A few shape optimization results for a biharmonic Steklov problem

TL;DR

This paper investigates shape optimization for a boundary-mass concentrated biharmonic Steklov problem, deriving shape derivatives, identifying critical domains, and establishing an isoperimetric inequality for the first eigenvalue.

Contribution

It introduces new shape derivative formulas, characterizes critical shapes as balls, and proves an isoperimetric inequality for the first eigenvalue in this context.

Findings

Balls are critical domains under volume constraint.

Derived Hadamard-type formulas for shape derivatives.

Proved an isoperimetric inequality for the first eigenvalue.

Abstract

We derive the equation of a free vibrating thin plate whose mass is concentrated at the boundary, namely a Steklov problem for the biharmonic operator. We provide Hadamard-type formulas for the shape derivatives of the corresponding eigenvalues and prove that balls are critical domains under volume constraint. Finally, we prove an isoperimetric inequality for the first positive eigenvalue.