On second order elliptic equations with a small parameter

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The Neumann problem with a small parameter (1εL₀+L₁)uε(x)=f(x) for x∈ G,.∂ uε/∂ γε(x)|∂ G=0 is considered in this paper. The operators L₀ and L₁ are self-adjoint second order operators. We assume that L₀ has a non-negative characteristic form and L₁ is strictly elliptic. The reflection is with respect to inward co-normal unit vector γε(x). The behavior of ε 0uε(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.