A nonlinear elliptic problem with terms concentrating in the boundary

TL;DR

This paper studies how solutions to a nonlinear reaction-diffusion equation behave when reaction and potential terms concentrate near a boundary segment, especially under oscillatory boundary conditions, showing convergence to a limit problem.

Contribution

It introduces a novel analysis of solution convergence for nonlinear elliptic equations with boundary-concentrated terms and oscillatory boundary behavior.

Findings

Solutions converge to a limit problem as the neighborhood shrinks.

Reaction and potential terms translate into boundary flux and potential conditions.

Oscillatory boundary behavior influences the limit problem's boundary conditions.

Abstract

In this paper we investigate the behavior of a family of steady state solutions of a nonlinear reaction diffusion equation when some reaction and potential terms are concentrated in a $\epsilon$-neighborhood of a portion $\Gamma$ of the boundary. We assume that this $\epsilon$-neighborhood shrinks to $\Gamma$ as the small parameter $\epsilon$ goes to zero. Also, we suppose the upper boundary of this $\epsilon$-strip presents a highly oscillatory behavior. Our main goal here is to show that this family of solutions converges to the solutions of a limit problem, a nonlinear elliptic equation that captures the oscillatory behavior. Indeed, the reaction term and concentrating potential are transformed into a flux condition and a potential on $\Gamma$, which depends on the oscillating neighborhood.