Concentration on Surfaces for a Singularly Perturbed Neumann Problem in Three-Dimensional Domains

TL;DR

This paper proves the existence of solutions to a singularly perturbed elliptic Neumann problem in three-dimensional domains that concentrate along surfaces close to a specified hypersurface, with the concentration behavior depending on the small parameter.

Contribution

It establishes the existence of surface-concentrating solutions near a hypersurface intersecting the boundary at right angles, extending understanding of pattern formation in elliptic PDEs.

Findings

Solutions concentrate along surfaces close to the hypersurface

Concentration is exponentially small away from the surface

The concentrating surface collapses to the original hypersurface as epsilon approaches zero

Abstract

We consider the following singularly perturbed elliptic problem $$ \varepsilon^2\triangle\tilde{u}-\tilde{u}+\tilde{u}^p=0, \ \tilde{u}>0\quad \mbox{in} \ \Omega,\ \ \ \frac{\partial\tilde{u}}{\partial \mathbf{n}}=0 \quad \mbox{on}\ \partial\Omega, $$ where $\Omega$ is a bounded domain in $\mathbb{R}^3$ with smooth boundary, $\varepsilon$ is a small parameter, $\mathbf{n}$ denotes the inward normal of $ \partial\Omega$ and the exponent $p>1$. Let $\Gamma$ be a hypersurface intersecting $\partial\Omega$ in the right angle along its boundary $\partial\Gamma$ and satisfying a {\em non-degenerate condition}. We establish the existence of a solution $u_\varepsilon$ concentrating along a surface $\tilde{\Gamma}$ close to $\Gamma$, exponentially small in $\varepsilon$ at any positive distance from the surface $\tilde{\Gamma}$, provided $\varepsilon$ is small and away from certain {\em critical numbers}. The concentrating surface $\tilde{\Gamma}$ will collapse to $\Gamma$ as $\varepsilon\rightarrow 0$.