On second order elliptic equations with a small parameter

TL;DR

This paper analyzes the asymptotic behavior of solutions to a Neumann boundary problem involving second order elliptic operators with a small parameter, describing the limit via a differential equation on a tree structure.

Contribution

It introduces a novel approach to describe the limit of solutions using diffusion process analysis and differential equations on a tree, with explicit operators and conditions.

Findings

Limit of solutions described by an ODE on a tree

Explicit calculation of differential operators on tree edges

Effective characterization of boundary behavior as epsilon approaches zero

Abstract

The Neumann problem with a small parameter $$(\dfrac{1}{\epsilon}L_0+L_1)u^\epsilon(x)=f(x) \text{for} x\in G, .\dfrac{\partial u^\epsilon}{\partial \gamma^\epsilon}(x)|_{\partial G}=0$$ is considered in this paper. The operators $L_0$ and $L_1$ are self-adjoint second order operators. We assume that $L_0$ has a non-negative characteristic form and $L_1$ is strictly elliptic. The reflection is with respect to inward co-normal unit vector $\gamma^\epsilon(x)$. The behavior of $\lim\limits_{\epsilon\downarrow 0}u^\epsilon(x)$ is effectively described via the solution of an ordinary differential equation on a tree. We calculate the differential operators inside the edges of this tree and the gluing condition at the root. Our approach is based on an analysis of the corresponding diffusion processes.