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Monotonicity of positive solutions to semilinear elliptic equations with mixed boundary conditions in triangles

Rui Li, Ruofei Yao

Abstract

We study positive solutions of semilinear elliptic equations in a planar triangular domain under mixed boundary conditions, consisting of homogeneous Dirichlet boundary conditions on one side and homogeneous Neumann boundary conditions on the remaining two sides. Using the method of moving planes, we prove that if the Neumann vertex is non-obtuse, then every positive solution is strictly increasing in the direction of the inward unit normal to the Dirichlet side. If the Neumann vertex is obtuse, we show that monotonicity instead holds in the direction of the outward normal direction to the longer Neumann side, under certain technical conditions. Furthermore, by applying the maximum principle, we demonstrate that these monotonicity properties extend to the first mixed eigenfunction of the Laplacian: its unique global maximum lies on the longer Neumann side and coincides with the Neumann vertex precisely when that vertex is non-obtuse or when the Neumann sides have equal length. This answers a question raised in the Polymath7 research thread 1 regarding the location of extrema for mixed boundary eigenfunctions in triangles.

Monotonicity of positive solutions to semilinear elliptic equations with mixed boundary conditions in triangles

Abstract

We study positive solutions of semilinear elliptic equations in a planar triangular domain under mixed boundary conditions, consisting of homogeneous Dirichlet boundary conditions on one side and homogeneous Neumann boundary conditions on the remaining two sides. Using the method of moving planes, we prove that if the Neumann vertex is non-obtuse, then every positive solution is strictly increasing in the direction of the inward unit normal to the Dirichlet side. If the Neumann vertex is obtuse, we show that monotonicity instead holds in the direction of the outward normal direction to the longer Neumann side, under certain technical conditions. Furthermore, by applying the maximum principle, we demonstrate that these monotonicity properties extend to the first mixed eigenfunction of the Laplacian: its unique global maximum lies on the longer Neumann side and coincides with the Neumann vertex precisely when that vertex is non-obtuse or when the Neumann sides have equal length. This answers a question raised in the Polymath7 research thread 1 regarding the location of extrema for mixed boundary eigenfunctions in triangles.
Paper Structure (6 sections, 23 theorems, 181 equations, 4 figures)

This paper contains 6 sections, 23 theorems, 181 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Some preliminaries
  3. The proof of monotonicity for non-obtuse Neumann vertex
  4. The symmetry property in an isosceles triangle
  5. The proof of monotonicity for an obtuse Neumann vertex
  6. The eigenfunction with obtuse Neumann vertex

Key Result

Theorem 1.1

Let ${f} \in \operatorname{Lip}_{\mathrm{loc}}(\mathbb{R})$, and let ${u}$ be a positive solution of LY101 in a planar triangle $\Omega$, where $\Gamma_{D}$ is one side of the triangle $\Omega$. If the two Neumann sides have equal lengths, then ${u}$ is symmetric with respect to the angle bisector a

Figures (4)

  • Figure 1: The moving lines $\color{red}{T}_{\lambda, \vartheta}$ and moving domains $\color{green}{D}_{\lambda, \vartheta, \vartheta_{1}}$
  • Figure 2: The domains $\color{blue}\mathcal{D}_{1}$, $\color{cyan}\mathcal{D}_{2}$ and $\color{orange}\mathcal{D}_{3}$
  • Figure 3: The moving domain ${D}_{\Upsilon, \vartheta}$ when the slope of ${T}_{\Upsilon, \vartheta}$$\vartheta$ is positive and large
  • Figure 4: The case for $\gamma > \pi/2$ and $\alpha \geq \pi/4$

Theorems & Definitions (42)

  • Conjecture 1: Pol12
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1
  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2
  • ...and 32 more