Homogenization of a Differential Operator on a Fine Periodic Net of Curves

Abstract

Here we deal with the homogenization of an ordinary differential operator on an ε-periodic net of curves in ℝⁿ. The role of the operator homogenized is played by a partial differential operator acting on functions in n variables. As a model we take the problem of the propagation of heat in a fine wire-cloth. On its segments there are given some ordinary second order differential equations. At the nodes, the sum of the flows rates is equal to zero. Finally, we impose the homogeneous Dirichlet condition at the boundary points.