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On the eigenvalues of the biharmonic operator on annuli

Davide Buoso, Riccardo Molinarolo

TL;DR

This work analyzes the simplicity of the first eigenvalue (fundamental tone) of the biharmonic operator under Dirichlet and Navier boundary conditions on radially symmetric domains, including balls, punctured balls, and spherical shells. The authors solve the eigenvalue problems by separation of variables, expressing radial parts through ultraspherical Bessel functions and enforcing boundary conditions to obtain explicit determinant conditions, while addressing the special case of the punctured ball via $H^2$-capacity arguments. A key result is that for $N\ge3$ the fundamental tone is simple and radial on these domains, whereas in $N=2$ a threshold inner radius can force the fundamental tone to be radial or to involve nonradial angular modes; a Rayleigh quotient decomposition by angular momentum $\ell$ underpins these conclusions. The Navier problem exhibits analogous behavior, with simplicity in higher dimensions and possible multiplicity in two dimensions. Overall, the paper links spectral properties of the biharmonic operator to dimension, domain geometry, and capacity theory, with implications for plate buckling and vibration models.

Abstract

We show that the fundamental tone of the bilaplacian with Dirichlet or Navier boundary conditions on radially symmetric domains is always simple in dimension $N\ge3$. In dimension $N=2$ we show that it is simple if the inner radius is big enough.

On the eigenvalues of the biharmonic operator on annuli

TL;DR

This work analyzes the simplicity of the first eigenvalue (fundamental tone) of the biharmonic operator under Dirichlet and Navier boundary conditions on radially symmetric domains, including balls, punctured balls, and spherical shells. The authors solve the eigenvalue problems by separation of variables, expressing radial parts through ultraspherical Bessel functions and enforcing boundary conditions to obtain explicit determinant conditions, while addressing the special case of the punctured ball via -capacity arguments. A key result is that for the fundamental tone is simple and radial on these domains, whereas in a threshold inner radius can force the fundamental tone to be radial or to involve nonradial angular modes; a Rayleigh quotient decomposition by angular momentum underpins these conclusions. The Navier problem exhibits analogous behavior, with simplicity in higher dimensions and possible multiplicity in two dimensions. Overall, the paper links spectral properties of the biharmonic operator to dimension, domain geometry, and capacity theory, with implications for plate buckling and vibration models.

Abstract

We show that the fundamental tone of the bilaplacian with Dirichlet or Navier boundary conditions on radially symmetric domains is always simple in dimension . In dimension we show that it is simple if the inner radius is big enough.

Paper Structure

This paper contains 5 sections, 2 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Radially symmetric domains
  4. Fundamental tones
  5. Useful properties of Bessel functions

Key Result

Theorem 1

For $N \geq 3$, the fundamental tone of the ball $B$, of a spherical shell or of the punctured ball $B_a$ for $a \in [0,1)$, is always simple and associated with a radial eigenfunction, both for the Dirichlet problem eq:weakdirev and for the Navier problem eq:weaknavev (for any $\sigma$ such that th

Figures (1)

  • Figure 1: The smallest eigenvalue associated with $\ell=0$ (bold) and $\ell=1$ (dashed) for $a\in[0,0.012]$ and for different values of $\sigma$.

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • proof