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The p-Laplace equation in domains with multiple crack section via pencil operators by Alvarez-Caudevilla, Pablo; Galaktionov, Victor A. is a scholarly article available to read on EtoBox.

What is The p-Laplace equation in domains with multiple crack section via pencil operators about?

The p-Laplace equation $$ \n \cdot (|\n u|^n \n u)=0 \whereA n>0, $$ in a bounded domain $\O \subset \re^2$, with inhomogeneous Dirichlet conditions on the smooth boundary $\p \O$ is considered. In addition, there is a finite collection of curves $$\Gamma = \Gamma_1\cup...\cup\Gamma_m \subset \O, \quad \{on which we assume homogeneous Dirichlet boundary conditions} \quad u=0, $$ modeling a multiple crack formation, focusing at the origin $0 \in \O$. 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.

Author

Alvarez-Caudevilla, Pablo; Galaktionov, Victor A.

Published

2013

Language

EN