A classical approach for the $p$-Laplacian in oscillating thin domains

Roughness‐induced effects on the thermomicropolar fluid flow through a thin domain

“…which is the second estimate (24). By using the Poincaré inequality, we get the first estimate in (24).…”

“…which is the second estimate (24). By using the Poincaré inequality, we get the first estimate in (24). For the microrotation, from (33), we get the estimates of the microrotation (25).…”

“…We extend each unknown in the following: –From the boundary conditions satisfied by $$\tilde{\bf u}^\varepsilon$$ and $$\tilde{w}^{\varepsilon }$$, we extend them by zero in $$\Omega \setminus \widetilde{\Omega }^{\varepsilon }$$ and denote the extensions by $$ \tilde{\bf U}^\varepsilon$$ and $$\tilde{W}^{\varepsilon }$$, respectively. –For the temperature $$\tilde{T}^\varepsilon$$, we use the extension operator described in Ref. [24, Lemma 2.3] called $$\mathcal {P}^\varepsilon$$, which allows us to extend functions from $$H^1(\widetilde{\Omega }^\varepsilon )$$, which are zero on the lateral boundaries, to $$H^1(\Omega )$$. Moreover, this extension satisfies …”

“…-For the temperature T𝜀 , we use the extension operator described in Ref. [24,Lemma 2.3] called  𝜀 , which allows us to extend functions from 𝐻 1 ( Ω𝜀 ), which are zero on the lateral boundaries, to 𝐻 1 (Ω). Moreover, this extension satisfies…”

The p-Laplacian in thin channels with locally periodic roughness and different scales*

“…where g is a positive, bounded and periodic function satisfying some regularity hypothesis and ε > 0 is a small parameter which goes to zero. Thereby, in the limit ε → 0, the open set Q ε degenerates to the unit interval presenting oscillatory behaviour on the upper boundary (see for instance [1,3,5,[18][19][20][21][22] where similar approach are performed). The periodic rough boundary considered above is certainly a first step, but usually not enough, since most of the irregularities present in real applications are not periodic.…”