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ASYMPTOTIC BEHAVIOR,
WITH RESPECT TO THE DOMAIN,
OF SOLUTIONS
OF PARTIAL
DIFFERENTIAL EQUATIONS

Institut de Recherche
d'Informatique
et d'Automatique

Domaine de Voluceau
Rocquencourt
B.P. 105 78150 - Le Chesnay
France
Tél.: 954 90 20

Olivier PIRONNEAU
Christian SAGUEZ

Rapport de Recherche N° 218
Février 1977

3ep.
ASYMPTOTIC BEHAVIOR, WITH RESPECT TO
THE DOMAIN, OF SOLUTIONS OF PARTIAL
DIFFERENTIAL EQUATIONS

O. Pironneau, C. Saguez

Résumé :

Dans cet article on étudie le comportement asymptotique des solutions d'équations aux dérivées partielles par rapport à la convergence de Hausdorff des domaines. Les résultats sont appliqués à quelques problèmes de la physique, à des problèmes d'optimum design et à des problèmes de contrôle dans les inéquations variationnelles.

Abstract :

This paper studies the asymptotic behavior of solutions of partial differential equations with
respect to the convergence of the domains in the sense of Hausdorff. Applications of the results
to physical problems, optimum design problems and optimal control of variational inequalities
are studied.
## 0. INTRODUCTION.

We shall consider the following typical problem : let {$\Omega_n$} be a family of open sets of $\mathbb{R}^N$ and {$u_n$} the solutions of the corresponding Dirichlet problems :

$$ - \Delta u_n = f \text{ in } \Omega_n ; u_n = 0 \text{ on } \partial \Omega_n. $$

What happens to {$u_n$} when {$\Omega_n \to \Omega$} ?

Such problems have been studied by NECAS [15], when {$\Omega_n \subset \Omega_{n+1}$} $\forall n$, LIONS [9], PIRONNEAU [16] and others, when {$\partial \Omega_n \to \partial \Omega$} strongly. Here we shall consider a much weaker type of convergence for {$\Omega_n$} namely with respect to the Hausdorff metric. This type of study aims at replacing P.D.E.'s on domains with very irregular boundaries by P.D.E.'s on simpler domains.

For example, we shall see that the flow around a ciliated microorganism is very close to the flow around the outer envelope of the cilia. In particular we have obtained an interesting formula for the computation of the momentum on the organism due to the flow. Other applications may be found in fluid mechanics (e.g. meteorology : how to take mountains into account) and in other fields such as acoustics...

We shall also see that this method can be applied to the theory of control of variational inequalities : it is shown that, with the Hausdorff metric, the free boundary of a variational inequality depends continuously on the control. At last the above results show that the problems of optimum design are numerically stable ; they can also be used to obtain existence theorems for certain optimum design problems.

The theoretical results of this study can essentially be formulated as follows : let {$\Omega_n$} $n \ge 1$ be a family of open sets of $\mathbb{R}^N$, included in $\mathcal{O}$, open bounded, and let {$u_n$} $n \ge 1$ be the solutions of a (elliptic, parabolic or hyperbolic "classical") P.D.E. in $\Omega_n$. Let $\mathcal{O}-\Omega_n$ be an accumulation point (w/r Hausdorff) of {$\mathcal{O}-\Omega_n$} and let $u$ be an accumulation point of {$u_n$} "in the weak sense of the P.D.E."
Let $\ell$ be an accumulation point in $L^\infty(\mathcal{O})$ weak star of the sequence of characteristic functions of $\mathcal{O}-\Omega_n$. If $\ell$ is such that $\{x|\ell(x) \neq 0\} = \mathcal{O} - \Omega$, and if $\Omega$ is an open set, then $u$ is a solution of the P.D.E. in $\Omega$. In certain cases one can show that $\{u_n\}$ converges in a strong sense. Computations of correctors are sketched in the cases where $\Omega_n$ have (space-wise) periodic boundaries and the period (but not the amplitude) tends to zero.

The plan of the paper is the following :

I - Elliptic problems of order 2 with Dirichlet conditions.

. Position of the problem

. Main results

. Applications

. Multiple scales and correctors

. Illustration

II - *Other elliptic problems.*

. Higher order operators

. Stationary Navier-Stokes equation

### III - *Applications*

. Ciliated microorganisms

. Applications to optimum design problems

. Applications to variational inequalities

IV - *Parabolic and hyperbolic problems*

. Parabolic problems

. Parabolic problems in non cylindrical domains

. Second order hyperbolic problems
# I - ELLIPTIC PROBLEMS OF ORDER 2 WITH DIRICHLET CONDITIONS.

