Homogenization of a vertical oscillating Neumann condition

TL;DR

This paper studies the homogenization of PDEs with oscillating Neumann boundary conditions, revealing a novel anisotropic pinning effect and establishing a comparison principle for the resulting boundary condition.

Contribution

It introduces a new singularly anisotropic pinned Neumann condition and proves homogenization using viscosity solutions, extending rate-independent pinning phenomena to higher-dimensional PDEs.

Findings

Discovery of a pinning effect at zero tangential slope.

Establishment of a comparison principle for the new boundary condition.

Demonstration of rate-independent pinning in multi-dimensional PDEs.

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.