Abstract

We study the asymptotic behavior of the solution to the diffusion equation in a planar domain, perforated by tiny sets of different shapes with a constant perimeter and a uniformly bounded diameter, when the diameter of a basic cell, \(\varepsilon \), goes to 0. This makes the structure of the heterogeneous domain aperiodical. On the boundary of the removed sets (or the exterior to a set of particles, as it arises in chemical engineering), we consider the dynamic unilateral Signorini boundary condition containing a large-growth parameter \(\beta (\varepsilon )\). We derive and justify the homogenized model when the problem’s parameters take the “critical values”. In that case, the homogenized problem is universal (in the sense that it does not depend on the shape of the perforations or particles) and contains a “strange term” given by a non-linear, non-local in time, monotone operator H that is defined as the solution to an obstacle problem for an ODE operator. The solution of the limit problem can take negative values even if, for any \(\varepsilon \), in the original problem, the solution is non-negative on the boundary of the perforations or particles.

中文翻译：

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具有临界直径的非周期等周平面均匀化：动态单边边界条件的通用非局部奇异项

摘要

我们研究平面域中扩散方程解的渐近行为，平面域由具有恒定周长和均匀有界直径的不同形状的微小集合穿孔，当基本单元的直径 \(\ varepsilon \)时，为0。这使得异构域的结构成为非周期性的。在移除集合的边界上（或一组粒子的外部，如化学工程中出现的那样），我们考虑包含大增长参数 \( \beta (\varepsilon )\) 的动态单边 Signorini 边界条件。当问题的参数取“临界值”时，我们推导并证明同质化模型的合理性。在这种情况下，均质化问题是普遍的（在某种意义上，它不依赖于穿孔或颗粒的形状），并且包含由非线性、非时间局部、单调算子 H 给出的“奇怪项”它被定义为 ODE 算子障碍问题的解。极限问题的解可以取负值，即使对于原始问题中的任何\(\varepsilon \)，解在穿孔或颗粒的边界上是非负的。