Elliptic Equations in Weak Oscillatory Thin Domains: Beyond Periodicity with Boundary-Concentrated Reaction Terms

We are keen on examining the behavior of the solutions to certain Elliptic Partial Differential Equations which are posed in a oscillating thin domain $R^{\epsilon}$. This domain is a slender region in $\mathbb{R}^{2}$ displaying oscillations along its boundary. It is described as the region between two oscillatory functions, that is,

(1)

$$R^{\epsilon}=\Big{\{}(x,y)\in\mathbb{R}^{2}\;|\;x\in I\subset\mathbb{R},\;-\epsilon k^{1}(x,\epsilon)<y<\epsilon k^{2}(x,\epsilon)\Big{\}},$$

where $I=[a,b]$ is any interval in $\mathbb{R}$ and the functions $k^{1},k^{2}$ satisfy certain hypotheses, see (H) in the next section. These two functions may present periodic oscillation, see Figure 1 for a example, but also more intricate situations beyond the classical periodic setting, see Figure 2 for an example with quasi-periodic oscillating boundaries.

We are interested in analyzing the following linear elliptic equation with reaction terms concentrated in a very narrow oscillating neighborhood of the upper oscillating boundary

(2)

$$\left\{\begin{gathered}-\Delta w^{\epsilon}+w^{\epsilon}=\frac{1}{\epsilon^{\gamma}}\chi_{\theta^{\epsilon}}f^{\epsilon}\quad\textrm{ in }R^{\epsilon},\\ \frac{\partial w^{\epsilon}}{\partial\nu^{\epsilon}}=0\quad\textrm{ on }\partial R^{\epsilon},\end{gathered}\right.$$

where $f^{\epsilon}\in L^{2}(R^{\epsilon})$, and $\nu^{\epsilon}$ is the unit outward normal to $\partial R^{\epsilon}$. Let be $\chi_{\theta^{\epsilon}}$ the characteristic function of the narrow strip defined by

(3)

$$\theta^{\epsilon}=\Big{\{}(x,y)\in\mathbb{R}^{2}\;|\;x\in I\subset\mathbb{R},\;\epsilon[k^{2}(x,\epsilon)-\epsilon^{\gamma}H_{\epsilon}(x)]<y<\epsilon k^{2}(x,\epsilon)\Big{\}},$$

where $\gamma>0$ and $H_{\epsilon}:I\subset\mathbb{R}\to\mathbb{R}$ is a $C^{1}$ quasi-periodic and bounded function.

It is important to highlight that we will obtain the limiting problem explicitly, assuming very general conditions on oscillatory functions. Afterward, we will study this limiting problem in interesting specific cases, such as quasi-periodic and even almost periodic functions. The authors firmly believe that analyzing problems in homogenization beyond the classical periodic setting is compelling from both an applied and a purely mathematical standpoint. In the real world, many materials and structures defy perfect periodicity and many real world problems involve multiple scales of heterogeneity or complexity.

Homogenization, a powerful mathematical tool in the realm of materials science and engineering, has traditionally been confined to the study of materials with perfect periodic structures, see [1, 13, 18, 19]. However, a growing body of research suggests that venturing beyond this classical periodic setting offers a rich landscape of possibilities, both in terms of practical applications and as a playground for mathematical exploration. Therefore, over the years, different homogenization techniques have been adapted and created to study problems beyond the periodic setting. In this context, the almost periodic framework offers an appropriate answer to certain limitations presented by periodic models. To illustrate how this topic is at the forefront of research, we can cite several articles, knowing that there are many more, arranged from the most classic to the most recent, see [2, 16, 23, 25, 28].

In the literature, there exists a multitude of studies dedicated to exploring the impact that oscillatory boundaries and thickness of the domain have on the solution behaviors of partial differential equations posed in thin domains. For further insight into this subject, one may refer to the following selected references and the studies cited within them [4, 5, 10, 20, 27]. However, the volume of research that avoids the consideration of periodic boundaries is significantly less. In this regard, it is worth mentioning some articles have dealt with bounded domains with locally periodic boundaries, see [9, 12]. Equations set in thin domains with the upper and lower boundaries oscillating at different periods have also been studied. This means that the requirements for traditional homogenization are not met, as there is no cell that describes the domain. In fact, the behavior of the two boundaries is reflected in the limit. While [11] focuses on boundaries with strong oscillations, [3] explores weak oscillations, yielding results characteristic of quasi-periodic functions

On the other hand, numerous studies have also focused on singular elliptic and parabolic problems, characterized by potential and reactive terms that are localized within a narrow region close of the boundary. First, equations in fixed and bounded domains were studied, see [6, 21, 22]. Recently, problems combining both singular situations have been studied. That is, thin domains with oscillating boundaries with terms concentrated on the boundary, [7, 8, 24]. However, all of these fall within the assumptions of periodicity.

