Blow-up scaling and global behaviour of solutions of the bi-laplace equation via pencil operators

As the main problem, the bi-Laplace equation Δ²u = 0 (Δ = D_x² + D_y²) in a bounded domain Ω ℝ², with inhomogeneous Dirichlet or Navier-type conditions on the smooth boundary ∂Ω is considered. In addition, there is a finite collection of curves Γ = Γ₁ ∪. ∪ Γ_m ⊂ Ω on which we assume homogeneous Dirichlet conditions u = 0, focusing at the origin 0 ∈Ω (the analysis would be similar for any other point). This makes the above elliptic problem overdetermined. Possible types of the behaviour of solution u(x, y) at the tip 0 of such admissible multiple cracks, being a singularity point, are described, on the basis of blow-up scaling techniques and spectral theory of pencils of non self-adjoint operators. Typical types of admissible cracks are shown to be governed by nodal sets of a countable family of harmonic polynomials, which are now represented as pencil eigenfunctions, instead of their classical representation via a standard Sturm-Liouville problem. Eventually, for a fixed admissible crack formation at the origin, this allows us to describe all boundary data, which can generate such a blow-up crack structure. In particular, it is shown how the co-dimension of this data set increases with the number of asymptotically straight-line cracks focusing at 0.