## I.1 Position of the problem :

Let $\mathcal{O}$ be a bounded open set in $\mathbb{R}^N$, with regular boundary. Let $\{F_n\}$ be a family of closed sets in $\bar{\mathcal{O}}$. Let

$$ (1.1) \qquad \Omega_n = \mathcal{O} - F_n $$

<figure><img src="image_2.png" /><figcaption>Figure 1</figcaption></figure>

Let $\{f_n\}$ be a family of functions of $L^2(\mathcal{O})$. Let

$$ (1.2) \qquad a(u,v) = \int_{\mathcal{O}} a_{ij}(x) \frac{\partial u}{\partial x_i} \frac{\partial v}{\partial x_j} dx + \int_{\mathcal{O}} a_o(x) uvdx $$

where $a_o, a_{ij} \in L^\infty(\mathcal{O})$, are such that there exists $\alpha > 0$ with

$$ (1.3) \qquad a_{ij} \xi_i \xi_j \geq \alpha |\xi|^2, \quad a_o \geq 0 \quad \forall \xi \in \mathbb{R}^N $$

(repeated indices are summed from 1 to N)

Let $u_n \in H^1(\Omega_n)$ be the solution of the Dirichlet problem :

$$ (1.4) \qquad Au_n = f_n \quad \text{in } \Omega_n ; \quad u_n|_{\partial\Omega_n} = 0 $$

where

$$ (1.5) \qquad A = - \frac{\partial}{\partial x_j} a_{ij} \frac{\partial}{\partial x_i} + a_o $$
The $\hat{\wedge}$ will denote the extension in $\mathcal{O}$ by zero. Then $\hat{u}_n$ is the solution of :

$$
(1.6) \quad a(\hat{u}_n, \hat{v}) = (f_n, \hat{v}) , \quad \forall v \in H_o^1(\Omega_n) ; u_n \in H_o^1(\Omega_n)
$$

We shall study {$\hat{u}_n$} when $n \to +\infty$

I.2) Main result :

Hypotheses :

H1 $f_n \to f$ in $L^2(\mathcal{O})$ weakly

H2 i) $F_n \to F$ with respect to the Hausdorff metric$^{(*)}$

ii) $F$ is closed and such that : $\phi \in H_o^1(\mathcal{O})$, $\phi=0$ a.e. in $F \Rightarrow \phi \in H_o^1(\mathcal{O}-F)$

H3 Let $X_n = X_F$ be the characteristic function of $F_n$ in $\mathcal{O}$

i) $X_n \to \ell$ in $L^1(\mathcal{O})$ weak star

ii) $\{x|\ell(x)>0\} = F$ a.e.

**Theorem 1** : If H1-H3 hold, {$\hat{u}_n$} converges strongly in $H^1(\mathcal{O})$ toward $\hat{u}$
where the restriction u of $\hat{u}$, to $\Omega = \mathcal{O}-F$, is the solution in $H^1_0(\Omega)$ of

$$
(1.7) \quad \text{Au} = f \text{ in } \Omega
$$

and $\hat{u}=0$ a.e. in $F$

Proof :

From (1.6) one can prove that

$$
(1.8) \quad \| \hat{u}_n \|_{H_o^1(\mathcal{O})} \le C
$$

Therefore there exists $\hat{u} \in H_o^1(\mathcal{O})$ and a subsequence, also denoted by{$\hat{u}_n$},
such that

<footnote>(*) The Hausdorff distance between A and B is defined as :</footnote>

<footnote>$$
\delta(A,B) = \sup[\rho(A,B), \rho(B,A)] \text{ where } \rho(A,B) = \sup_{x \in A} d(x,B)
$$
(for more details see DELLACHERIE [5])
$$</footnote>
$$
(1.9) \qquad \hat{u}_n \to \hat{u} \text{ in } H_o^1(\mathcal{O}) \text{ weakly (therefore strongly in } L^2(\mathcal{O}))
$$

Hence from H3 i)

$$
0 = X_n \hat{u}_n \to \ell \hat{u} \text{ weakly in } L^2(\mathcal{O})
$$

which from H3 ii) implies that

$$
(1.10) \qquad \hat{u}=0 \quad \text{a.e. in } F
$$

Now from H2 i) and lemma I.3.1 in Pironneau [ ] (or lemma 2)

$$
\forall v \in H_o^1(\Omega), \exists \{v_n\} \quad v_n \in H_o^1(\Omega_n) \text{ such that}
$$

$$
\hat{v}_n \to \hat{v} \text{ strongly in } H_o^1(\mathcal{O})
$$

Therefore from (1.9) :

$$
a(\hat{u}, \hat{v}) - (f, \hat{v}) = \lim_{n \to +\infty} [a(\hat{u}_n, \hat{v}_n) - (f_n, \hat{v}_n)] = 0 \quad \forall v \in H_o^1(\Omega)
$$

which shows the weak convergence of {$\hat{u}_n$}. The strong convergence follows
from the coercivity of a and the fact that : ($f_n, u_n$) $\to$ (f, u) implies
$a(u_n, u_n)$ $\to$ a(u, u). From (1.10) and H2 ii), $u \in H_o^1(\mathcal{O}-F)$ which completes the
proof. $\blacksquare$

