[Paper Review] Asymptotic analysis of a boundary-value problem with the nonlinear boundary multiphase interactions in a perforated domain

Taras Mel’nyk, O. A. Sivak|ArXiv.org|Jun 13, 2008

Advanced Mathematical Modeling in Engineering8 references3 citations

TL;DR

This paper presents an asymptotic analysis of a second-order elliptic boundary-value problem with nonlinear Robin-type boundary conditions on periodically perforated domains, where two distinct sets of holes impose different nonlinear interactions. Using a variant of Zhikov's approach without extension operators, the authors prove convergence of the solution and energy integral as the perforation parameter ε → 0, derive asymptotic approximations with sharp error estimates in H¹, and establish homogenized limits involving effective coefficients and averaged nonlinearities on the macroscopic scale.

ABSTRACT

We consider a boundary-value problem for the second order elliptic differential operator with rapidly oscillating coefficients in a domain $Ω_ε$ that is $ε-$periodically perforated by small holes. The holes are divided into two $ε-$periodical sets depending on the boundary interaction at their surfaces. Therefore, two different nonlinear Robin boundary conditions $σ_ε(u_ε) + εκ_{m} (u_ε) = εg^{(m)}_ε, m=1, 2,$ are given on the corresponding boundaries of the small holes. The asymptotic analysis of this problem is made as $ε o0,$ namely the convergence theorem both for the solution and for the energy integral is proved without using extension operators, the asymptotic approximations both for the solution and for the energy integral are constructed and the corresponding error estimates are obtained.

Motivation & Objective

• To analyze the asymptotic behavior of solutions to a nonlinear elliptic boundary-value problem in a domain perforated with small holes.
• To study the impact of two distinct nonlinear Robin-type boundary conditions on different sets of holes in a periodic microstructure.
• To derive a homogenized limit problem for the solution and energy integral as the perforation parameter ε → 0.
• To construct asymptotic approximations for the solution and energy integral with rigorous error estimates in the H¹ norm.
• To establish convergence results and error bounds without relying on Sobolev extension operators, using Zhikov’s method and unfolding techniques.

Proposed method

• The problem is formulated in a perforated domain Ωε with ε-periodically distributed holes, divided into two sets with distinct nonlinear boundary conditions.
• The differential operator has rapidly oscillating, 1-periodic coefficients a_ij(x/ε), satisfying ellipticity and symmetry conditions.
• Weak solutions are defined via integral identities involving flux terms and nonlinear boundary terms εκm(uε) on ∂Bε(m).
• The analysis uses a modified version of Zhikov’s approach, avoiding Sobolev extension operators, and adapts techniques from [13] on thick junctions.
• The homogenization process involves correctors based on periodic solutions to cell problems in the perforated unit cell Q₀.
• Error estimates are derived by decomposing the solution into macroscopic and oscillatory components and estimating residual terms via periodicity and averaging.

Experimental results

Research questions

• RQ1 How does the solution uε of the boundary-value problem behave as the perforation parameter ε tends to zero?
• RQ2 What is the form of the homogenized limit problem for the solution and energy integral in the presence of two distinct nonlinear boundary interactions?
• RQ3 Can convergence and error estimates be established without using extension operators in Sobolev spaces?
• RQ4 How do the nonlinear boundary conditions κ₁(uε) and κ₂(uε) on different hole sets affect the macroscopic behavior?
• RQ5 What are the asymptotic approximations for the solution and energy integral, and what are the corresponding error bounds in H¹(Ω)?

Key findings

• The solution uε converges to a limit v₀ in H¹(Ω) as ε → 0, with convergence rate O(ε¹/²) under the given assumptions.
• The energy integral ∫Ωε a_ij^ε ∂x_j uε ∂x_i uε dx converges to the homogenized energy ∫Ω â_ij ∂x_j v₀ ∂x_i v₀ dx with error O(ε¹/² + ε‖f₀‖L² + ‖fε − f₀‖L² + ∑‖gε⁽ᵐ⁾ − g₀⁽ᵐ⁾‖L²).
• Asymptotic approximations for uε and the energy integral are constructed using corrector functions derived from periodic cell problems in the perforated unit cell Q₀.
• The error in the H¹-norm for the solution approximation is bounded by O(ε¹/² + ‖fε − f₀‖L² + ∑‖gε⁽ᵐ⁾ − g₀⁽ᵐ⁾‖L²), proving convergence without extension operators.
• The nonlinear boundary terms εκm(uε) contribute to the macroscopic limit through averaged coefficients involving |Sm| and the mean of qm = |Sm|/|Q₀|.
• The homogenized problem features an effective elliptic operator with coefficients â_ij derived from the periodic structure and cell problems, and nonlinear boundary conditions with averaged nonlinearities.