On the Eigenvalues of the Biharmonic Steklov Problem on a Thin Set

TL;DR

This paper studies how the eigenvalues of a biharmonic operator with Steklov boundary conditions behave on thin tubular neighborhoods of planar domains, showing they tend to zero as the neighborhood thickness decreases.

Contribution

It provides the first asymptotic analysis of biharmonic Steklov eigenvalues on thin sets, revealing their convergence to zero as the set becomes thinner.

Findings

Eigenvalues tend to zero as the neighborhood thickness approaches zero.

The asymptotic behavior is characterized for biharmonic operators with Steklov conditions.

Results contribute to understanding spectral properties on thin geometries.

Abstract

This paper investigates the asymptotic behavior of the eigenvalues of the biharmonic operator on a thin set with Steklov boundary condition. The thin set is taken to be a tubular neighborhood of a planar smooth domain. We show that, as the thickness of this neighborhood tends to zero, all eigenvalues of the biharmonic operator with Steklov boundary condition converge to zero.