**Remark 1** : If {f_n} is bounded in L^2(Ω), then one can always extract a subsequence such that H1, H2 i) (see DELLACHERIE [5]) and H3 i) are satisfied.

**Remark 2** : Note that H2 i) is too strong. It can be replaced by :
H2'i) $\forall x \in F \quad \exists \{x_n\} \quad x_n \in F_n$ such that $x_n \to x$.

**Remark 3** : Note that hypotheses H1-H3 are sufficient but not necessary in theorem 1 (see example below)
### I.3) Applications.

Let $O = ]0,1[ \times ]0,1[$. Let $P = [0,1] \times [b,1]$, $b \in ]0,1[$.

Let $P_1$ be a closed set of $P$ having at most $n$'s connex components with continuous boundaries and such that $\bar{P}_1 = P_1$. We shall consider :

$$ (1.11) \qquad F_{\varepsilon} = \{(k\varepsilon + \varepsilon y_1, y_2) | (y_1, y_2) \in P_1, k \in \mathbb{N}, k\varepsilon < 1\} $$

<figure><img src="image_3.png" /><figcaption>Figure 2</figcaption></figure>

Geometry of example I.2. The quantity $\ell(y)$ is shown as the length of $\equiv$.

#### Lemma 1 : (LIONS [11])

If $F_{\varepsilon}$ is given by (1.11) then $\{X_{\varepsilon}\}_{\varepsilon}$ converges in $L^{\infty}(O)$ weak star toward $\ell(\cdot)$; where $\ell(y)$ is the Lebesgue measure of $P_1 \cap \{y_2=y\}$.

#### Proof : It suffices to show that

$$ (1.12) \qquad I_{\varepsilon} = \int \chi_{\varepsilon}(x_1, x_2) \phi(x_1) \psi(x_2) dx \rightarrow \int \ell(x_2) \phi(x_1) \psi(x_2) dx $$

$\forall \phi, \psi$ continuous with compact support in $]0,1[$.

Let $K(x) = F_{\varepsilon} \cap \{x_1=x\}$, then

$$ I_{\varepsilon} = \int_{0}^{1} \phi(x_1) dx_1 \int_{K(x_1)} \psi(x_2) dx_2 $$
But K is periodic of period $\epsilon$, therefore

$$k(y_1) = \int_{K(\epsilon y_1)} \psi(x_2) dx_2 \text{ is periodic of period 1 and}$$
$$I_{\epsilon} = \int_{0}^{1} \phi(x_1) k\left(\frac{x_1}{\epsilon}\right) dx_1.$$

Therefore

$$I_{\epsilon} \rightarrow \int_{0}^{1} \phi(x_{1}) \overline{k} dx$$

where

$$(1.13) \qquad \bar{k} = \int_{0}^{1} k(y_1) dy_1 = \int_{0}^{1} dy_1 \int_{K'(y_1)} \psi(x_2) dx_2 = \int_{0}^{1} \psi(x_2) \ell(x_2) dx_2$$

and where

$$(1.14) \qquad K'(y_1') = P_1 \cap \{y_1 = y_1'\} . \blacksquare$$

**Corollary 1 :**

If $F_{\epsilon}$ is given by (1.11) then the conclusion of theorem 1 holds with $F = \bigcup_{\epsilon} F_{\epsilon}$, provided that hypothesis H1 holds.

**Proof :** From Lemma 1, hypothesis H3 is satisfied with $F = \bigcup_{\epsilon} F_{\epsilon} = [0,1] \times [b,1]$. On the other hand every point of F is the limit of a sequence of points in $F_{\epsilon}$; therefore from Remark 2 theorem 1 holds if H1 holds.

**Corollary 2 :**

If all $F_{\epsilon}$ are described by a finite number of local maps $\{(O_i, \phi_i)\}_{i=1}^m$ such that $\phi_i^{-1}(\bar{O}_i \cap F_{\epsilon})$ is in the form of (1.11) then theorem 1 holds with $F = \bigcup_{\epsilon} F_{\epsilon}$. The reader will extend easily this result to $R^N$.

