Homogenization of a Vertical Oscillating Neumann Condition

Abstract

We homogenize the Laplace and heat equations with the Neumann data oscillating in the “vertical" u-variable. These are simplified models for interface motion in heterogeneous media, particularly capillary contact lines. The homogenization limit reveals a pinning effect at zero tangential slope, leading to a novel singularly anisotropic pinned Neumann condition. The singular pinning creates an unconstrained contact set generalizing the contact set in the classical thin obstacle problem. We establish a comparison principle for the heat equation with this new type of boundary condition. The comparison principle enables a proof of homogenization via the method of half-relaxed limits from viscosity solution theory. Our work also demonstrates – for the first time in a PDE problem in multiple dimensions – the emergence of rate-independent pinning from gradient flows with wiggly energies. Prior limit theorems of this type, in rate independent contexts, were limited to ODEs and PDEs in one dimension.