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

Pablo Álvarez-Caudevilla; V. A. Galaktionov

doi:10.3934/cpaa.2016.15.261

Article Contents

2016, Volume 15, Issue 1: 261-286. Doi: 10.3934/cpaa.2016.15.261

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

Pablo Álvarez-Caudevilla^1, and
V. A. Galaktionov^2,

1.
Universidad Carlos III de Madrid, Av. Universidad 30, 28911-Leganés
2.
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY

Received: February 28, 2015

Revised: June 30, 2015

Published: December 2015

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Abstract

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

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Mathematics Subject Classification: 31A30, 35A20, 35C11, 35G15.

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