1 Introduction

The systematic study of mixed variational problems goes back more than 50 years (Babus̆ka, 1971; Brezzi 1974) and since then it has been revealed as a powerful technique for the study of partial differential equations. Moreover, its associated finite element methods, the mixed ones, constitute a fundamental tool for the numerical study of these problems (Boffi 2008; Garralda-Guillem and Ruiz Galán 2019). In this article we consider a variant of a system of mixed variational equations, when we introduce a certain perturbation of one of the equations and also allow the domain to contain holes, that is, the domain of the problem is perforated, the latter situation motivated by its enormous applications. The problem posed admits both a direct and an inverse approach, and we deal with both here. For the first of them, we design a Galerkin scheme, and for the second we establish a generalization of the classical collage theorem (Barnsley 1989) that, in this context, allows numerically approximating some considered inverse problems. Thus, for estimating some parameters in the model problem from known data (in practice, observations) we use a target element in a Banach space associated with the perturbed mixed problem and use the stability in a sense of the direct problem. In particular, we generalize previous works along these lines for ordinary and partial differential equations over solid and perforated domains (Berenguer et al. 2016; Kunze et al. 2004; Kunze and La Torre 2018, 2017, 2016, 2015; Kunze et al. 2015, 2010, 2009; Kunze and Vrscay 1999).

The paper is structured around 5 sections. In Sect. 2 we describe the Mixed Variational Equation considered and the stability conditions that will allow us to deal with a suitable inverse problem. In Sect. 3 we introduce the perforated domains considered and in Sects. 4 and 5 we analyze the relationship between the solutions of the direct and inverse problems on solid domains and on perforated domains, when the holes are small enough in a certain sense. We also illustrate the results with a numerical example. Finally, in Sect. 6 we include some conclusions.

2 Collage-type inverse problems for mixed variational equations

We discuss here a more general version of the classical system of mixed variational equations corresponding to the mixed variational formulation of a differential problem which includes a kind of perturbation. The perturbation term is modelled by means of a new bilinear form, that has to be interpreted to be small in some sense.

Suppose that E and F are real Hilbert spaces, \(a: E \times E \longrightarrow \mathbb {R}\), \(b: E \times F \longrightarrow \mathbb {R}\) and \(c: F \times F \longrightarrow \mathbb {R}\) are bounded and bilinear and \(x^*:E\rightarrow \mathbb {R}\) and \(y^*:F\rightarrow \mathbb {R}\) are bounded and linear. Our problem reads as follows: Find \((x_0,y_0) \in E\times F\) such that

figure a

We use the following general result for a family of such problems that include a stability property, (2.1), which will be essential for our purposes, since it will allow us to deal with a suitable inverse problem. Furthermore, such a stability condition (2.1) is a Generalized Collage Theorem that extends those in Berenguer et al. (2016) and Kunze et al. (2009) in the Hilbertian framework.

Theorem 2.1

Let E and F be real Hilbert spaces, \(\Lambda \) be a nonempty set and for each \(\lambda \in \Lambda \), let \(a_\lambda : E \times E \longrightarrow \mathbb {R}\), \(b_\lambda : E \times F \longrightarrow \mathbb {R}\) and \(c_\lambda : F \times F \longrightarrow \mathbb {R}\) be bounded and bilinear and let \(K_\lambda :=\{x \in E: \ b_\lambda (x,\cdot )=0\}\) in such a way that

  1. (i)

    \(x \in K_\lambda \ \wedge \ a_{\lambda }(x,\cdot )_{|{K}_\lambda }=0 \Rightarrow \ x=0\)

and for some \(\alpha _\lambda ,\beta _\lambda >0\) there hold

  1. (ii)

    \(x \in K_\lambda \ \Rightarrow \ \alpha _\lambda \Vert x \Vert \le \Vert a_{\lambda }(\cdot ,x)_{|{K}_\lambda } \Vert \) and

  2. (iii)

    \(y \in F \ \Rightarrow \ \beta _\lambda \Vert y \Vert \le \Vert b_{\lambda }(\cdot ,y) \Vert \).

If

$$\begin{aligned} \rho _\lambda :=\max \left\{ \frac{1}{\alpha _\lambda }, \frac{1}{\beta _\lambda } \left( 1+\frac{\Vert a_\lambda \Vert }{\alpha _\lambda } \right) , \frac{1}{\beta _\lambda ^2}\Vert a_\lambda \Vert \left( 1+\frac{\Vert a_\lambda \Vert }{\alpha _{\lambda }} \right) \right\} \end{aligned}$$

and in addition

  1. (iv)

    \(\Vert c_\lambda \Vert < \displaystyle \frac{1}{\rho _\lambda }\),

then for each \(\lambda \in \Lambda \) and \((x_\lambda ^*,y_\lambda ^*) \in E^* \times F^*\) there exists a unique \((x_\lambda ,y_\lambda ) \in E \times F\) such that

figure b

Furthermore, if \((x,y) \in E \times F\), then

$$\begin{aligned} \max \{ \Vert x_\lambda -x\Vert ,\Vert y_\lambda -y\Vert \} \le \frac{\rho _\lambda }{1-\rho _\lambda \Vert c_\lambda \Vert } \left( \Vert x_\lambda ^*-a_\lambda (x,\cdot )-b_\lambda (\cdot ,y) \Vert +\Vert y_\lambda ^* -b_\lambda (x,\cdot )-c_{\lambda }(y, \cdot )\Vert \right) . \end{aligned}$$
(2.1)

Proof

Let \(\lambda \in \Lambda \). The existence and uniqueness of solution for problem (\(P_{\lambda }\)) is a well-known fact (see, for instance Boffi 2008, Proposition 4.3.2), but we give a sketch of the proof in order to derive also the control of the norms in (2.1) in a precise way. So, let us endow the product space \(E \times F\) with the norm

$$\begin{aligned} \Vert (x,y) \Vert :=\max \{\Vert x \Vert , \Vert y \Vert \}, \qquad (x \in E, \ y \in F) \end{aligned}$$

and its dual space \(E^* \times F^*\) with the corresponding dual norm, that is,

$$\begin{aligned} \Vert (x^*,y^*) \Vert :=\Vert x^* \Vert + \Vert y^* \Vert , \qquad (x^* \in E^*, \ y^* \in F^*). \end{aligned}$$