The paper is organized as follows: In Section 2, we establish the necessary hypotheses, define the notation, and present the central result of the article, which reveals the explicit limit problem under general hypotheses about the oscillating boundaries. In Section 3, adapting the technique introduced in [3], we will show that after appropiate change of variables the solutions of the problem (2) can be aproximated by a one-dimensional problem with oscillating coefficients and with a force term reflecting the reaction terms on the thin neighborhood of the upper boundary. In Section 4, we proceed to analyze the limit of the one-dimensional equation derived in the previous section, thereby proving the main result of the paper. In Section 5, we derive the limit problem for specific cases of oscillating functions, including quasi-periodic and almost-periodic types. Finally, Section 6, we show some numerical evidences about the proved results.

Let $k^{i}$ be a function, $i=1,2$,

$$\begin{array}[]{rccl}k^{i}:&I\times(0,1)&\longrightarrow&\mathbb{R}^{+}\\ &(x,\epsilon)&\longrightarrow&k^{i}(x,\epsilon)=k_{\epsilon}^{i}(x),\end{array}$$

such that

• (H.1)

  $k_{\epsilon}^{i}$ is a $C^{1}$ function and

  (4)

  $$\epsilon\Big{|}\frac{d}{dx}k^{i}_{\epsilon}(x)\Big{|}\buildrel\epsilon\to 0\over{\longrightarrow}0\hbox{ uniformly in }I.$$

• (H.2)

  There exist two positive constants independent of $\epsilon$ such that

  (5)

  $$0<C^{i}_{1}\leq k_{\epsilon}^{i}(\cdot)\leq C^{i}_{2}.$$

• (H.3)

  There exists a function $K^{i}$ in $L^{2}(I)$ such that

  $$k^{i}_{\epsilon}\stackrel{{\scriptstyle\epsilon\to 0}}{{\rightharpoonup}}K^{i}\quad\hbox{ w}-L^{2}(I).$$

• (H.4)

  There exists a function $P$ in $L^{2}(I)$ such that

  $$\frac{1}{k^{1}_{\epsilon}+k^{2}_{\epsilon}}\stackrel{{\scriptstyle\epsilon\to 0}}{{\rightharpoonup}}P\quad\hbox{ w}-L^{2}(I).$$

Indeed, the thickness of the domain has order $\epsilon$ and we say that the domain presents weak oscillations due to the convergence (4).

We will assume that the function which defines the narrow strip is a quasi-periodic smooth function. There are constants $H_{0},H_{1}\geq 0$ such that

(6)

$$H_{0}\leq H_{\epsilon}(x)\leq H_{1}\,\,\mbox{for all}\,\,x\in I\,\,\mbox{and}\,\,\epsilon>0.$$

First, we present the homogenized problem for the general case of functions satisfying the hypotheses (H). Subsequently, we direct our attention to several intriguing examples that provide insights into this matter. In particular, our overarching framework allows us consider quasi-periodic or almost periodic functions. See Figure 2 where quasi-periodic functions are considered. That is the case where

(7)

$$k^{1}_{\epsilon}(x)=h(x/\epsilon^{\alpha}),\quad k^{2}_{\epsilon}(x)=g(x/\epsilon^{\beta}),$$

with $0<\alpha,\beta<1$ and the functions $g,h\,:\mathbb{R}\to\mathbb{R}$ are $C^{1}$ quasi-periodic functions. Note that the extensively studied structure known as the periodic setting is included within this framework.

