On Second Order Elliptic Equations with a Small Parameter
Mark Iosifovich Freidlin, Wenqing Hu · Communications in Partial Differential Equations · 2013
The Neumann problem with a small parameter 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 γϵ(x). The behavior of lim ϵ↓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.