According to conditions (i), (ii) and (iii) and to Gatica (2014, Theorem 2.1), the bounded and linear operator \(S_\lambda : E \times F \longrightarrow E^* \times F^*\) defined at each \((x,y) \in E \times F\) as

$$\begin{aligned} S_\lambda (x,y):=(a_\lambda (x,\cdot )+b_\lambda (\cdot ,y),b_\lambda (x,\cdot )) \end{aligned}$$

is an isomorphism. But, in view of Atkinson and Han (2009, Theorem 2.3.5), in order to state the existence of a unique solution for the perturbed mixed system (\(P_{\lambda }\)) it is enough to show that

$$\begin{aligned} \Vert S_{\lambda }^{-1} \Vert < \frac{1}{\Vert c_{\lambda } \Vert }, \end{aligned}$$
(2.2)

inequality which is valid, since in view of Garralda-Guillem and Ruiz Galán (2014, Theorem 3.6) and (iv), we have that

$$\begin{aligned} \begin{array}{rl} \Vert S_\lambda ^{-1}\Vert &{} = \displaystyle \sup _{\Vert x^*\Vert +\Vert y^*\Vert \le 1} \Vert S_\lambda ^{-1}(x^*,y^*)\Vert \\ &{} \le \displaystyle \sup _{\Vert x^*\Vert +\Vert y^*\Vert \le 1} \max \left\{ \displaystyle \frac{\Vert x^* \Vert }{\alpha _\lambda }+\frac{1}{\beta _\lambda }\left( 1+\frac{\Vert a_\lambda \Vert }{\alpha _\lambda }\right) \Vert y^*\Vert , \frac{1}{\beta _\lambda } \left( 1+ \frac{\Vert a_\lambda \Vert }{\alpha _\lambda } \right) \left( \Vert x^*\Vert +\frac{\Vert a_\lambda \Vert }{\beta _\lambda } \Vert y^*\Vert \right) \right\} \\ &{} \le \displaystyle \sup _{\Vert x^*\Vert +\Vert y^*\Vert \le 1} \max \left\{ \displaystyle \frac{1}{\alpha _\lambda },\frac{1}{\beta _\lambda }\left( 1+\frac{\Vert a_\lambda \Vert }{\alpha _\lambda }\right) , \frac{1}{\beta _\lambda } \left( 1+ \frac{\Vert a_\lambda \Vert }{\alpha _\lambda } \right) , \frac{\Vert a_\lambda \Vert }{\beta _\lambda ^2} \left( 1+ \frac{\Vert a_\lambda \Vert }{\alpha _\lambda } \right) \right\} \\ &{}\quad (\Vert x^*\Vert +\Vert y^*\Vert ) \\ &{} \le \rho _{\lambda } \\ &{} < \displaystyle \frac{1}{\Vert c_\lambda \Vert }. \end{array} \end{aligned}$$

Furthermore, according to (2.2) and Atkinson and Han (2009, Theorem 2.3.5) or Garralda-Guillem and Ruiz Galán (2014, Theorem 3.6) once again, we arrive at

$$\begin{aligned} \max \{ \Vert x_\lambda \Vert ,\Vert y_\lambda \Vert \} \le \frac{\rho _{\lambda }}{1-\rho _{\lambda } \Vert c_\lambda \Vert } \left( \Vert x^*\Vert +\Vert y^*\Vert \right) , \end{aligned}$$
(2.3)

where \((x_\lambda ,y_\lambda ) \in E \times F\) is the unique solution of (\(P_{\lambda }\)). To conclude, given \((\hat{x}_\lambda ,\hat{y}_\lambda ) \in E \times F\), since \((x_\lambda -\hat{x}_\lambda ,y_\lambda -\hat{y}_\lambda )\) is the unique solution of the perturbed mixed problem

$$\begin{aligned} \left\{ \begin{array}{c} a_\lambda (x_\lambda -\hat{x}_\lambda ,\cdot )+b_\lambda (\cdot ,y_\lambda -\hat{y}_\lambda )=x_\lambda ^*-a_\lambda (\hat{x}_{\lambda },\cdot )-b_\lambda (\cdot ,\hat{y}_\lambda ) \\ b_\lambda (x_\lambda -\hat{x}_\lambda ,\cdot )+c_\lambda (y_\lambda -\hat{y}_\lambda ,\cdot )=y_\lambda ^*-b_\lambda (\hat{x}_\lambda ,\cdot )-c_\lambda (\hat{y}_\lambda ,\cdot ) \end{array} \right. , \end{aligned}$$

then, according to inequality (2.3),

$$\begin{aligned}{} & {} \max \{ \Vert x_\lambda -\hat{x}_\lambda \Vert ,\Vert y_\lambda -\hat{y}_\lambda \Vert \} \le \frac{\rho _{\lambda }}{1-\rho _{\lambda } \Vert c_\lambda \Vert } \Big ( \Vert x_\lambda ^*-a_\lambda (\hat{x}_\lambda ,\cdot ) \\ {}{} & {} \quad \quad \quad -b_\lambda (\cdot ,\hat{y}_\lambda ) \Vert +\Vert y_\lambda ^* -b_\lambda (\hat{x}_\lambda ,\cdot )-c_\lambda (\hat{y}_\lambda ,\cdot )\Vert \Big ). \end{aligned}$$

Finally, the arbitrariness of \(\lambda \in \Lambda \) yields (2.1). \(\square \)

It is worth mentioning that if

$$\begin{aligned} \alpha := \inf _{\lambda \in \Lambda } \alpha _\lambda>0, \quad \beta := \inf _{\lambda \in \Lambda } \beta _\lambda>0, \quad \delta := \sup _{\lambda \in \Lambda } \Vert a_\lambda \Vert , \quad \gamma := \inf _{\lambda \in \Lambda } \Vert c_\lambda \Vert >0 \end{aligned}$$

and

$$\begin{aligned} \rho :=\max \left\{ \frac{1}{\alpha }, \frac{1}{\beta } \left( 1+\frac{\delta }{\alpha } \right) , \frac{ \delta }{\beta ^2} \left( 1+\frac{\delta }{\alpha }\right) \right\} , \end{aligned}$$

then

$$\begin{aligned}{} & {} \inf _{\lambda \in \Lambda } \max \{ \Vert x_\lambda -x \Vert , \Vert y_\lambda -y \Vert \} \le \frac{\rho }{1-\rho \gamma } (\Vert x_\lambda ^* -a_\lambda (x,\cdot )\\ {}{} & {} \qquad \quad -b_\lambda (\cdot ,y) \Vert + \Vert y_\lambda ^* - b_\lambda (x, \cdot )-c_{\lambda }(y, \cdot ) \Vert ). \end{aligned}$$