**Remark 4 :** The above results still holds when $u_n$, solution of (1.4), satisfies a non-homogeneous boundary condition of the type

$$(1.15) \qquad u_n|_{\partial \Omega_n} = z|_{\partial \Omega_n}, \quad z \in H^1(\partial \Omega)$$
### I.4) Multiple scales and correctors.

In this section, we shall show briefly how the result of the corollary 2 can be derived from the theory of homogenization (LIONS [10], BENSOUSSAN-LIONS-PAPANICOLAU [3]). This method can also be used to study ${1/\epsilon}$ ($u_\epsilon - u$) (correctors). For the sake of simplicity we shall take $\mathcal{O} = ]0,1[ \times ]0,1[ $ and

$$ (1.16) \qquad F_{\epsilon} = \{(x_1, x_2) | 1 \ge x_2 \ge g(\frac{x_1}{\epsilon}), x_1 \in ]0,1[\} $$

where $g$ is continuous periodic of period $T$, strictly positive and less than 1. Then

$$ (1.17) \qquad b = \min_{x_1 \in [0,T]} g(x_1) $$

We shall also suppose that $f_\epsilon(x_1,x_2) = f(\frac{x_1}{\epsilon}, x_1, x_2)$; $f(\cdot, x_1, x_2)$ T-periodic.
Let $u_\epsilon$ be the solution of (1.4). In the new variables

$$ (1.18) \qquad X_1 = x_1; \quad X_2 = x_2/g\left(\frac{x_1}{\epsilon}\right) \quad (X_1, X_2) \in D = ]0,1]^2 $$

$u_\epsilon(X_1, X_2)$ is the solution of a Dirichlet problem whose differential operator has periodic coefficients in :

$$ (1.19) \qquad Y_1 = X_1/\epsilon $$

Therefore let us suppose that $u_\epsilon$ can be developed as :

$$ (1.20) \qquad u_{\epsilon}(X) = w_{0}(X, Y_{1}) + \epsilon w_{1}(X, Y_{1}) + \dots $$

where $w_i(\cdot, Y_1)|_{\partial D} = 0$; $w_i(X, \cdot)$ T-periodic and $w_i \in H^1(D \times ]0, T[)$

By identification in $\epsilon$, one found that in the old variables one must have
$w_0(X, Y_1) = u(x_1, x_2)$ and

$$ (1.21) \qquad \left\{ \begin{array}{l} Au = \frac{1}{T} \int_{0}^{T} f(x_1, x_2, y_1) dy_1 = \bar{f} \quad \text{in } ]0,1[ \times ]0,b[ = \Omega \\ u|_{\partial \Omega} = 0 \qquad u|_{\partial \mathcal{O} - \Omega} = 0 \end{array} \right. $$
(1.21) is identical to theorem 1.

One may proceed to compute $w_1, w_2, \dots$. However the reader will check that $w_1 \equiv 0$ and $w_2$ satisfies a partial differential equation whose solution is not in $H^1(\mathcal{O})$. Further investigation shows that this P.D.E. is the $\varepsilon$-limit of a well-posed P.D.E. which has a boundary layer. Therefore the proper development of $u_\varepsilon$ is :

$$ (1.22) \quad \left\{ \begin{aligned} u_\varepsilon(x) &= u(x) + \varepsilon^2 w_2^0(x, \frac{x_1}{\varepsilon}) + \varepsilon w_2^1(x, \frac{x_1}{\varepsilon}, \frac{x_2^{-b}}{\varepsilon}) + \\ & \qquad + \varepsilon^3 w_3^0(x, \frac{x_1}{\varepsilon}) + \varepsilon^2 w_3^1(x, \frac{x_2^{-b}}{\varepsilon}) + \dots \end{aligned} \right. $$

where $w_2^0$ is $\frac{x_1}{\varepsilon}$-periodic and $w_2^1$ is $\frac{x_1}{\varepsilon}$-periodic if $y_2 = \frac{x_2^{-b}}{\varepsilon} \le 0$. Then from (1.4) $w_2^0$ must satisfy :

$$ (1.23) \quad -a_{11} \frac{\partial^2 w_2^0}{\partial y_1^2} = f - \bar{f} \quad (y_1 = \frac{x_1}{\varepsilon}) $$

and $w_2^1$ must satisfy :

$$ (1.24) \quad \left\{ \begin{aligned} & -a_{ij} \frac{\partial^2 w_2^1}{\partial y_i \partial y_j} = -\delta(y_2=b) \frac{\partial u}{\partial x_2} (x_1, b) \\ & w_2^1(x, y_1, \pm\infty) = 0 \\ & w_2^1(x, 0, y_2) = w_2^1(x, T, y_2) = 0 \end{aligned} \right. $$

where $\delta(y_2=b)$ is the Dirac function on the line $y_2=b$. ($\langle \delta(y_2=b), v(y_1, y_2) \rangle =$ $v(y_1, b)$)

