $u$ along sections of the moving line. Using a planar sub-spherical sector as an example, we illustrate how these new innovative techniques and the moving plane method can be combined to derive symmetry and monotonicity results, particularly when the amplitude is less than or equal to $2\pi/3$.