Therefore, in order to approximate the solution of the corresponding inverse problem we solve the optimization problem

$$\begin{aligned} \min _{\lambda \in \Lambda } ( \Vert x^*_\lambda -a_\lambda (x,\cdot )-b_\lambda (\cdot ,y) \Vert + \Vert y^*_\lambda - b_\lambda (y, \cdot )-c_{\lambda }(y, \cdot ) \Vert ). \end{aligned}$$
(2.4)

3 Perforated domains

We address next a modification of the problem (P) that tries to model situations from different engineering or material sciences in which perforated domains appear. We will understand by perforated domains, those in which holes appear. We illustrate this type of problem with the following example.

Example 3.1

Let \(\Omega =(0,1)^2\), \(\Gamma =\partial \Omega \), \(\delta \in \mathbb {R}\) and \(f \in H_0^1(\Omega )\), and let us consider the boundary value problem: Find \(\psi \in H^2(\Omega )\) such that

$$\begin{aligned} \left\{ \begin{array}{l} \Delta ^2\psi +\delta \psi =f \hbox {in } \Omega \\ \psi |_{\scriptscriptstyle \Gamma } = 0 \\ \Delta \psi |_{\scriptscriptstyle \Gamma } =0 \end{array} \right. . \end{aligned}$$
(3.1)

Now, we study the same type of problem in a perforated domain described as follows. Let us denote by \(\Omega _B\) a collection of circular holes \(\cup _{j=1}^m B(x_j,\rho _j)\) where \(x_j\in \Omega \), \(\rho _j>0\) and the holes \(B(x_j,\rho _j)\) are nonoverlapping and lie strictly inside \(\Omega \). We will consider \(\varepsilon =\max _j \rho _j\) and denote by \(\Omega _\varepsilon \) the closure of the set \(\Omega {\setminus } \Omega _B\).

Let \(\Omega _\varepsilon \), \(\Gamma _\varepsilon =\partial \Omega _\varepsilon \), \(\delta \in \mathbb {R}\) and \(f \in H_0^1(\Omega _\varepsilon )\), and let us consider the boundary value problem: Find \(\psi \in H^2(\Omega _\varepsilon )\) such that

$$\begin{aligned} \left\{ \begin{array}{ll} \Delta ^2\psi +\delta \psi =f &{} \hbox {in } \Omega _\varepsilon \\ \psi |_{\scriptscriptstyle \Gamma _\varepsilon } = 0 &{} \\ \Delta \psi |_{\scriptscriptstyle \Gamma _\varepsilon } =0 &{} \end{array} \right. . \end{aligned}$$
(3.2)

Following classical passages, this problem can be written as follows: Find \((x_{0\varepsilon },y_{0\varepsilon }) \in E_\varepsilon \times F_\varepsilon \) such that

figure c

This system adopts the form of (\(P_{\lambda }\)) with \(\textrm{card}(\Lambda )=1\), the real Hilbert spaces \(E_\varepsilon =F_\varepsilon :=H_0^1(\Omega _\varepsilon )\), the continuous bilinear forms \(a_\varepsilon : E_\varepsilon \times E_\varepsilon \longrightarrow \mathbb {R}\), \(b_\varepsilon : E_\varepsilon \times F_\varepsilon \longrightarrow \mathbb {R}\) and \(c_\varepsilon : F_\varepsilon \times F_\varepsilon \longrightarrow \mathbb {R}\) defined for each \(x_1,x_2 \in E_\varepsilon ,\) and \(y_1,y_2\in F_\varepsilon \), as

$$\begin{aligned}{} & {} a_\varepsilon (x_1,x_2):= \int _{\Omega _\epsilon } x_1 x_2,\\{} & {} b_\varepsilon (x_1,y_1):=- \int _{\Omega _\epsilon } \nabla x_1 \nabla y_1, \end{aligned}$$

and

$$\begin{aligned} c_\varepsilon (y_1,y_2):=-\delta \int _{\Omega _\epsilon } y_1 y_2, \end{aligned}$$

and the continuous linear functionals \(x_\varepsilon ^*:=0 \in E_\varepsilon ^*\) and \(y_\varepsilon ^* \in F_\varepsilon ^*\) given by

$$\begin{aligned} y^*_\varepsilon (y):=- \int _{\Omega _\epsilon } fy, \qquad (y \in F_\varepsilon ). \end{aligned}$$

The next two sections are devoted to study the relations between the solutions of problems (P) and (\(P_\varepsilon \)) and the corresponding inverse problems when such problems are close in a certain sense.

4 Mixed variational problems on perforated domains: the direct problem

We introduce an abstract formulation of the problem above, considering two sequences of spaces \(\{E_{\varepsilon _n}\}_{n\in \mathbb {N}}\), \(\{F_{\varepsilon _n}\}_{n\in \mathbb {N}}\) which we note \(\{E_n\}_{n\in \mathbb {N}}\) and \(\{F_n\}_{n\in \mathbb {N}}\) respectively.

Let E, F, \(\{E_n\}_{n\in \mathbb {N}}\), \(\{F_n\}_{n\in \mathbb {N}}\) be real Hilbert spaces, \(a:E\times E \longrightarrow \mathbb {R}\), \(b:E\times F \longrightarrow \mathbb {R}\) and \(c:F\times F \longrightarrow \mathbb {R}\) be bounded bilinear forms, and for \(n\in \mathbb {N}\), let \(a_n: E_n \times E_n \longrightarrow \mathbb {R}\), \(b_n: E_n \times F_n \longrightarrow \mathbb {R}\) and \(c_n: F_n \times F_n \longrightarrow \mathbb {R}\) be bounded bilinear forms. Let \(x^*:E \longrightarrow \mathbb {R}\) and \(y^*:F \longrightarrow \mathbb {R}\) be bounded linear functionals and for \(n\in {\mathbb N}\), let \(x_n^*:E_n \longrightarrow \mathbb {R}\) and \(y_n^*:F_n \longrightarrow \mathbb {R}\) be bounded linear functionals.

