Homogenization of Variational Inequalities for the p-Laplace Operator in Perforated Media Along Manifolds

Abstract

We address homogenization problems of variational inequalities for the p-Laplace operator in a domain of $\mathbb {R}^n$ ($n\ge$ 3, $p\in [2,n)$) periodically perforated by balls of radius $O(\varepsilon ^\alpha )$ where $\alpha >1$ and $\varepsilon$ is the size of the period. The perforations are distributed along a $(n-1)$-dimensional manifold $\gamma$, and we impose constraints for solutions and their fluxes (associated with the p-Laplacian) on the boundary of the perforations. These constraints imply that the solution is positive and that the flux is bounded from above by a negative, nonlinear monotonic function of the solution multiplied by a parameter $\varepsilon ^{-\kappa }$, $\kappa \in \mathbb {R}$ and $\varepsilon$ is a small parameter that we shall make to go to zero. We analyze different relations between the parameters $p, \, n, \, \varepsilon , \, \alpha$ and $\kappa$, and obtain homogenized problems which are completely new in the literature even for the case $p=2$.

Appendix

In this section, we introduce some results useful for proofs. The first result provides the existence and uniqueness of the solution of the functional equation (13) arising in the homogenized problem (11)–(12a) while the second one simplifies the computations throughout the paper. The proofs of both results can be found in [5] (cf. Propositions 2.2 and 2.3, respectively).

Lemma 1

Let $p\ge 2.$ Let $\varrho$ be a strictly positive constant and let $\sigma$ be the function $\sigma (x,u)$ defined from $\overline{\Omega }\times \mathbb {R}$ into $\mathbb {R}$ which is assumed to be a continuously differentiable function in $\overline{\Omega }\times \mathbb {R}$ satisfying (1)–(2). Then, the equation

$$\begin{aligned} |H|^{p-2}H =\varrho \, \sigma (x,\tau -H ) \end{aligned}$$

has a unique solution $H(x,\tau )$ which is a continuously differentiable function in $\overline{\Omega }\times (\mathbb {R}\setminus \{0\})$ and continuous in $\overline{\Omega }\times \mathbb {R},$ and satisfies $H(x,0)=0$ and

$$\begin{aligned}&(|H(x,u)|^{p-2}H(x,u)-|H(x,v)|^{p-2}H(x,v))(u-v)\ge \widetilde{k}_{1}|u-v|^{p}, \end{aligned}$$

(52)

$$\begin{aligned}&\quad |H(x,u)|\le |u|, \end{aligned}$$

(53)

for all $x\in \overline{\Omega }, \,u,v\in \mathbb {R}$ and a certain constant $\widetilde{k}_{1}>0$.

Lemma 2

Let $p\ge 2$. Let $v\in W^{1,\infty }(\Omega )$, $\varphi \in W^{1,p}(\Omega , \partial \Omega )$ and $\eta _\varepsilon \in W^{1,p}(\Omega , \partial \Omega )$ such that $\Vert {\nabla \eta _\varepsilon }\Vert _{L^m(\Omega )}\rightarrow 0$, as $\varepsilon \rightarrow 0$, for $m\in [1,p)$. Then,

$$\begin{aligned}&\lim _{\varepsilon \rightarrow 0} \int _{\Omega _\varepsilon } \Big (|\nabla (v+\eta _\varepsilon )|^{p-2}\nabla (v+\eta _\varepsilon )- |\nabla v|^{p-2}\nabla v\Big )\nabla \varphi \, dx \nonumber \\&\quad = \lim _{\varepsilon \rightarrow 0} \int _{\Omega _\varepsilon } |\nabla \eta _\varepsilon |^{p-2}\nabla \eta _\varepsilon \nabla \varphi \, dx. \end{aligned}$$

(54)

In addition, (54) also holds in the case where $\varphi$ depends on $\varepsilon$, namely $\varphi \equiv \varphi _\varepsilon$, with $\Vert {\nabla \varphi _\varepsilon }\Vert _{L^p(\Omega )}$ bounded independently of $\varepsilon$.

Finally, we introduce the following auxiliary estimates where the constant K does not depend on $\varepsilon$ nor on the functions w:

Lemma 3

Let $P^{j}_{\varepsilon }$ be the center of the ball $G^{j}_{\varepsilon }$ and let $T^{j}_{\varepsilon /4}$ denote the ball of radius $\varepsilon /4$ with center $P^{j}_{\varepsilon }$, $j\in \Upsilon _{\varepsilon }$. Then,

$$\begin{aligned} \left| \sum \limits _{j\in \Upsilon _{\varepsilon }}\int \limits _{\partial {T^{j}_{\varepsilon /4}}} w \, ds- 2^{2-2n} \omega _{n}\int \limits _{\gamma } w \, d\hat{x}\right| \le K\varepsilon ^{1/2}\Vert w \Vert _{H^{1}(\Omega )}, \quad w\in H_0^1(\Omega ). \end{aligned}$$

See Lemma 1 in [16] for the proof.

Lemma 4

Let $\Pi _{\varepsilon }=\Omega \cap \{-{\varepsilon }/2<x_{1}<{\varepsilon }/2\}$. Then,

$$\begin{aligned} \left| \frac{1}{\varepsilon }\int \limits _{\Pi _\varepsilon } w^2 \, dx-\int \limits _{\gamma } w^2 \, d\hat{x}\right| \le K\varepsilon ^{1/2} \Vert \nabla w \Vert ^2_{L^2(\Omega )}, \quad w\in H_0^1(\Omega ). \end{aligned}$$

See Lemma 2.6 in [8] for precise references for the proof.

Lemma 5

Let $\Pi _{\varepsilon }=\Omega \cap \{-{\varepsilon }/2<x_{1}<{\varepsilon }/2\}$. Let $w\in W^{1,p}(\Omega )$, $2\le p <n$. Then,

$$\begin{aligned}&\Vert w\Vert _{L^p(G_\varepsilon )}^p \le K \left( a_\varepsilon \Vert w\Vert _{L^p(S_\varepsilon )}^p +a_\varepsilon ^p\Vert \nabla w\Vert _{L^p(G_\varepsilon )}^p\right) , \quad \text{ and }\\&\quad \Vert w\Vert _{L^p(\Pi _\varepsilon \setminus G_\varepsilon )}^p \le K \big (a_\varepsilon ^{1-n}\varepsilon ^{n} \Vert w\Vert _{L^p(S_\varepsilon )}^p +a_\varepsilon ^{p-n}\varepsilon ^{n}\Vert \nabla w\Vert _{L^p(\Omega )}^p\big ). \end{aligned}$$

See Theorem 5.1 in [8] and Lemma 2.6 in [5] related to the proofs of the first and the second inequality respectively.