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

Ruofei Yao

TL;DR

This work studies symmetry and monotonicity of positive solutions to $\Delta u + f(u)=0$ in planar sub-spherical sectors under mixed Dirichlet–Neumann boundaries. It advances the moving-plane method by developing a maximum principle tailored to mixed boundary problems and by incorporating Hartman–Wintner angular-derivative analysis to secure non-vanishing tangential derivatives on the Neumann part. Through a careful analysis of angle relations and a continuation of moving planes in multiple directions, the authors establish axial symmetry and monotonicity for all admissible openings with $\beta\le 2\pi/3$, including detailed handling of challenging regimes $\pi/3<\beta<\pi/2$ via double-domain techniques. The results extend classical symmetry theory to mixed-boundary elliptic problems in symmetric domains, providing new tools for qualitative PDE analysis in sectors and related geometries.

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 $2π/3$.

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

TL;DR

This work studies symmetry and monotonicity of positive solutions to in planar sub-spherical sectors under mixed Dirichlet–Neumann boundaries. It advances the moving-plane method by developing a maximum principle tailored to mixed boundary problems and by incorporating Hartman–Wintner angular-derivative analysis to secure non-vanishing tangential derivatives on the Neumann part. Through a careful analysis of angle relations and a continuation of moving planes in multiple directions, the authors establish axial symmetry and monotonicity for all admissible openings with , including detailed handling of challenging regimes via double-domain techniques. The results extend classical symmetry theory to mixed-boundary elliptic problems in symmetric domains, providing new tools for qualitative PDE analysis in sectors and related geometries.

Abstract

In this paper, we investigate the symmetry properties of positive solutions 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 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 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 .
Paper Structure (20 sections, 26 theorems, 258 equations, 8 figures)

This paper contains 20 sections, 26 theorems, 258 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Problem setting and main result
  4. Ideas of the proof
  5. Organization of the paper
  6. Some preliminaries
  7. The maximum principle with mixed boundary conditions
  8. The monotonicity near the Dirichlet boundary
  9. The zero set
  10. The relation of angles
  11. The start of the method of moving planes
  12. The moving lines and moving domains
  13. The moving plane method with a priori boundary condition
  14. The case for $\lambda \geq \lambda_{C}$
  15. The case $\vartheta_{A}(\lambda) \geq \beta$
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let ${f} \in \mathrm{Lip}_{\mathrm{loc}}(\mathbb{R})$, and let $\Sigma$, $\Gamma_{D}$, and $\Gamma_{N}$ be as defined above. Suppose that $0 < \beta < \alpha \leq \pi$ and Then any solution ${u}$ of Yao0105 satisfies the following properties:

Figures (8)

  • Figure 1: Sub-spherical sector (left), spherical sector (middle) and super-spherical sector (right).
  • Figure 2: The relation of the angles.
  • Figure 3: The moving lines ${T}_{\lambda, \pi/2}$ and the moving domains ${D}_{\lambda, \pi/2}$.
  • Figure 4: The moving lines ${T}_{\lambda, \vartheta}$ and the moving domains $D_{\lambda, \vartheta, \vartheta_{1}}$.
  • Figure 5: The function of $\zeta(\vartheta)$.
  • ...and 3 more figures

Theorems & Definitions (58)

  • Theorem 1.1
  • Remark 1
  • Lemma 1
  • proof
  • Remark 2
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4: Maximum principle with small volume
  • ...and 48 more