The variational formulation of (2) is the following: find $w^{\epsilon}\in H^{1}(R^{\epsilon})$ such that

$$\int_{R^{\epsilon}}\Big{\{}\displaystyle\frac{\partial w^{\epsilon}}{\partial x}\displaystyle\frac{\partial\varphi}{\partial x}+\displaystyle\frac{\partial w^{\epsilon}}{\partial y}\displaystyle\frac{\partial\varphi}{\partial y}+w^{\epsilon}\varphi\Big{\}}dxdy=\frac{1}{\epsilon^{\gamma}}\int_{\theta^{\epsilon}}f^{\epsilon}\varphi dxdy,\,\forall\varphi\in H^{1}(R^{\epsilon}).$$

Observe that, for fixed $\epsilon>0$, the existence and uniqueness of solution to problem (2) is guaranteed by Lax-Milgram Theorem. Then, we will analyze the behavior of the solutions as the parameter $\epsilon$ tends to zero. Particularly, by adapting the procedure demonstrated in [3], we initially implement a suitable change of variables. This enables us to substitute, in a certain sense, the original problem (2) posed in a 2-oscillating thin domain into a simpler problem with oscillating coefficients posed in an interval of $\mathbb{R}$. Notice that, this fact is in agreement with the intuitive idea that the family of solutions $w^{\epsilon}$ should converge to a function of just one variable as $\epsilon$ goes to zero since the domain shrinks in the vertical direction. Subsequently, by using the previous results and adapting well-known techniques in homogenization we obtain explicitly the homogenized limit problem for the general case.

Due to the order of the height of the thin domains $R^{\epsilon}$ it makes sense to consider the following measure in thin domains

$$\rho_{\epsilon}(\mathcal{O})=\frac{1}{\epsilon}|\mathcal{O}|,\;\forall\,\mathcal{O}\subset R^{\epsilon}.$$

The rescaled Lebesgue measure $\rho_{\epsilon}$ allows us to preserve the relative capacity of a measurable subset $\mathcal{O}\subset R^{\epsilon}$. Moreover, using the previous measure we introduce the spaces $L^{p}(R^{\epsilon},\rho_{\epsilon})$ and $W^{1,p}(R^{\epsilon},\rho_{\epsilon})$, for $1\leq p<\infty$ endowed with the norms obtained rescaling the usual norms by the factor $\frac{1}{\epsilon}$, that is,

	$\displaystyle|||\varphi|||_{L^{p}(R^{\epsilon})}$	$\displaystyle=\epsilon^{-1/p}||\varphi||_{L^{p}(R^{\epsilon})},\quad\forall\varphi\in L^{p}(R^{\epsilon}),$
	$\displaystyle|||\varphi|||_{W^{1,p}(R^{\epsilon})}$	$\displaystyle=\epsilon^{-1/p}||\varphi||_{W^{1,p}(R^{\epsilon})},\quad\forall\varphi\in W^{1,p}(R^{\epsilon}).$

It is very common to consider this kind of norms in works involving thin domains, see e.g. [20, 27, 26].

Then, assuming that

(8)

$$\frac{1}{\epsilon^{\gamma+1}}\int_{-\epsilon k^{1}_{\epsilon}(x)}^{\epsilon k^{2}_{\epsilon}(x)}\chi_{\theta^{\epsilon}}(x,y)f^{\epsilon}(x,y)dy\stackrel{{\scriptstyle\epsilon\to 0}}{{\rightharpoonup}}f_{0}(x)\quad\hbox{w}-L^{2}(I),$$

the main result obtained is the following:

Finally, we will get the boundary limiting problem for interesting particular cases such as quasi-periodicity or almost periodic framework.

In this section, we focus on the study of the elliptic problem (2) and begin by performing a change of variables that simplifies the domain in which the original problem is posed. As a matter of fact, we will be able to reduce the study of (2) in the thin domain $R^{\epsilon}$ to the study of an elliptic problem with oscillating coefficients capturing the geometry and oscillatory behavior of the open sets where the concentrations take place in the lower dimensional fixed domain $I$. This dimension reduction will be the key point to obtain the correct limiting equation. In order to state the main result of this section, let us first make some definitions. We will denote by

(9)

$$\displaystyle\eta^{i}(\epsilon)=\sup_{x\in I}\Big{|}\epsilon\frac{d}{dx}k^{i}_{\epsilon}(x)\Big{|}>0,\quad i=1,2\quad\hbox{ and }\quad\eta(\epsilon)=\eta^{1}(\epsilon)+\eta^{2}(\epsilon).$$

