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arxiv: 1506.00134 · v1 · pith:G2VFHH5Snew · submitted 2015-05-30 · 🧮 math.AP · math.DG

Boundary concentrations on segments

classification 🧮 math.AP math.DG
keywords quadcurvaturemeanomegapartialboundaryeqnarrayfunction
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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}.

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