English

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 p>2p>2 and Ω\Omega is a smooth and bounded domain in R2\R^2. 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}.

Keywords

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

R2 v1 2026-06-22T09:44:22.582Z