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Homogenization of a vertical oscillating Neumann condition

William M Feldman, Zhonggan Huang

TL;DR

This work establishes a multi-dimensional homogenization framework for Laplace and heat equations with Neumann data oscillating in the vertical variable, revealing a singular, anisotropic pinned Neumann boundary in the limit. The authors develop a viscosity-solution-based approach, anchored by half-relaxed limits and a robust parabolic comparison principle, to derive the homogenized boundary condition $∂_t u=Δu$ in the bulk and $∂_1 u ∈ [L_*(∇'u),L^*(∇'u)]$ on the boundary, with explicit coefficients in both general and laminar cases. Central constructs include plane-like correctors, a pinning interval, and a foliation of steady states into extremal solutions, which together explain rate-independent hysteresis and the appearance of free facets. The results extend the theory of gradient flows with wiggly energies to multi-dimensional PDEs, providing rigorous links between microscopic boundary oscillations and macroscopic, rate-independent interface dynamics with potential applications to capillary contact lines and heterogeneous boundary interactions. The paper also proves a parabolic comparison principle for the homogenized problem, enabling a solid framework for convergence via half-relaxed limits and establishing long-time behavior toward extremal steady states under monotone boundary data.

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.

Homogenization of a vertical oscillating Neumann condition

TL;DR

This work establishes a multi-dimensional homogenization framework for Laplace and heat equations with Neumann data oscillating in the vertical variable, revealing a singular, anisotropic pinned Neumann boundary in the limit. The authors develop a viscosity-solution-based approach, anchored by half-relaxed limits and a robust parabolic comparison principle, to derive the homogenized boundary condition in the bulk and on the boundary, with explicit coefficients in both general and laminar cases. Central constructs include plane-like correctors, a pinning interval, and a foliation of steady states into extremal solutions, which together explain rate-independent hysteresis and the appearance of free facets. The results extend the theory of gradient flows with wiggly energies to multi-dimensional PDEs, providing rigorous links between microscopic boundary oscillations and macroscopic, rate-independent interface dynamics with potential applications to capillary contact lines and heterogeneous boundary interactions. The paper also proves a parabolic comparison principle for the homogenized problem, enabling a solid framework for convergence via half-relaxed limits and establishing long-time behavior toward extremal steady states under monotone boundary data.

Abstract

We homogenize the Laplace and heat equations with the Neumann data oscillating in the ``vertical" -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.

Paper Structure

This paper contains 34 sections, 49 theorems, 407 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Pinning, contact set and rate-independent system
  3. Literature
  4. Organization of the paper
  5. Acknowledgments
  6. Preliminaries
  7. Notations
  8. Touching, crossing and half relaxed limits
  9. General steady states
  10. The global energy minimizers
  11. Plane-like correctors
  12. Homogenization of general viscosity solutions
  13. The homogenized coefficients in the laminar case
  14. Homogenization of extremal solutions in laminar media
  15. A characterization of the extremal homogenized steady states in the laminar case
  16. ...and 19 more sections

Key Result

Theorem 1.1

Suppose $f$ is 1-periodic and regular on $\mathbb{R}$. Then a sequence $u^\varepsilon$ of solutions to eq.pNNep, with a fixed Dirichlet condition on the parabolic boundary $\partial_{\textup{p}}^{ +} (B_1^+\times[0,\infty))$, converge locally uniformly on $(x,t) \in \overline{B_1^+}\times[0,\infty)$

Figures (2)

  • Figure 1: The graph of $u^\varepsilon$ over $B_1^+$ is a moving interface interacting with an inhomogeneous medium via a Neumann condition at the boundary $B_1'$.
  • Figure 2: This figure formally illustrates the critical points of the energy $E_\varepsilon$ as defined in \ref{['eq.theenergyepsilonforlaminarcase']} and their homogenization. On the left and right are the extremal solutions, and they homogenize exactly to the extremal solutions of $L_\ast(\nabla' u)\le \partial_1 u \le L^\ast(\nabla' u)$. The global energy minimizers $u_\textup{glb}^\varepsilon$ homogenizes exactly to the standard Neumann problem $\partial_1 u = \langle f\rangle$.

Theorems & Definitions (124)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 114 more