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

As the main problem, the bi-Laplace equation Delta(2)u = 0 (Delta = D-x(2) + D-y(2)) in a bounded domain Q subset of R-2, with in homogeneous Dirichlet or Navier-type conditions on the smooth boundary partial derivative Omega is considered. In addition, there is a finite collection of curves Gamma = Gamma U-1...U Gamma(m) subset of Omega, on which we assume homogeneous Dirichlet conditions u = 0, focusing at the origin 0 epsilon 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.

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