The $$p\,$$ p -Laplacian equation in a rough thin domain with terms concentrating on the boundary
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The p‑Laplacian equation in a rough thin domain with terms concentrating on the boundary Ariadne Nogueira1 · Jean Carlos Nakasato1 Received: 1 August 2019 / Accepted: 11 January 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this work, we use reiterated homogenization and unfolding operator approach to study the asymptotic behavior of the solutions of the p-Laplacian equation with Neumann boundary conditions set in a rough thin domain with concentrated terms on the boundary. We study weak, resonant and high roughness, respectively. In the three cases, we deduce the effective equation capturing the dependence on the geometry of the thin channel and the neighborhood where the concentrations take place. Keywords p-Laplacian · Neumann boundary condition · Thin domains · Homogenization Mathematics Subject Classification 35B25 · 35B40 · 35J92
1 Introduction In this work, we are interested in analyzing the asymptotic behavior of solutions of a quasilinear elliptic problem posed in a family of thin domains R𝜀 with forcing terms concentrated on a neighborhood O𝜀 ⊂ R𝜀 of the boundary 𝜕R𝜀 . We assume { ( )} x , 0 < 𝜀 ≪ 1, R𝜀 = (x, y) ∈ ℝ2 ∶ 0 < x < 1, 0 < y < 𝜀g 𝛼 (1) 𝜀 for any 𝛼 > 0 , where (𝐇𝐠 ) g ∶ ℝ → ℝ is a strictly positive, Lipschitz, Lg-periodic function. Moreover, we define
g0 = min g(x) x∈ℝ
and
g1 = max g(x) x∈ℝ
* Ariadne Nogueira [email protected] Jean Carlos Nakasato [email protected] 1
Depto. de Matemática Aplicada, Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, São Paulo, SP, Brazil
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so that 0 < g0 ≤ g(x) ≤ g1 for all x ∈ ℝ. On the other hand, we set the narrow strip O𝜀 by ( )] { [ ( ) ( )} x x x O𝜀 = (x, y) ∈ ℝ2 ∶ 0 < x < 1, 𝜀 g 𝛼 − 𝜀𝛾 h 𝛽 < y < 𝜀g 𝛼 𝜀 𝜀 𝜀 with 𝛾 , 𝛽 and 𝛼 > 0 , and h ∶ ℝ → ℝ being a positive function of class C1 , Lh-periodic with bounded derivatives. Notice that parameters 𝛼 and 𝛽 set, respectively, the roughness order of the upper boundary of the thin domain R𝜀 and the singular shape of the 𝜀𝛾-neighborhood O𝜀 , whereas their profile is given by positive and periodic functions g and h. Finally, the parameter 𝛾 > 0 only establishes the order of the Lebesgue measure of O𝜀 with respect to R𝜀. We first analyze the solutions of the problem
∫R 𝜀
{|∇u𝜀 |p−2 ∇u𝜀 ∇𝜑 + |u𝜀 |p−2 u𝜀 𝜑}dxdy =
1 f 𝜀 𝜑dxdy, 𝜀𝛾 ∫O𝜀
𝜑 ∈ W 1,p (R𝜀 ),
(2)
which is the variational formulation of the quasilinear equation
⎧ 1 p−2 𝜀 𝜀 ⎪ −Δp u𝜀 + �u𝜀 � u𝜀 = 𝜀𝛾 𝜒O𝜀 f in R . ⎨ 𝜕u𝜀 ⎪ �∇u𝜀 �p−2 𝜀 = 0 on 𝜕R𝜀 𝜕𝜈 ⎩
(3)
Here, 𝜈 𝜀 denotes the unit outward normal to the boundary 𝜕R𝜀 and, for 1 < p < ∞ , Δp ⋅ is the p-Laplacian differential operator. Consider 𝜒O𝜀 the characteristic function of the set O𝜀 , � and we take forcing terms f 𝜀 ∈ Lp (R𝜀 ) for p′ > 0 , with 1∕p� + 1∕p = 1. Notice that R𝜀 ⊂ (0, 1) × (0, 𝜀g1 ) for all 𝜀 > 0 , degenerating to the unit interval as 𝜀 → 0 . Hence, it is reasonable to expect that the family of solutions u𝜀
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