Observe that from hypothesis (H.1) we have $\eta(\epsilon)\buildrel\epsilon\to 0\over{\longrightarrow}0$

Also, we denote by $K_{\epsilon}(x)=k^{1}_{\epsilon}(x)+k_{\epsilon}^{2}(x)$ (that is $\epsilon K_{\epsilon}(x)$ is the thickness of the thin domain $R^{\epsilon}$ at the point $x\in I$), so from (5) there exist constants $K_{0},K_{1}>0$ independent of $\epsilon>0$ such that

(10)

$$0<K_{0}\leq K_{\epsilon}(x)\leq K_{1}\,\,\mbox{for all}\,\,I\subset\mathbb{R}.$$

Now we will consider the following one dimensional problem

(11)

$$\left\{\begin{gathered}-\frac{1}{K_{\epsilon}(x)}(K_{\epsilon}(x)\hat{w}^{\epsilon}_{x})_{x}+\hat{w}^{\epsilon}=\frac{1}{\epsilon^{\gamma}}\hat{f}^{\epsilon}\quad\textrm{ in }I,\\ \hat{w}^{\epsilon}_{x}(a)=\hat{w}^{\epsilon}_{x}(b)=0\end{gathered}\right.$$

where

(12)

$$\displaystyle\hat{f}^{\epsilon}(x)=\frac{1}{\epsilon K_{\epsilon}(x)}\int_{-\epsilon k^{1}_{\epsilon}(x)}^{\epsilon k^{2}_{\epsilon}(x)}\chi_{\theta^{\epsilon}}(x,y)f^{\epsilon}(x,y)dy.$$

The key result of this section is the following

In order to prove this result, we will need to obtain first some preliminary lemmas. We start transforming equation (2) into an equation in the modified thin domain

(13)

$$R_{a}^{\epsilon}=\Big{\{}(\bar{x},\bar{y})\in\mathbb{R}^{2}\;|\;\bar{x}\in I,\;0<\bar{y}<\epsilon\,K_{\epsilon}(\bar{x})\Big{\}}.$$

(see Figure 3) where it can be seen that we have transformed the oscillations of both boundaries into oscillations of just the boundary at the top. For this, we considering the following family of diffeomorphisms

$$\begin{array}[]{rl}L^{\epsilon}:R_{a}^{\epsilon}&\longrightarrow R^{\epsilon}\\ (\bar{x},\bar{y})&\longrightarrow(x,y):=(\bar{x},\,\bar{y}\,-\,\epsilon\,k^{1}_{\epsilon}(\bar{x})).\end{array}$$

Notice that the inverse of this diffeomorphism is $(L^{\epsilon})^{-1}(x,y)=(x,\,y\,+\,\epsilon\,k^{1}_{\epsilon}(x))$. Moreover, from the structure of these diffeomorphisms and hypothesis (H.1) we easily get that there exists a constant $C$ such that the Jacobian Matrix of $L^{\epsilon}$ and $(L^{\epsilon})^{-1}$ satisfy

(14)

$$\|JL^{\epsilon}\|_{L^{\infty}},\|J(L^{\epsilon})^{-1}\|_{L^{\infty}}\leq C.$$

Moreover, we also have $det(JL^{\epsilon})(\bar{x},\bar{y})=det(J(L^{\epsilon})^{-1})(x,y)=1$.

We will show that the study of the limit behavior of the solutions of (2) is equivalent to analyze the behavior of the solutions of the following problem

(15)

$$\left\{\begin{gathered}-\Delta v^{\epsilon}+v^{\epsilon}=\frac{1}{\epsilon^{\gamma}}\chi_{\theta^{\epsilon}_{1}}f_{1}^{\epsilon}\quad\textrm{ in }R_{a}^{\epsilon},\\ \frac{\partial v^{\epsilon}}{\partial\nu^{\epsilon}}=0\quad\textrm{ on }\partial R_{a}^{\epsilon},\end{gathered}\right.$$

where the function $\chi_{\theta^{\epsilon}_{1}}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is the characteristic function of the narrow strip $\theta^{\epsilon}_{1}$ given by

