Biharmonic Neumann eigenvalues on domains: geometric bounds and related results
We will consider an eigenvalue problem for the biharmonic operator on bounded domains of Riemannian manifolds with Neumann boundary conditions.
We will discuss the classical boundary conditions and the corresponding variational formulation, and we will present a few properties of the eigenvalues. In particular we will show examples of domains for which, unlike the Euclidean case or the Dirichlet boundary conditions case, negative eigenvalues may appear.
Then we will present our main result, namely upper bounds compatible with the Weyl's law for all the eigenvalues under a given lower bound on the Ricci curvature. Joint work with B. Colbois (Université de Neuchâtel).