Operator estimates for non-periodically perforated domains with Dirichlet and nonlinear Robin conditions: vanishing limit

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

“…In the case when the Robin condition is present and is linear, these are operator estimates describing the norm resolvent convergence of the perturbed operator to the homogenized one. Similar operator estimates were already established in the case of only Dirichlet condition on the boundaries of the cavities, when the solution vanishes in the limit [21], and also in the case of the Neumann condition on the boundaries of the cavities [22], when the cavities disappear under the homogenization. And in the present paper, we show that the operator estimates are also present in a much more nontrivial situation, when a strange term appears.…”

“…This is a standard estimate for the L 2 -norm of the trace of a function from W 1 2 -norm, and we additionally state that it is uniform on 𝜀 and k. The latter statement is implied by a similar estimate for u ∈ B R 3 (0)∖𝜔 k,𝜀 established in the proof of Lemma 3.5 in [21]. The mentioned estimate from [21] was established by passing to local variables (𝜏, s) in the vicinity of the boundaries 𝜕 ωk,𝜀 , the existence of which is ensured by Assumption A.1, and then the proof follows standard lines. In this proof, also Lemma 3.2 from [21] is to be used, which provides an estimate…”

“…Proof. Assumption A.2 guarantees that estimate (2.5) in [21] is satisfied, and hence, by Lemma 3.7 in [21], there exists…”

“…The mentioned estimate from [21] was established by passing to local variables (𝜏, s) in the vicinity of the boundaries 𝜕 ωk,𝜀 , the existence of which is ensured by Assumption A.1, and then the proof follows standard lines. In this proof, also Lemma 3.2 from [21] is to be used, which provides an estimate…”

“…Operator estimates in such case were obtained, and the convergence rates were shown to be order sharp. The case of the Neumann condition on the boundaries of the cavities was treated in [22]. It was shown that in this case, the cavities disappear under the homogenization, and appropriate operator estimates were established.…”