(16)

$$\theta^{\epsilon}_{1}=\Big{\{}(\bar{x},\bar{y})\in\mathbb{R}^{2}\;|\;\bar{x}\in I\subset\mathbb{R},\;\epsilon[K_{\epsilon}(\bar{x})-\epsilon^{\gamma}H_{\epsilon}(\bar{x})]<\bar{y}<\epsilon K_{\epsilon}(\bar{x})\Big{\}}.$$

The vector $\nu^{\epsilon}$ is the outward unit normal to $\partial R_{a}^{\epsilon}$ and

(17)

$$f_{1}^{\epsilon}=f^{\epsilon}\circ L^{\epsilon}.$$

To achieve a better understanding of the behavior from the nonlinear concentrated term, we will analize the concentrating integral. We are now interested in analyzing concentrated integrals on the narrow strips $\theta_{1}^{\epsilon}$, to this end we will adapt results from [8] at Section 3. The next Lemma is about concentrating integrals, some estimates will be given in different Lebesgue and Sobolev-Bochner spaces.

We would like to obtain a uniform boundedness for the solutions of the problem (15) for any $\epsilon>0$ and $\gamma>0$, so from now on we will assume that $f^{\epsilon}$ satisfies the following hypothesis

(21)

$$\frac{1}{\epsilon^{\gamma}}|||f^{\epsilon}|||^{2}_{L^{2}(\theta^{\epsilon})}\leq c,$$

for some positive constant $c$ independent of $\epsilon>0$.

The next result discusses the relationship between the solutions of problems (2) and (15).

Now we define a transformation on the thin domain $R^{\epsilon}_{a}$, which will map $R^{\epsilon}_{a}$ into the fixed rectangle $Q=I\times(0,1)$. This transformation is given by

$$\begin{array}[]{rl}S^{\epsilon}:Q&\longrightarrow R_{a}^{\epsilon}\\ (x,y)&\longrightarrow(\bar{x},\bar{y}):=(x\,,\,y\,\epsilon K_{\epsilon}(x)).\end{array}$$

We recall that $K_{\epsilon}(x)=k_{\epsilon}^{2}(x)+k_{\epsilon}^{1}(x)$.

Using the chain rule and standard computations it is not difficult to see that there exist $c,C>0$ such that if $v^{\epsilon}\in H^{1}(R_{a}^{\epsilon})$ and $u^{\epsilon}=v^{\epsilon}\circ S^{\epsilon}\in H^{1}(Q)$ then the following estimates hold

(24)

$$c|||v^{\epsilon}|||^{2}_{L^{2}(R^{\epsilon}_{a})}\leq\|u^{\epsilon}\|^{2}_{L^{2}(Q)}\leq C|||v^{\epsilon}|||^{2}_{L^{2}(R^{\epsilon}_{a})},$$

(25)

$$c\Big{|}\Big{|}\Big{|}\frac{\partial v^{\epsilon}}{\partial\bar{y}}\Big{|}\Big{|}\Big{|}^{2}_{L^{2}(R^{\epsilon}_{a})}\leq\frac{1}{\epsilon^{2}}\Big{\|}\frac{\partial u^{\epsilon}}{\partial y}\Big{\|}^{2}_{L^{2}(Q)}\leq C\Big{|}\Big{|}\Big{|}\frac{\partial v^{\epsilon}}{\partial\bar{y}}\Big{|}\Big{|}\Big{|}^{2}_{L^{2}(R^{\epsilon}_{a})},$$

(26)

$$c|||\nabla v^{\epsilon}|||^{2}_{L^{2}(R^{\epsilon}_{a})}\leq\|\nabla u^{\epsilon}\|^{2}_{L^{2}(Q)}\leq C|||\nabla v^{\epsilon}|||^{2}_{L^{2}(R^{\epsilon}_{a})}.$$

Now, under this change of variables and defining $u^{\epsilon}=v^{\epsilon}\circ S^{\epsilon}$ where $v^{\epsilon}$ satisfies (15) and $f^{\epsilon}_{2}=f^{\epsilon}_{1}\circ S^{\epsilon}$, where $f_{1}^{\epsilon}$ is defined in (17), problem (15) becomes

