In this talk we will consider the eigenvalues of the biharmonic operator subject to various boundary conditions, namely Dirichlet, Neumann, Navier and Steklov ones. Note that such problems arise in the theory of linear elasticity, within the so-called Kirchhoff-Love model for the vibration of a plate. We will show that the eigenvalues of the Bilaplacian are analytic with respect to the shape, and compute Hadamard-type formulas for their differentials. Then, using the Lagrange Multiplier Theorem, we are able to show that the ball is a critical domain under volume constraint for any eigenvalue (under any of the boundary conditions considered). In the last part of the talk we will focus on eigenvalue shape optimization results for Neumann and Steklov problems. We will provide isoperimetric inequalities in quantitative form for the fundamental tones (i.e., the first non-trivial eigenvalues), and discuss the limiting cases. Based on the papers [1, 2, 3, 4].

References:
[1] D. Buoso, Analyticity and criticality results for the eigenvalues of the bihar- monic operator, to appear on the proceedings of the conference "Geometric Properties for Parabolic and Elliptic PDE's - 4th Italian-Japanese Workshop".
[2] D. Buoso, L.M. Chasman, L. Provenzano, On the stability of some isoperi- metric inequalities for the fundamental tones of free plates, submitted.
[3] D. Buoso, L. Provenzano, A few shape optimization results for a biharmonic Steklov problem, J. Differential Equations, 259 (2015), no. 5, 1778-1818.
[4] D. Buoso, L. Provenzano, On the eigenvalues of a biharmonic Steklov problem, in: Integral Methods in Science and Engineering: Theoretical and Computational Advances, Birkhäuser, 2015.