Boundary Concentrations on Segments

We consider the following singularly perturbed Neumann problem ε∆u− u+ u = 0 , u > 0 in Ω, ∂u ∂ν = 0 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 segment of the boundary which contains a strict local minimum point of the mean curvature function and has the same mean curvature at the two end points. We find a continuum limit of ODE systems governing the interactions of spikes and show that the derivative of mean curvature function acts as friction force. Our construction is partly motivated by the construction of CMC surfaces on broken geodesics by Butscher and Mazzeo [10].