We consider the problem (P) and for \(n\in {\mathbb N}\), the following problems: find \((x_{0n},y_{0n}) \in E_n\times F_n\) such that

figure d

We write

$$\begin{aligned} K:=\{x \in E: \ b(x,\cdot )=0\}, \end{aligned}$$

and for \(n\in {\mathbb N}\)

$$\begin{aligned} K_n:=\{x \in E_n: \ b_n(x,\cdot )=0\}. \end{aligned}$$

Then, we suppose now that the bounded bilinear forms in problems (P) and (\(P_n\)) verify assumption (i) of Theorem 2.1 for cardinal of \(\Lambda \) equal to 1, and assumptions (ii), (iii) and (iv) of the same result in the following way: For some \(\alpha ,\beta >0\) and \(\alpha _n,\beta _n>0\), \((n \in \mathbb {N})\), there hold

  1. (ii)

    • \(x \in K \ \Rightarrow \ \alpha \Vert x \Vert \le \Vert a(\cdot ,x)_{|{K}} \Vert \),

    • \(x \in K_n \ \Rightarrow \ \alpha _n \Vert x \Vert \le \Vert a_n(\cdot ,x)_{|{K_n}} \Vert \),

  2. (iii)

    • \(y \in F \ \Rightarrow \ \beta \Vert y \Vert \le \Vert b(\cdot ,y) \Vert \),

    • \(y \in F_n \ \Rightarrow \ \beta _n \Vert y \Vert \le \Vert b_n(\cdot ,y) \Vert \).

and noting

$$\begin{aligned} \rho :=\max \left\{ \frac{1}{\alpha }, \frac{1}{\beta } \left( 1+\frac{\Vert a\Vert }{\alpha } \right) , \frac{1}{\beta ^2}\Vert a\Vert \left( 1+\frac{\Vert a\Vert }{\alpha } \right) \right\} \end{aligned}$$

and for \(n \in {\mathbb N}\)

$$\begin{aligned} \rho _n:=\max \left\{ \frac{1}{\alpha _n}, \frac{1}{\beta _n} \left( 1+\frac{\Vert a_n\Vert }{\alpha _n} \right) , \frac{1}{\beta _n^2}\Vert a_n\Vert \left( 1+\frac{\Vert a_n\Vert }{\alpha _n} \right) \right\} , \end{aligned}$$
  1. (iv)

    • \(\Vert c \Vert < \displaystyle \frac{1}{\rho }\),

    • for \(n \in {\mathbb N}\), \(\Vert c_n \Vert < \displaystyle \frac{1}{\rho _n}\).

In view of Theorem 2.1 these assumptions ensures the existence and uniqueness of solution for problems (P) and (\(P_n\)), noted \((x_0,y_0)\in E\times F\) and \((x_{0n},y_{0n})\in E_n \times F_n\) respectively. Moreover, we have the following control of the norms:

$$\begin{aligned} \max \{ \Vert x_0\Vert _E, \Vert y_0\Vert _F\} \le \frac{\rho }{1-\rho \Vert c\Vert } \left( \Vert x^*\Vert +\Vert y^*\Vert \right) , \end{aligned}$$
(4.1)

and for \(n\in {\mathbb N}\)

$$\begin{aligned} \max \{ \Vert x_{0n}\Vert _{E_n}, \Vert y_{0n}\Vert _{F_n}\} \le \frac{\rho _n}{1-\rho _n \Vert c_n\Vert } \left( \Vert x_n^*\Vert +\Vert y_n^*\Vert \right) . \end{aligned}$$
(4.2)

The next result establishes the relation between the solutions of problems (P) and (\(P_n\)) when such problems are close in a certain sense:

Theorem 4.1

With the previous notations and assumptions, let us suppose that

  1. (a)

    The Hilbert spaces E, F, \(\{E_n\}_{n\in \mathbb {N}}\), \(\{F_n\}_{n\in \mathbb {N}}\) verify:

    • The sequences \(\{E_n\}_{n\in \mathbb {N}}\) and \(\{F_n\}_{n\in \mathbb {N}}\) are increasing i.e., if \(n,m\in \mathbb {N}\), \(n<m\), then \(E_n \subset E_{m}\subset E\) and \(F_n \subset F_{m}\subset F\).

    • \(\overline{\bigcup _{n\in {\mathbb N}}E_n}=E\) and \(\overline{\bigcup _{n\in {\mathbb N}}F_n}=F\).

    • There exist \(\gamma _E, \gamma _F>0\) such that for each \(n \in {\mathbb N}\), \(x\in E_n\), \(y\in F_n\),

      $$\begin{aligned} \Vert x\Vert _E\le \gamma _E \Vert x\Vert _{E_n}, \quad \text {and} \quad \Vert y\Vert _F\le \gamma _F \Vert y\Vert _{F_n}. \end{aligned}$$
  2. (b)

    There exist three sequences \(\{\mu _n\},\{\eta _n\},\{\delta _n\}\), with

    $$\begin{aligned} \lim _{n\rightarrow \infty }\mu _n=\lim _{n\rightarrow \infty }\eta _n=\lim _{n\rightarrow \infty }\delta _n=0, \end{aligned}$$

    such that for \(n\in {\mathbb N}\), \(x_1,x_2 \in E_n\) and \(y_1,y_2 \in F_n\) we have:

    • \(|a(x_1,x_2)-a_n(x_1,x_2)|\le \mu _n \Vert x_1\Vert _{E_n} \Vert x_2\Vert _{E_n}\),

    • \(|b(x_1,y_1)-b_n(x_1,y_1)|\le \eta _n \Vert x_1\Vert _{E_n} \Vert y_1\Vert _{F_n}\),

    • \(|c(y_1,y_2)-c_n(y_1,y_2)|\le \delta _n \Vert y_1\Vert _{F_n} \Vert y_2\Vert _{F_n}\).

  3. (c)

    The sequences of funcionals \(\{x_n^*\}_{n \in {\mathbb N}}\) and \(\{y_n^*\}_{n \in {\mathbb N}}\), converge to \(x^*\) and \(y^*\), respectively, in the \(w^*\)–topology.

