On Shape Optimization for Fourth Order Steklov eigenvalue Problems

Abstract

We study three types of fourth-order Steklov eigenvalue problems. For the first two of them, we derive the asymptotic expansion of the eigenvalues on Euclidean annular domains \(\mathbb {B}^n_1\setminus \overline{\mathbb {B}^n_\epsilon }\) as \(\epsilon \rightarrow 0\), in turn yielding some interesting results regarding the shape optimization of the eigenvalues. For these two problems, we also compute the respective spectra on cylinders over closed Riemannian manifolds. For the third problem, we obtain a sharp upper bound for its first non-zero eigenvalue on star-shaped and mean convex Euclidean domains.