On shape optimization for fourth order Steklov eigenvalue problems

TL;DR

This paper investigates the spectral properties of fourth-order Steklov eigenvalue problems, deriving asymptotic expansions and bounds to inform shape optimization in Euclidean and Riemannian domains.

Contribution

It provides new asymptotic formulas for eigenvalues on annular domains and sharp bounds on eigenvalues for star-shaped domains, advancing understanding of shape optimization for these problems.

Findings

Asymptotic expansion of spectra on Euclidean annuli as inner radius shrinks to zero.

Spectra computation on cylinders over Riemannian manifolds.

Sharp upper bounds for the first non-zero eigenvalue on star-shaped, convex domains.

Abstract

We study three types of fourth-order Steklov eigenvalue problems. For the first two of them, we derive the asymptotic expansion of their spectra on Euclidean annular domains $\mathbb{B}^n_1\setminus \overline{\mathbb{B}^n_\epsilon}$ as $\epsilon \to 0$, leading to conclusions on shape optimization. For these two problems, we also compute their spectra on cylinders over closed Riemannian manifolds. Last, for the third problem, we obtain a sharp upper bound for its first non-zero eigenvalue on star-shaped and mean convex Euclidean domains.