Then, the sequences of solutions \((\{x_{0n},y_{0n})\}_{n \in {\mathbb N}}\) of problems (\(P_n\)) converge in the w–topology on \(E\times F\), except partials, to \((x_0,y_0)\), solution of problem (P).

Proof

From assumption c) we have that there exist \(M_E,M_F\ge 0\) such that

$$\begin{aligned} \Vert x_n^*\Vert \le M_E, \quad \Vert y_n^*\Vert \le M_F, \quad (n\in {\mathbb N}). \end{aligned}$$
(4.3)

We can deduce from this fact and from (4.2) that the boundness of the sequences \(\{x_{0n}\}_{n \in {\mathbb N}}\), \(\{y_{0n}\}_{n \in {\mathbb N}}\) depends on the boundness of \(\{\rho _n\}\) or that of \(\{\Vert a_n\Vert \}\) and \(\{\Vert c_n\Vert \}\). But the sequence \(\{\Vert a_n\Vert \}\) is bounded, since

$$\begin{aligned} \Vert a_n\Vert{} & {} \displaystyle =\sup _{x_1, x_2\in E_n} \frac{|a_n(x_1,x_2)|}{\Vert x_1\Vert _{E_n}\Vert x_2\Vert _{E_n} } \\{} & {} \displaystyle \le \sup _{x_1,x_2\in E_n} \frac{|a_n(x_1,x_2)-a(x_1,x_2)|}{\Vert x_1\Vert _{E_n}\Vert x_2\Vert _{E_n} }+\sup _{x_1,x_2\in E_n} \frac{|a(x_1,x_2)|}{\Vert x_1\Vert _{E_n}\Vert x_2\Vert _{E_n} }, \end{aligned}$$

and taking into account assumption b) for the first term and assumption a) for the second, we have that the last sum is less or equal that

$$\begin{aligned} \sup _{x_1,x_2\in E_n} \frac{\mu _n \Vert x_1\Vert _{E_n}\Vert x_2\Vert _{E_n}}{\Vert x_1\Vert _{E_n}\Vert x_2\Vert _{E_n} }+ \sup _{x_1,x_2\in E_n} \gamma _E \gamma _E \frac{|a(x_1,x_2)|}{\Vert x_1\Vert _{E}\Vert x_2\Vert _{E} }\le \mu _n + \gamma _E^2 \Vert a\Vert . \end{aligned}$$

Similar arguments show the boundness of \(\{\Vert c_n\Vert \}\).

We deduce that \(\{x_{0n}\}_{n \in {\mathbb N}}\), \(\{y_{0n}\}_{n \in {\mathbb N}}\) are bounded an then they have partial subsequences \(\{x_{0n_k}\}_{n \in {\mathbb N}}\), \(\{y_{0n_k}\}_{n \in {\mathbb N}}\) which converge weakly. We note \(x_1\) and \(y_1\) the limits of such subsequences. We prove finally that \((x_1,y_1)\in E\times F\) is solution of problem (P) and then from the uniqueness of the solution we have the result. For this purpose, for each \(x\in E\), according to a), the continuity of the bilinear forms and the density of \(\cup _{n \in \mathbb {N}} E_n\) and \(\cup _{n \in \mathbb {N}} F_n\), we can suppose that there exists \(n_k\) such that \(x\in E_{n_k}\). Then,

$$\begin{aligned} |a(x_1,x)+b(x,y_1)-x^*(x)|{} & {} \le |a(x_1,x)-a(x_{0n_k},x)|+ |b(x,y_1)-b(x,y_{0n_k})| \\{} & {} \quad + |a(x_{0n_k},x)+b(x,y_{0n_k})-x^*(x)|. \end{aligned}$$

From the weak continuity of a and b in each variable we deduce that the first two terms in the sum converge to 0. For the third one, we have that

$$\begin{aligned} |a(x_{0n_k},x)+b(x,y_{0n_k})-x^*(x)|{} & {} \le |a(x_{0n_k},x)-a_{n_k}(x_{0n_k},x)| \\{} & {} \quad +|a_{n_k}(x_{0n_k},x)+b_{n_k}(x,y_{0n_k})-x_{n_k}^*(x)| \\{} & {} \quad +|b(x,y_{0n_k})-b_{n_k}(x,y_{0n_k})| \\{} & {} \quad +|x_{n_k}^*(x)-x^*(x)|. \end{aligned}$$

In view the assumption (b) we deduce that the first and third terms tend to 0. The second is 0 because \((x_{0n_k},y_{0n_k})\) is the solution of the problem (\(P_{n_k}\)), and the last term tends to 0 from (c).

A similar reasoning proves that, given \(y\in F\), \(b(x_1,y)+c(y_1,y)=y^*(y)\), which concludes the proof. \(\square \)

5 Mixed variational problems on perforated domains: the inverse problem

We deal now with inverse problems associated with problems in perforated domains and we analyse the relationship between the minimizers of an inverse problem defined on a solid domain and the minimizers of an inverse problem defined on a perforated domain when the holes are small enough.

Let \(\Lambda \) a compact set of \(\mathbb {R}^n\). Let E, F, \(\{E_n\}_{n\in \mathbb {N}}\), \(\{F_n\}_{n\in \mathbb {N}}\) be real Hilbert spaces, \(a^\lambda :E\times E \longrightarrow \mathbb {R}\), \(b^\lambda :E\times F \longrightarrow \mathbb {R}\) and \(c^\lambda :F\times F \longrightarrow \mathbb {R}\) be bounded bilinear forms, and for \(n\in \mathbb {N}\), let \(a^\lambda _n: E_n \times E_n \longrightarrow \mathbb {R}\), \(b^\lambda _n: E_n \times F_n \longrightarrow \mathbb {R}\) and \(c^\lambda _n: F_n \times F_n \longrightarrow \mathbb {R}\) be bounded bilinear forms. Let \(x_\lambda ^*:E \longrightarrow \mathbb {R}\) and \(y_\lambda ^*:F \longrightarrow \mathbb {R}\) be bounded linear functionals, and for \(n\in {\mathbb N}\), \(x_{\lambda n}^*:E_n \longrightarrow \mathbb {R}\) and \(y_{\lambda n}^*:F_n \longrightarrow \mathbb {R}\) be bounded linear functionals.

