Boundary concentrations on segments
Analysis of PDEs
2015-06-02 v1 Differential Geometry
Abstract
We consider the following singularly perturbed Neumann problem \begin{eqnarray*} \ve^2 \Delta u -u +u^p = 0 \, \quad u>0 \quad {\mbox {in}} \quad \Omega, \quad {\partial u \over \partial \nu}=0 \quad {\mbox {on}} \quad \partial \Omega, \end{eqnarray*} where and is a smooth and bounded domain in . We construct a new class of solutions which consist of large number of spikes concentrating on a {\bf segment} of the boundary which contains a local minimum point of the mean curvature function and has the same mean curvature at the end points. We find a continuum limit of ODE systems governing the interactions of spikes and show that the mean curvature function acts as {\em friction force}.
Cite
@article{arxiv.1506.00134,
title = {Boundary concentrations on segments},
author = {Weiwei Ao and Hardy Chan and Juncheng Wei},
journal= {arXiv preprint arXiv:1506.00134},
year = {2015}
}
Comments
36 pages