Boundary concentrations on segments

2015/05/30 by Weiwei Ao, Hardy Chan, Ao, Weiwei +3
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG

paper · pdf · doi:10.48550/arxiv.1506.00134

36 pages

arxiv created 2015/05/30 · arxiv updated 2015/06/02

Abstract

We consider the following singularly perturbed Neumann problem \ve² Δu -u +u^p = 0 u>0 \mbox in Ω, ∂ u \over ∂ ν=0 \mbox on ∂ Ω, where p>2 and Ω is a smooth and bounded domain in \R². 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.

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