For \(\lambda \in \Lambda \) we consider the family of problems (\(P_{\lambda }\)) described on Theorem 2.1 and for each \(n\in \mathbb {N}\) the family of problems: Find \((x_{\lambda n},y_{\lambda n}) \in E_n\times F_n\) such that

figure e

If we suppose that all the bilinear forms verify assumptions (i), (ii), (iii) and (iv) of Theorem 2.1, it follows that for each \(\lambda \in \Lambda \) and for each \(n\in \mathbb {N}\), problems (\(P_{\lambda }\)) and (\(P_n^\lambda \)) have a unique solution \((x_\lambda ,y_\lambda )\) and (\(x_{\lambda n},y_{\lambda n})\) repectively. Moreover, given a target element \((x,y)\in E\times F\), Theorem 2.1 states that

$$\begin{aligned}{} & {} \inf _{\lambda \in \Lambda }\max \left\{ \Vert x_\lambda -x\Vert _E,\Vert y_\lambda -y\Vert _F\right\} \nonumber \\{} & {} \quad \le \inf _{\lambda \in \Lambda }\dfrac{\rho _\lambda }{1-\rho _\lambda \gamma } \left( \Vert x^*_\lambda -a^\lambda (x,\cdot )-b^\lambda (\cdot ,y)\Vert + \Vert y^*_\lambda - b^\lambda (x,\cdot ) -c^\lambda (y,\cdot )\Vert \right) \end{aligned}$$
(5.1)

with \(\gamma := \inf _{\lambda \in \Lambda } \Vert c^\lambda \Vert >0\). Then, in order to solve the inverse problem, we must solve the optimization problem

$$\begin{aligned} \min _{\lambda \in \Lambda } \left( G^\lambda (x,y)+S^\lambda (x,y) \right) , \end{aligned}$$

where, \(G^\lambda (x,y)=\Vert x^*_\lambda -a^\lambda (x,\cdot )-b^\lambda (\cdot ,y)\Vert \) and \(S^\lambda (x,y)=\Vert y^*_\lambda - b^\lambda (x,\cdot )-c^\lambda (y,\cdot )\Vert \), for a given \((x,y)\in E\times F\). With the same arguments as above, given a target element \((x_n,y_n)\in E_n\times F_n\) in order to approximate the solution of the inverse problem (\(P_n^\lambda \)) we must minimize the collage distance, that is, solve the optimization problem

$$\begin{aligned} \min _{\lambda \in \Lambda } \left( G^\lambda _n(x_n,y_n)+S^\lambda _n(x_n,y_n) \right) , \end{aligned}$$

where \(G^\lambda _n(x_n,y_n)=\Vert x^*_{\lambda n}-a^\lambda _n(x,\cdot )-b^\lambda _n(\cdot ,y)\Vert \) and \(S^\lambda _n (x_n,y_n)=\Vert y^*_{\lambda n} - b^\lambda _n (x,\cdot )-c^\lambda _n(y,\cdot )\Vert \).

Our goal is to show that solutions of inverse problems (\(P_{\lambda }\)) and (\(P_n^\lambda \)) are arbitrary closed when problems are closed enough in the sense established in the next result.

Theorem 5.1

With the above notation, suppose that the Hilbert spaces E, F, \(\{E_n\}_{n\in \mathbb {N}}\), \(\{F_n\}_{n\in \mathbb {N}}\) verify:

  1. (i)

    The sequences \(\{E_n\}_{n\in \mathbb {N}}\) and \(\{F_n\}_{n\in \mathbb {N}}\) are increasing sequences, i.e. if \(n,m\in \mathbb {N}\), \(n<m\), then \(E_n \subset E_{m}\subset E\) and \(F_n \subset F_{m}\subset F\).

  2. (ii)

    There exist two sequences of projections \(\pi _n: E\longrightarrow E_n\), and \(Q_n:F \longrightarrow F_n\), such that for \(x\in E\) and \(y\in F\),

    $$\begin{aligned} \lim _{n \rightarrow \infty } \Vert x-\pi _n(x)\Vert =\lim _{n \rightarrow \infty } \Vert y-Q_n(y)\Vert =0. \end{aligned}$$
  3. (iii)

    The bilinear forms and functionals are given by \(a^\lambda _n=a^\lambda _{|E_n\times E_n}\), \(b^\lambda _n=b^\lambda _{|E_n\times F_n}\), \(c^\lambda _n=c^\lambda _{|F_n\times F_n}\), \(x^*_{\lambda n}=x^*_{\lambda _{|E_n}}\), \(y^*_{\lambda n }=y^*_{\lambda _{|F_n}}\).

  4. (iv)

    For all \(n\in \mathbb {N}\), \(x\in E\), \(y\in F\), \(x_n\in E_n\) and \(y_n\in F_n\), the functions \(G^\lambda (x,y)\), \(G^\lambda _n(x_n,y_n)\), \(S^\lambda (x,y)\) and \(S^\lambda _n (x_n,y_n):\Lambda \rightarrow \mathbb {R}^{+}\) are continuous.

Let \(\{\lambda _n\}\) a sequence of minimizers of \(G^\lambda _n(\pi _n(x),Q_n(y))+S^\lambda _n (\pi _n(x),Q_n(y))\) over \(\Lambda \). Then there exists \(\lambda ^*\in \Lambda \) and a partial subsequence of \(\{\lambda _n\}\), which we will note \(\{\lambda _n\}\) as well, in order to simplify the notation, such that \(\{\lambda _{n}\}\rightarrow \lambda ^*\), with \(\lambda ^*\) a minimizer of \(G^\lambda (x,y)+ S^\lambda (x,y)\) over \(\Lambda \).

