Operator estimates for homogenization of the Robin Laplacian in a perforated domain

Operator estimates for Neumann sieve problem

“…Originally this toolbox was developed to handle convergence of the Laplace-Beltrami operator on manifolds which shrink to a graph. It also has shown to be effective for homogenization problems in perforated domains, see [2,30,31]. The main ingredient of the proof is a construction of a suitable operator from H 1 (Ω \ Γ) to H 1 (Ω ε ) (these spaces are the domains of the sesquilinear forms associated with the operators A ε and A ).…”

“…The case when holes are distributed in the entire domain was addressed in [43,46,47] (Neumann conditions on the boundary of holes), [2,31] (Dirichlet conditions, also [2] treats Neumann holes), [30] (linear Robin conditions), [11,17] (Dirichlet and nonlinear Robin conditions). In [30,43,44] the holes are distributed periodically (in [31] -locally periodically), and are identical, while in [2,11,17] quite general assumptions on sizes and location of holes are imposed. The surface distribution of holes was treated in [15,18,19,24].…”

“…Further we will apply(3.19) for the largest p satisfying(3.18). For n = 4 we are not able to choose the largest p (in the dimension 4 the embedding H 2 → L p holds for any p < ∞, but not for p = ∞), and in this case we need the following estimate on the constant C 4,p in the right-hand-side of(3.19) [30, Lemma 4.4]:…”

Operator estimates for non‐periodically perforated domains with Dirichlet and nonlinear Robin conditions: Strange term

“…In [11][12][13][14][15][16], the classical homogenization results were improved, and operator estimates were established for several cases of periodic and almost periodic perforation in arbitrary domains. The case of the Neumann condition was addressed in [14][15][16], the sizes of the cavities were of the same order as the distances between them, and the perforation was purely periodic.…”

“…All cavities were of the same shapes up to an arbitrary rotation, and its location in the periodicity cell was also quite arbitrary. In [11, 12], the perforation was pure periodic, and it was made by small balls with the Dirichlet or Neumann [11] or Robin [12] condition on the boundaries. The main results of the cited papers were the formulation of the homogenized problems and various operator estimates; their order sharpness was not established.…”