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 their spectra on Euclidean annular domains ${\mathbb{B}}_1^n\setminus \overline{{\mathbb{B}}_{\hat{I}}^n\mu }$ as $\hat{I}\mu \to 0$, 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.

37 pages; all comments are welcome! In v2, we rephrase some statements of our results in the Introduction section