Proof

Let \(M,N,R,\mu \) and \(\nu \) given by

$$\begin{aligned}{} & {} M=\sup _{ \lambda \in \Lambda } \left\{ \Vert a ^\lambda \Vert \right\} , \quad N=\sup _{\lambda \in \Lambda } \left\{ \Vert b ^\lambda \Vert \right\} , \quad \ R=\sup _{\lambda \in \Lambda } \left\{ \Vert c ^\lambda \Vert \right\} , \quad \mu =\sup _{ \lambda \in \Lambda } \left\{ \Vert x^*_\lambda \Vert \right\} , \\ {}{} & {} \nu =\sup _{ \lambda \in \Lambda } \left\{ \Vert y^*_\lambda \Vert \right\} . \end{aligned}$$

Then on the one hand, given \((x,y)\in E\times F\),

$$\begin{aligned} \begin{array}{rl} G^\lambda _n (\pi _n x,Q_n y) + S^\lambda _n (\pi _n x,Q_n y) &{} \le G^\lambda (\pi _n x,Q_n y) + S^\lambda (\pi _n x,Q_n y)) \\ &{} = G^\lambda (\pi _n x-x+x,Q_n y-y+y) \\ &{} \quad + S^\lambda (\pi _n x-x+x,Q_n y-y+y) \\ &{} \le \Vert \phi ^\lambda -a^\lambda (\pi _n x-x+x,\cdot )-b^\lambda (\cdot ,Q_n y-y+y)\Vert \\ &{} \quad + \Vert \psi ^\lambda - b^\lambda (\pi _n x-x+x,\cdot )-c^\lambda (Q_n y-y+y,\cdot )\Vert \\ &{} \le \Vert \phi ^\lambda -a^\lambda (x,\cdot )-b^\lambda (\cdot ,y)\Vert + \Vert \psi ^\lambda - b^\lambda (x,\cdot )-c^\lambda (y,\cdot )\Vert \\ &{} \quad +\Vert a^\lambda (\pi _n x-x,\cdot )-b^\lambda (\cdot ,Q_n y-y)\Vert \\ &{} \quad + \Vert b^\lambda (\pi _n x-x,\cdot )-c^\lambda (Q_n y-y,\cdot )\Vert \\ &{} \le G^\lambda (x,y) + S^\lambda (x,y)+ \Vert a^\lambda \Vert \Vert \pi _n x-x \Vert + \Vert b^\lambda \Vert \Vert Q_n y-y\Vert \\ &{} \quad + \Vert b^\lambda \Vert \Vert \pi _n x-x\Vert + \Vert c^\lambda \Vert \Vert Q_n y-y \Vert \\ &{} \le G^\lambda (x,y) + S^\lambda (x,y) \\ &{} \quad +\max \left\{ \Vert \pi _n x-x \Vert _E , \Vert Q_n y-y \Vert _F \right\} (M+2N+R). \end{array}\nonumber \\ \end{aligned}$$
(5.2)

And, on the other hand,

$$\begin{aligned} \begin{array}{rl} G^\lambda (\pi _n x,Q_n y) &{} = \Vert \phi ^\lambda -a^\lambda (\pi _n x,\cdot )-b^\lambda (\cdot ,Q_n y)\Vert \\ &{} \le \Vert \phi ^\lambda \circ \pi _n -a^\lambda (\pi _n x,\pi _n (\cdot ))-b^\lambda (\pi _n (\cdot ) ,Q_n y) \Vert \\ &{} \quad + \Vert a^\lambda (\pi _n x,\pi _n(\cdot ))- a^\lambda (\pi _n x,\cdot ) \Vert + \Vert b^\lambda (\pi _n (\cdot ) ,Q_n y ) -b^\lambda (\cdot ,Q_n y )\Vert \\ &{} \quad + \Vert \phi ^\lambda -\phi ^\lambda \circ \pi _n \Vert \\ &{} \displaystyle \le G^\lambda _n (\pi _n x,Q_n y) \\ &{} \quad \displaystyle + M \Vert \pi _n x \Vert \sup _{v\in E,\Vert v \Vert _E=1} \Vert v-\pi _n v \Vert _E + N \Vert Q_n y \Vert \sup _{v\in E,\Vert v \Vert _E=1} \Vert v-\pi _n v \Vert _E \\ &{} \quad + \mu \displaystyle \sup _{v\in E,\Vert v \Vert _E=1} \Vert v-\pi _n v \Vert _E \\ &{} =G^\lambda _n (\pi _n x,Q_n y)+(M \Vert \pi _n x \Vert +N \Vert Q_n y \Vert + \mu )\displaystyle \sup _{v\in E,\Vert v \Vert _E=1} \Vert v-\pi _n v \Vert _E, \end{array} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{rl} S^\lambda (\pi _n x,Q_n y) &{} =\Vert \psi ^\lambda -b^\lambda (\pi _n x,\cdot )-c^\lambda (Q_n y,\cdot )\Vert \\ &{} \le \Vert \psi ^\lambda \circ Q_n- b^\lambda (\pi _n x,Q_n(\cdot ))-c^\lambda (Q_n y,Q_n (\cdot ))\Vert +\Vert b^\lambda (\pi _n x,Q_n(\cdot ))- b^\lambda (\pi _n x,\cdot )\Vert \\ &{} \quad +\Vert c^\lambda (Q_n y,Q_n (\cdot )) -c^\lambda (Q_n y,\cdot ) \Vert +\Vert \psi ^\lambda -\psi ^\lambda \circ Q_n\Vert \\ &{} \le S^\lambda _n(\pi _n x,Q_n y)+N \Vert \pi _n x\Vert \displaystyle \sup _{w\in F,\Vert w\Vert _F=1} \Vert w-Q_n w \Vert _F \\ &{} \displaystyle \quad + R \Vert Q_n y\Vert \sup _{w\in F,\Vert w\Vert _F=1} \Vert w-Q_n w \Vert _F +\nu \sup _{w\in F,\Vert w\Vert _F=1} \Vert w-Q_n w \Vert _F \\ &{} = S^\lambda _n(\pi _n x,Q_n y)+ \displaystyle (N \Vert \pi _n x\Vert +R \Vert Q_n y\Vert + \nu ) \sup _{w\in F,\Vert w\Vert _F=1} \Vert w-Q_n w \Vert _F . \end{array} \end{aligned}$$

Then,

$$\begin{aligned} \begin{array}{rl} G^\lambda (\pi _n x,Q_n y)+S^\lambda (\pi _n x,Q_n y) &{} \le G^\lambda _n (\pi _n x,Q_n y) +S^\lambda _n (\pi _n x,Q_n y) \\ &{} \quad + \displaystyle (M \Vert \pi _n x \Vert +N \Vert Q_n y \Vert + \mu )\displaystyle \sup _{v\in E,\Vert v \Vert _E=1} \Vert v-\pi _n v \Vert _E \\ &{} \quad + \displaystyle (N \Vert \pi _n x\Vert +R \Vert Q_n y\Vert + \nu ) \sup _{w\in F,\Vert w\Vert _F=1} \Vert w-Q_n w \Vert _F. \end{array} \end{aligned}$$
(5.3)

