On Second Order Elliptic Equations with a Small Parameter

The Neumann problem with a small parameter (Formula presented.) is considered in this paper. The operators L₀and L₁are self-adjoint second order operators. We assume that L₀has a non-negative characteristic form and L₁is strictly elliptic. The reflection is with respect to inward co-normal unit vector γ^ε(x). The behavior of lim_ε↓0u^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.