On second order elliptic equations with a small parameter

Abstract

The Neumann problem with a small parameter $$(\frac{1}{\epsilon }{L}_0+L_1)u^{\epsilon }(x)=f(x)\text{for}x\in G,.\frac{\partial u^{\epsilon }}{\partial {\gamma }^{\epsilon }}(x){|}_{\partial G}=0$$ 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 $\underset{\epsilon \downarrow 0}{\lim }{u}^{\epsilon }(x)$ 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.

28 pages, 1 figure, revised version

References in corpus (1)

• Averaging of Hamiltonian flows with an ergodic component

Cited by in corpus (2)

• Wave propagation for reaction-diffusion equations on infinite random trees
• On the long-time behavior of a perturbed conservative system with degeneracy