Narrow Results

We consider the buckling eigenvalue problem for a clamped plate in the annulus. We identify the first eigenvalue in dependence of the inner radius, and study the number of nodal domains of the corresponding eigenfunctions. Moreover, in order to investigate the asymptotic behavior of eigenvalues and eigenfunctions as the inner radius approaches the outer one, we provide an analytical study of the buckling problem in rectangles with mixed boundary conditions.

For an arbitrary nonempty, open set Ω ⊂ ℝⁿ, n ∈ ℕ of finite (Euclidean) volume, we consider the minimally defined higher-order Laplacian , m ∈ ℕ, and its Krein–von Neumann extension A_K,Ω,m in L²(Ω). with N(λ, A_K,Ω,m), λ > 0, denoting the eigenvalue counting function corresponding to the strictly positive eigenvalues of A_K,Ω,m, we derive the bound

The proof relies on variational considerations and exploits the fundamental link between the Krein–von Neumann extension and an underlying (abstract) buckling problem.