Effective transmission conditions for second-order elliptic equations on networks in the limit of thin domains

Pierre-Louis Lions,Panagiotis E Souganidis

doi:10.5802/crmath.83

Abstract

We consider star-shaped tubular domains consisting of a number of non intersecting semi-infinite strips of small thickness that are connected by a central region of diameter proportional to the thickness of the strips. At the thin-domain limit, the region reduces to a network of half-lines with the same end point (junction). We show that the solutions of uniformly elliptic partial differential equations set on the domain with Neumann boundary conditions converge, in the thin-domain limit, to the unique solution of a second-order partial differential equation on the network satisfying an effective Kirchhoff-type transmission condition at the junction. The latter is found by solving an “ergodic”-type problem at infinity obtained after a first-order blow up at the junction.

Highlights

The aim of the paper is to study the asymptotic behavior of solutions of uniformly elliptic partial differential equation set on thin tubular domains around a fixed network with one junction
Since the limiting function is not smooth at the junction, the problem can be thought as a singular perturbation one, the transmission condition providing the necessary balance for the derivatives
The first claim follows by an argument similar to the one at the beginning of the proof since it involves only the equation satisfied on each Gi. For (53), we first observe that the boundary condition is immediate from the analogous property of v , while arguing as if v where smooth, we find that in W R0

Summary

Introduction

The aim of the paper is to study the asymptotic behavior of solutions of uniformly elliptic partial differential equation (pde for short) set on thin tubular domains around a fixed network with one junction. Since the sought after condition involves derivatives of the limiting solution at the junction, it is natural to use a first-order blow up at the origin This leads to a problem in an unscaled domain. The proof of Theorem 2 is based on solving an ergodic-type problem with Neumann conditions in a truncated domain followed by a delicate analysis of what happens as the truncation is removed. This is the place where the compatibility condition arises. We assume that F is independent of y and uniformly elliptic, Lipschitz continuous, and 1−positively homogeneous, F (0, x) = 0.

The proof of Theorem 2

The proof of Theorem 1