Therefore, according to the compactness of \(\Lambda \), given a sequence of minimizers \(\{\lambda _n\}\) of \(G^\lambda _n(\pi _n x,Q_n y)+S^\lambda _n (\pi _n x,Q_n y)\) over \(\Lambda \), there exists a convergent partial subsequence, also noted \(\{\lambda _n\}\), i.e., there exists \(\lambda ^* \in \Lambda \) such that \(\{\lambda _{n}\}\rightarrow \lambda ^*\). To see that \(\lambda ^*\) is a minimizer of \(G^\lambda (x,y)+ S^\lambda (x,y)\) over \(\Lambda \), we compute

$$\begin{aligned} \begin{array}{rl} G^{\lambda ^*} (x,y)+S^{\lambda ^*}(x,y) &{} =\displaystyle \lim _{n \rightarrow +\infty } \left( G^{\lambda _n} (\pi _{n} x,Q_{n}y)+S^{\lambda _{n}}(\pi _{n} x,Q_{n} y) \right) \quad \text {(by (5.12))} \\ &{} \le \displaystyle \lim _{n \rightarrow +\infty } (G^{\lambda _{n}}_{n} (\pi _{n} x,Q_{n}y) \\ &{} \quad + S^{\lambda _{n}}_{n}(\pi _{n} x,Q_{n}y)+ (M \Vert \pi _{n} x \Vert +N \Vert Q_{n} y \Vert + \mu ) \displaystyle \sup _{v\in E,\Vert v\Vert _E=1} \Vert v-\pi _{n} v \Vert _E \\ &{} \quad +(N \Vert \pi _{\varepsilon _n} x\Vert +R \Vert Q_{n} y\Vert + \nu ) \displaystyle \sup _{w\in F,\Vert w\Vert _F=1} \Vert w-Q_{n} w \Vert _F) \quad \text {(minimizers)} \\ &{} \le \displaystyle \lim _{n \rightarrow +\infty } (G^{\lambda }_{n} (\pi _{n} x,Q_{n}y) + S^{\lambda }_{n}(\pi _{n} x,Q_{n}y) \\ &{} \quad + (M \Vert \pi _{n} x \Vert +N \Vert Q_{n} y \Vert + \mu )\displaystyle \sup _{v\in E,\Vert v\Vert _E=1} \Vert v-\pi _{n} v \Vert _E \\ &{} \quad +\displaystyle (N \Vert \pi _{n} x\Vert +R \Vert Q_{n} y\Vert + \nu ) \sup _{w\in F,\Vert w\Vert _F=1} \Vert w-Q_{n} w \Vert _F) \quad \text {(by (5.11))} \\ &{} \le \displaystyle \lim _{n \rightarrow +\infty } (G^\lambda (x,y) + S^\lambda (x,y) \\ &{} \quad \displaystyle + (M+2N+R) \max \left\{ \Vert \pi _{n} x-x \Vert _E , \Vert Q_{n} y-y \Vert _F \right\} \\ &{} \quad \displaystyle +(M \Vert \pi _{n} x \Vert +N \Vert Q_{n} y \Vert + \mu ) \sup _{v\in E,\Vert v\Vert _E=1} \Vert v-\pi _{n} v \Vert _E \\ &{} \displaystyle \quad +(N \Vert \pi _{n} x\Vert +R \Vert Q_{n} y\Vert + \nu ) \sup _{w\in F,\Vert w\Vert _F=1} \Vert w-Q_{n} w \Vert _F) \\ &{} = G^\lambda (x,y) + S^\lambda (x,y). \end{array} \end{aligned}$$

\(\square \)

Remark 5.2

Condition (ii) is not too restrictive. In fact, if we suppose that the Hilbert spaces E, F, \(\{E_n\}_{n\in \mathbb {N}}\), \(\{F_n\}_{n\in \mathbb {N}}\) are separable and verify (i) and \(\overline{\bigcup _{n\in {\mathbb N}}E_n}=E\) and \(\overline{\bigcup _{n\in {\mathbb N}}F_n}=F\), then condition (ii) is satisfied.

Finally we illustrate the above results considering the following example related to Example 3.1.

Example 5.3

We consider the problem (3.1) with \(\delta = -2\) and f(x, y) the function for which the solution \(\psi (x,y)\) to (3.1) is \(10^3(x(1-x)y(1-y))^4\).

The same problem in a porous domain is (3.2) and if we take \(w=-\Delta \psi \) the mentioned problem is equivalent to

$$\begin{aligned} \left\{ \begin{array}{l} \Delta \psi +w=0 \hbox { in } \Omega _\varepsilon \\ -\Delta \psi -2 \psi =f(x,y) \hbox { in } \Omega _\varepsilon \\ \psi |_{\Gamma _\varepsilon } = 0 \\ \Delta \psi |_{\Gamma _\varepsilon } =0 \end{array} \right. , \end{aligned}$$
(5.4)

which could be written as (\(P_\varepsilon \)).

Our purpose is to recover A, B and C in the perturbed mixed system

$$\begin{aligned} \left\{ \begin{array}{l} A \Delta \psi +B w =0 \hbox { in } \Omega _\varepsilon \\ -A \Delta w+C\psi = f(x,y) \hbox { in } \Omega _\varepsilon \end{array} \right. . \end{aligned}$$

Observe that the exact values are \(A=B=1\) and \(C=-2\).

We consider four holes which are randomly taken with different shapes (squares, circles, and ellipses). The Tables 1 and 2 show the results after running the collage codding approach over the perforated domains for different sizes, considering n = 10 y n = 20 respectively. We will denote r the circle radius, l the square side and a and b the ellipse major and minor axis respectively.

Table 1 Results with n = 10
Full size table
Table 2 Results with n = 20
Full size table

6 Conclusion

Some conditions for the existence of solution of a perturbed mixed variational system and that of an associated inverse problem have been given. Furthermore, some convergence results related to the impact of the size of the holes have been derived. The numerical results show that as hole diameter decreases, results improve.