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

Abstract

As the main problem, the bi-Laplace equation $\Delta^2u=0 (\Delta=D_x^2+D_y^2)$ in a bounded domain $\Omega \subset \re^2$, with inhomogeneous Dirichlet or Navier-type conditions on the smooth boundary $\partial \Omega$ is considered. In addition, there is a finite collection of curves $$\Gamma = \Gamma_1\cup...\cup\Gamma_m \subset \Omega, \quad \mbox{on which we assume homogeneous Dirichlet} \quad u=0,$$ focusing at the origin $0 \in \Omega$ (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.