Construction of solutions for a nonlinear elliptic problem on Riemannian manifolds with boundary

TL;DR

This paper demonstrates that stable critical points of the boundary's mean curvature on a Riemannian manifold can generate solutions to a singularly perturbed elliptic problem with Neumann boundary conditions.

Contribution

It establishes a link between boundary mean curvature critical points and solutions to a specific nonlinear elliptic problem on manifolds with boundary.

Findings

Stable critical points of boundary mean curvature generate solutions.

Solutions are for a singularly perturbed elliptic problem.

Results apply to smooth compact Riemannian manifolds with boundary.

Abstract

Let (M,g) be a smooth compact, n dimensional Riemannian manifold,with smooth n-1 dimensional boundary. We prove that the stable critical points of the mean curvature of the boundary generates solutions for a singularly perturbed elliptic problem with Neumann boundary conditions .