On the Friedrichs inequality in a domain perforated aperiodically along the boundary. Homogenization procedure. Asymptotics for parabolic problems

On Friedrichs-type inequalities in domains rarely perforated along the boundary

“…Using the same arguments as in [4], it follows that λ 1 ε > 0. This together with the variational formulation of the smallest eigenvalue of (2) lead to the following Friedrichs-type inequality for functions u H 1 (Ω) which vanish on T ε μ :…”

“…We will now consider how to estimate the difference between the best constants, K 0 and K ε , in the inequalities (7) and (4). First, observe that by Proposition 1 and (5) we have that…”

“…where the best constant is given by A geometrical proof of that K ε K 0 was presented in [4] for the case p = ∞. The goal of this article is to answer the following questions, in the case 0 <p < ∞: (1) Is there a Friedrichs-type inequality related to the limit Robin boundary value problem?…”

“…Similar questions, with different types of microheterogeneities in a neighborhood of the boundary, were studied in [1][2][3][4][5]. In [1] (see also [2]), domains with a periodical perforation along the boundary were considered and the precise asymptotics of the best constant in a Friedrichs-type inequality was established.…”

“…It was assumed that the size of perforation and the period were of the same order. Two different cases with non-periodical perforation were considered in [4,5]. The convergence of the constant, as the size of perforation tends to zero, to the constant in the limit inequality was proved.…”