Boundary concentrations on segments

TL;DR

This paper constructs solutions to a singularly perturbed Neumann problem in a 2D domain, where multiple spikes concentrate along a boundary segment influenced by curvature, revealing curvature's role as a friction force.

Contribution

It introduces a new class of boundary-concentrating spike solutions on segments with specific curvature properties, linking geometric features to solution behavior.

Findings

Spikes concentrate on boundary segments with constant mean curvature at endpoints.

Mean curvature functions as a friction force affecting spike interactions.

A continuum limit of spike interaction ODEs is derived.

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