Spectral optimization of inhomogeneous plates. (English) Zbl 1523.74105
Summary: This article is devoted to the study of spectral optimization for inhomogeneous plates. In particular, we consider the optimization of the first eigenvalue of a vibrating plate with respect to its thickness and/or density. We prove the existence of an optimal thickness, using fine tools hinging on topological properties of rearrangement classes. In the case of a circular plate, we provide a characterization of this optimal thickness by means of Talenti inequalities. Moreover, 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 the biharmonic operator.
MSC:
| 74P10 | Optimization of other properties in solid mechanics |
| 74K20 | Plates |
| 74E05 | Inhomogeneity in solid mechanics |
| 74H45 | Vibrations in dynamical problems in solid mechanics |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 35Q93 | PDEs in connection with control and optimization |
Keywords:
first eigenvalue optimization; two-phase problem; optimal thickness existence; H-convergence; rearrangement class; Talenti inequalityReferences:
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