On the concentration for a singularly perturbed problem with nonlinear Neumann boundary condition

TL;DR

This paper proves the existence of solutions to a perturbed Allen-Cahn equation with nonlinear boundary conditions, where the solutions' nodal sets concentrate around a capillary hypersurface in a Riemannian manifold.

Contribution

It introduces a new method to construct solutions with prescribed concentration behavior near capillary hypersurfaces, extending previous work to nonlinear boundary conditions.

Findings

Existence of solutions with nodal sets concentrating on capillary hypersurfaces.

Extension of concentration phenomena to nonlinear Neumann boundary conditions.

Construction inspired by Pacard and Ritoré's methods.

Abstract

In this work, we prove the existence of a family of solutions of the Allen-Cahn equation with nonlinear Neumann boundary condition under some constraints, whose nodal sets concentrate asymptotically to a given volume nondegenerate capillary hypersurface in a compact Riemannian manifold. Our construction is inspired by the works of Pacard and of Pacard and Ritor\'e.