## Abstract

As the main problem, the bi-Laplace equation Δ^{2}u = 0 (Δ = D_{x}^{2} + D_{y}^{2}) in a bounded domain Ω ℝ^{2}, with inhomogeneous Dirichlet or Navier-type conditions on the smooth boundary ∂Ω is considered. In addition, there is a finite collection of curves Γ = Γ_{1} ∪. ∪ 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.

Original language | English |
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Pages (from-to) | 261-286 |

Number of pages | 26 |

Journal | Communications in Pure and Applied Analysis |

Volume | 15 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1 Jan 2016 |

## Keywords

- Bi- and Laplace equations
- Harmonic polynomials
- Higher-order equations
- Nodal sets
- Pencil of non selfadjoint operators