On second order elliptic equations with a small parameter

Abstract

The Neumann problem with a small parameter $$(\dfrac{1}εL_0+L_1)u^ε(x)=f(x) \text{for} x\in G, .\dfrac{\partial u^ε}{\partial γ^ε}(x)|_{\partial G}=0$$ 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 $γ^ε(x)$. The behavior of $\lim\limits_{ε\downarrow 0}u^ε(x)$ 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.