(27)

$$\left\{\begin{gathered}-\frac{1}{K_{\epsilon}}\hbox{div}\big{(}B^{\epsilon}(u^{\epsilon})\big{)}+u^{\epsilon}=\frac{1}{\epsilon^{\gamma}}\chi_{\theta^{\epsilon}_{2}}f_{2}^{\epsilon}\quad\textrm{ in }Q,\\ B(u^{\epsilon})\cdot\eta=0\quad\textrm{ on }\partial Q,\\ u^{\epsilon}=v^{\epsilon}\circ S^{\epsilon}\textrm{ in }Q,\end{gathered}\right.$$

where the function $\chi_{\theta^{\epsilon}_{2}}:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is the characteristic function of the narrow strip $\theta^{\epsilon}_{2}$ given by

(28)

$$\theta^{\epsilon}_{2}=\Big{\{}(x,y)\in\mathbb{R}^{2}\;|\;x\in I,\;1-\epsilon^{\gamma}\frac{H_{\epsilon}(x)}{K_{\epsilon}(x)}<y<1\Big{\}}.$$

The vector $\eta$ denotes the outward unit normal vector field to $\partial Q$ and

$$B^{\epsilon}(u^{\epsilon})=\Big{(}K_{\epsilon}\frac{\partial u^{\epsilon}}{\partial x}-y\displaystyle\frac{dK_{\epsilon}}{dx}\frac{\partial u^{\epsilon}}{\partial y}\,,\,-y\displaystyle\frac{dK_{\epsilon}}{dx}\frac{\partial u^{\epsilon}}{\partial x}+\Big{(}\frac{y^{2}}{K_{\epsilon}}\Big{(}\displaystyle\frac{dK_{\epsilon}}{dx}\Big{)}^{2}+\frac{1}{\epsilon^{2}K_{\epsilon}}\Big{)}\frac{\partial u^{\epsilon}}{\partial y}\Big{)}.$$

Notice that

(29)

$$c|||f^{\epsilon}_{1}|||^{2}_{L^{2}(\theta^{\epsilon}_{1})}\leq\|f^{\epsilon}_{2}\|^{2}_{L^{2}(\theta_{2}^{\epsilon})}\leq C|||f^{\epsilon}_{1}|||^{2}_{L^{2}(\theta^{\epsilon}_{1})}.$$

Therefore from (18) and (29) we have

(30)

$$c|||f^{\epsilon}|||^{2}_{L^{2}(\theta^{\epsilon})}\leq\|f^{\epsilon}_{2}\|^{2}_{L^{2}(\theta_{2}^{\epsilon})}\leq C|||f^{\epsilon}|||^{2}_{L^{2}(\theta^{\epsilon})}.$$

In the new system of coordinates we obtain a domain which is neither thin nor oscillating anymore. In some sense, we have rescaled the neighborhood (16) into the strip $\theta_{2}^{\epsilon}\subset Q$ and substituted the thin domain $R_{a}^{\epsilon}$ for a domain $Q$ independent on $\epsilon$, at a cost of replacing the oscillating thin domain by oscillating coefficients in the differential operator.

In order to analyze the limit behavior of the solutions of (27) we establish the relation to the solutions of the following easier problem

(31)

$$\left\{\begin{gathered}-\frac{1}{K_{\epsilon}}\Big{[}\frac{\partial}{\partial x}\Big{(}K_{\epsilon}\frac{\partial w_{1}^{\epsilon}}{\partial x}\Big{)}+\frac{1}{\epsilon^{2}K_{\epsilon}}\frac{\partial^{2}w_{1}^{\epsilon}}{\partial y^{2}}\Big{]}+w_{1}^{\epsilon}=\frac{1}{\epsilon^{\gamma}}\chi_{\theta^{\epsilon}_{2}}f_{2}^{\epsilon}\quad\textrm{ in }Q,\\ K_{\epsilon}\frac{\partial w_{1}^{\epsilon}}{\partial x}\,\eta_{1}+\frac{1}{\epsilon^{2}K_{\epsilon}}\frac{\partial w_{1}^{\epsilon}}{\partial y}\,\eta_{2}=0\quad\textrm{ on }\partial Q,\end{gathered}\right.$$

where $\eta=(\eta_{1},\eta_{2})$ is the outward unit normal to $\partial Q$.

Before obtaining a priori estimates for the solutions of (31), it is necessary to prove the following result