Spectral optimization of inhomogeneous plates

TL;DR

This paper investigates spectral optimization for inhomogeneous plates, focusing on maximizing the first eigenvalue by adjusting thickness and density, with existence, characterization, and stability results provided.

Contribution

It establishes the existence of optimal thickness, characterizes it for circular plates using Talenti inequalities, and proves stability under linear relations between thickness and density.

Findings

Existence of optimal thickness proven.

Optimal thickness characterized for circular plates.

Stability result established with linear thickness-density relation.

Abstract

This article is devoted to the study of spectral optimisation for inhomogeneous plates. In particular, we optimise the first eigenvalue of a vibrating plate with respect to its thickness and/or density. Our result is threefold. First, we prove existence of an optimal thickness, using fine tools hinging on topological properties of rearrangement classes. Second, in the case of a circular plate, we provide a characterisation of this optimal thickness by means of Talenti inequalities. Finally, we prove a stability result when assuming that the thickness and the density of the plate are linearly related. This proof relies on H-convergence tools applied to biharmonic operators.