The p-Laplace equation in domains with multiple crack section via pencil operators

Abstract

The p-Laplace equation $$ \n \cdot (|\n u|^n \n u)=0 \whereA n>0, $$ in a bounded domain $Ø\subset \re^2$, with inhomogeneous Dirichlet conditions on the smooth boundary $\p Ø$ is considered. In addition, there is a finite collection of curves $$Γ= Γ_1\cup...\cupΓ_m \subset Ø, \quad \{on which we assume homogeneous Dirichlet boundary conditions} \quad u=0, $$ modeling a multiple crack formation, focusing at the origin $0 \in Ø$. 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". 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 $n=0$. Using a combination of analytic and numerical methods, saddle-node bifurcations in $n$ are shown to occur for those nonlinear eigenvalues/eigenfunctions.