Semilinear elliptic equations in thin regions with terms concentrating on oscillatory boundaries

TL;DR

This paper investigates how solutions to semilinear elliptic equations in oscillating thin regions behave, showing they can be approximated by solutions of a simpler one-dimensional model that includes key physical effects.

Contribution

It introduces a novel analysis of solution behavior in oscillating thin domains with concentrated reaction terms, establishing approximation by a one-dimensional limit problem.

Findings

Solutions exhibit upper and lower semicontinuity.

Solutions can be approximated by a one-dimensional equation.

The model captures essential physical processes.

Abstract

In this work we study the behavior of a family of solutions of a semilinear elliptic equation, with homogeneous Neumann boundary condition, posed in a two-dimensional oscillating thin region with reaction terms concentrated in a neighborhood of the oscillatory boundary. Our main result is concerned with the upper and lower semicontinuity of the set of solutions. We show that the solutions of our perturbed equation can be approximated with ones of a one-dimensional equation, which also captures the effects of all relevant physical processes that take place in the original problem.