The p-Laplace Equation in Domains with Multiple Crack Section via Pencil Operators

Abstract

Abstract The p-Laplace equation ∇ · (|∇u|p−2∇u) = 0, where p > 2, in a bounded domain Ω ⊂ ℝ2, with inhomogeneous Dirichlet conditions on the smooth boundary ∂Ω is considered. In addition, there is a finite collection of curves Γ = Γ1 ∪ ... ∪ Γm ⊂ Ω, on which we assume homogeneous Dirichlet conditions u = 0, modeling a multiple crack formation, focusing at the origin 0 ∈ Ω. This makes the above quasilinear 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 a “nonlinear eigenvalue problem” via spectral theory of pencils of non self-adjoint operators. Specially interesting is the application of those techniques to non-linear problems as the one considered here. To do so we introduce a very novel change of variable compared with the classical one introduced by Kondratiev for the analysis of non-smooth domains, such as domains with corner points, edges, etc, studying the behaviour of the solutions at those problematic points. Typical types of admissible cracks are shown to be governed by nodal sets of a countable family of nonlinear eigenfunctions, which are obtained via branching from harmonic polynomials that occur for p = 2. Using a combination of analytic and numerical methods, saddle-node bifurcations in p are shown to occur for those nonlinear eigenvalues/ eigenfunctions.