Calculus of Variations and Geometric Measure Theory

E. Davoli - I. Mazari - U. Stefanelli

Spectral optimisation of inhomogeneous plates

created by davoli on 23 Jul 2021
modified on 30 Nov 2022


Accepted Paper

Inserted: 23 jul 2021
Last Updated: 30 nov 2022

Journal: SIAM Journal on Control and Optimization (SICON)
Year: 2022


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.