Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Whispering Gallery Modes for Semilinear Dirichlet Eigenvalue Problems

Zhengjiang Lin

Abstract

We study the boundary localization phenomenon, known as whispering gallery modes, for weak solutions to semilinear Dirichlet eigenvalue problems in the unit ball $B_1 \subseteq \mathbb{R}^d$ ($d \geq 2$) of the form \[ \begin{cases} -Δu + f(u) = λu & \text{in } B_1,\\ u = 0 & \text{on } \partial B_1. \end{cases} \] Here, $f = F'$ where $F$ is a nonnegative $C^2$-function with superquadratic polynomial growth. We prove the existence of a sequence of solutions $(u_n, λ_n)$ with $λ_n \to +\infty$ such that, for any $τ\in (0,1)$, \[ \lim_{n \to \infty} \frac{E_τ(u_n)}{E_1(u_n)} = 0, \] where $E_ρ(u) = \int_{B_ρ} \bigl( \frac{1}{2} |\nabla u|^2 + F(u) \bigr) \, dx$ is the energy over the ball of radius $ρ$. This establishes that the energy of these high-eigenvalue solutions concentrates near the boundary, extending the classical whispering gallery mode phenomenon from linear Laplacian eigenfunctions to the semilinear setting. As a direct application, the case $F(u) = u^4 /4$ yields, after suitable scaling, a sequence of high-eigenvalue boundary-concentrating solutions, providing a nonlinear analogue of whispering gallery modes for the Allen-Cahn equations. The approach combines spectral properties of the linear Laplacian with local nonlinear bifurcation and is expected to adapt to related interior localization phenomena, such as bouncing ball modes, in other geometries where linear eigenfunctions exhibit analogous localization behaviors (e.g., filled ellipses or elliptical annuli).

Whispering Gallery Modes for Semilinear Dirichlet Eigenvalue Problems

Abstract

We study the boundary localization phenomenon, known as whispering gallery modes, for weak solutions to semilinear Dirichlet eigenvalue problems in the unit ball () of the form Here, where is a nonnegative -function with superquadratic polynomial growth. We prove the existence of a sequence of solutions with such that, for any , where is the energy over the ball of radius . This establishes that the energy of these high-eigenvalue solutions concentrates near the boundary, extending the classical whispering gallery mode phenomenon from linear Laplacian eigenfunctions to the semilinear setting. As a direct application, the case yields, after suitable scaling, a sequence of high-eigenvalue boundary-concentrating solutions, providing a nonlinear analogue of whispering gallery modes for the Allen-Cahn equations. The approach combines spectral properties of the linear Laplacian with local nonlinear bifurcation and is expected to adapt to related interior localization phenomena, such as bouncing ball modes, in other geometries where linear eigenfunctions exhibit analogous localization behaviors (e.g., filled ellipses or elliptical annuli).
Paper Structure (7 sections, 17 theorems, 107 equations)

This paper contains 7 sections, 17 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Proof of \ref{['thm:main bifurcation']}
  4. Proof of \ref{['lem:solve simplified 1']}
  5. Proof of \ref{['lem:solve simplified 2']}
  6. Proof of \ref{['thm:main bifurcation']}
  7. Proof of \ref{['thm:main whispering gallery linear eigen']}

Key Result

Theorem 1.1

Let $\Omega = B_1$ be the unit ball in ${\mathbb R}^d$ with $d \geq 2$. Assume that $F$ satisfies a:superlinearity and the Banach space $V$ is defined in def:space V. There is a family of solution pairs $\{ (u_{n},\lambda_n)\} \subseteq V \times {\mathbb R}$ to eq:eigen, such that $\lim_{n \to \inft where we let $B_{\tau}$ be the ball of radius $\tau$ and

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof : Proof of \ref{['thm:main whispering gallery']}
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • ...and 24 more