Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A not-so-strange term coming from somewhere

Giacomo Canevari, Kirill Cherednichenko, Arghir Zarnescu

TL;DR

This work develops a rigorous homogenisation framework for the Laplacian in periodically perforated domains with Robin boundary conditions on the holes, where the Robin term scales with the total hole surface. In a critical regime, the homogenised equation acquires a nonlinear zero-order term determined by a Steklov-type spectral problem in which the eigenvalue appears in both the bulk and boundary equations; the resulting "+strange term" connects to the classical capacitary term in the Dirichlet limit as the Robin parameter grows. The authors establish an abstract variational setting, prove the convergence and structure of the limit via a family of spectral problems parameterised by a boundary-penalty variable \(\kappa\), and derive trace inequalities essential for the homogenisation analysis. In a detailed model example, they obtain explicit expressions for the limit coefficient in the spherical-hole case and discuss subcritical and supercritical regimes, illustrating how the micro-geometry of perforations governs macroscopic behaviour. The results illuminate how energy balance and boundary effects interplay to bridge Dirichlet and Robin homogenisation theories, with potential applications to composite materials and surface-energy-influenced media.

Abstract

We consider Laplace's equation in a periodically perforated domain, with Robin boundary conditions on the holes and a Robin coefficient inversely proportional to the total surface area of the holes. We show that, in a certain critical regime, the homogenised equation contains an additional zero-order term, which depends nonlinearly on the Robin coefficient. This additional term is defined in terms of solutions to a Steklov-type spectral problem, where the eigenvalue appears both in the equation and the boundary conditions. The ``strange term'' for the problem with Robin conditions converges to zero as the Robin parameter tends to zero and, at least under suitable assumptions, it converges to the capacitary ``strange term'' for the Dirichlet problem as the Robin parameter tends to infinity.

A not-so-strange term coming from somewhere

TL;DR

This work develops a rigorous homogenisation framework for the Laplacian in periodically perforated domains with Robin boundary conditions on the holes, where the Robin term scales with the total hole surface. In a critical regime, the homogenised equation acquires a nonlinear zero-order term determined by a Steklov-type spectral problem in which the eigenvalue appears in both the bulk and boundary equations; the resulting "+strange term" connects to the classical capacitary term in the Dirichlet limit as the Robin parameter grows. The authors establish an abstract variational setting, prove the convergence and structure of the limit via a family of spectral problems parameterised by a boundary-penalty variable , and derive trace inequalities essential for the homogenisation analysis. In a detailed model example, they obtain explicit expressions for the limit coefficient in the spherical-hole case and discuss subcritical and supercritical regimes, illustrating how the micro-geometry of perforations governs macroscopic behaviour. The results illuminate how energy balance and boundary effects interplay to bridge Dirichlet and Robin homogenisation theories, with potential applications to composite materials and surface-energy-influenced media.

Abstract

We consider Laplace's equation in a periodically perforated domain, with Robin boundary conditions on the holes and a Robin coefficient inversely proportional to the total surface area of the holes. We show that, in a certain critical regime, the homogenised equation contains an additional zero-order term, which depends nonlinearly on the Robin coefficient. This additional term is defined in terms of solutions to a Steklov-type spectral problem, where the eigenvalue appears both in the equation and the boundary conditions. The ``strange term'' for the problem with Robin conditions converges to zero as the Robin parameter tends to zero and, at least under suitable assumptions, it converges to the capacitary ``strange term'' for the Dirichlet problem as the Robin parameter tends to infinity.

Paper Structure

This paper contains 17 sections, 26 theorems, 251 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Statement of the problem.
  3. The homogenisation limit (in the model example).
  4. An abstract setting and the model example
  5. Description of the abstract setting
  6. The model example fits into the abstract setting
  7. Properties of $\lambda_*(\kappa)$
  8. Analysis of Problem \ref{['eigenvalue']} at fixed $\varepsilon > 0$
  9. Analysis of Problem \ref{['eigenvalue']} as $\varepsilon\to 0$
  10. Proof of Theorem \ref{['th:lambda*']} and of Proposition \ref{['prop:lambda0']}
  11. The "strange term": proof of Theorem \ref{['th:strange_term']}
  12. Trace inequalities
  13. Proof of Theorem \ref{['th:strange_term']}
  14. The model example revisited
  15. The model example satisfies \ref{['hp:first']}--\ref{['hp:last']}
  16. ...and 2 more sections

Key Result

Theorem 1

Let $\mathscr{P}_\varepsilon$ be defined as in model_example--gamma_intro, with $\mu_\varepsilon$ as in mu_eps. Then, for any $f\in H^{-1}(\Omega)$ and any $\alpha\geq 0$, $\beta > 0$, the unique solution $u_\varepsilon$ to main_eq satisfies

Figures (1)

  • Figure 1: The domain and the unit periodicity cell $Y$.

Theorems & Definitions (63)

  • Theorem 1
  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Proposition 2.8
  • Proposition 2.9
  • ...and 53 more