### I.5) Illustration

Let $\mathcal{O} = \mathbb{R} \times \mathbb{R}^+$ ($\mathcal{O}$ is not bounded)

and

$$ (1.25) \quad \left\{ \begin{array}{l} F_\varepsilon = \{(x_1, x_2) | x_1 = n\varepsilon, n \in \mathbb{Z}, \\ \qquad 0 \le x_2 \le b\} \cup (\mathbb{R} \times \{0\}) \end{array} \right. $$

and let $u_\varepsilon$ be the solution of

<caption>Figure 3</caption>
$$
(1.26) \quad \left\{
\begin{array}{l}
- \Delta u_{\varepsilon} = 0 \text{ in } \mathcal{O}-F_{\varepsilon} \\
u_{\varepsilon}|_{F_{\varepsilon}} = 0 \quad u|_{\infty} = u_o x_2
\end{array}
\right.
$$

(u_ε is the stream function of a potential flow over the "comb" F_ε with
uniform speed at infinity).

By making use of the hodograph method and of the conformal mapping :

$\sin \frac{z}{2} = Z$, one can compute the exact solution of (1.26) (MORICE [13]) :

$$
(1.27) \quad u_{\epsilon} = \frac{2u_o \epsilon}{\pi} \quad \operatorname{Im} \left\{ \arg \sin \left[ \frac{\operatorname{ch} \frac{\pi}{2\epsilon} (x_1 + ix_2)}{\operatorname{ch} \frac{\pi b}{2\epsilon}} \right] \right\}
$$

when $\epsilon \to 0$, $u_\epsilon \to u$, uniformly with

$$
(1.28) \quad u = \begin{cases} 0 & \text{if } x_2 < b \\ x_2 - b & \text{if } x_2 \ge b \end{cases}
$$

Also : $F_{\epsilon} \rightarrow \mathbb{R} \times [0,b]$ in the sense of Hausdorff. However $\chi_{\epsilon}$ is identically zero and hypothesis H3 ii) is not satisfied in that case.

In order to simplify the notations, let $u_o=1$ and $\epsilon = 2\epsilon/\pi$. If $u_\epsilon$ is the
real part of the complex function in (1.27) then

$$
(1.29) \quad \sin \frac{v_{\epsilon}}{\epsilon} \quad \ch \frac{u_{\epsilon}}{\epsilon} = \cos \frac{x}{\epsilon} \quad \ch \frac{x_2}{\epsilon} / \ch \frac{b}{\epsilon}
$$

$$
(1.30) \quad \cos \frac{v_{\epsilon}}{\epsilon} \quad \sh \frac{u_{\epsilon}}{\epsilon} = - \sin \frac{x_1}{\epsilon} \quad \sh \frac{x_2}{\epsilon} / \ch \frac{b}{\epsilon}
$$

From (1.29), (1.30) it is easy to show that

$$
(1.31) \quad u_{\epsilon} = - \epsilon \sin \frac{x_1}{\epsilon} \exp \left( \frac{x_2 - b}{\epsilon} \right) + o(\epsilon) \text{ if } x_2 < b
$$

$$
(1.32) \quad u_{\epsilon} = x_2 - b + \epsilon \text{Log} \, 2 + \epsilon \cos 2 \frac{x_1}{\epsilon} \exp \left( - \frac{x_2 - b}{\epsilon} \right) + o(\epsilon) \text{ if } x_2 > b
$$

If (1.22) is used then

$$
(1.33) \quad u_{\epsilon} = (x_2 - b)^{+} + \epsilon w_2^{1}
$$
where $w_2^1(y)$ is a solution in $]0, 2\pi[ \times R$ of

$$ - \Delta_y w_2^1 = - \delta(y=b) , w_2^1(y_1, \pm\infty) = 0 , w_2^1 y_1\text{-periodic if } y_2 > b $$

Therefore $w(y_1, y_2') = w_2^1(y_1, y_2'+b) - y_2'^+$ is the solution of (1.26) over a period in the domain of figure 4.

Therefore from (1.29), with $b = +\infty$, $y_2' = y_2 - b$, $\epsilon=1$:

$$ (1.34) \qquad \begin{cases} \sin v \ch \omega = \cos y_1 \exp y_2' \\ \cos v \sh \omega = - \sin y_1 \exp y_2' \end{cases} $$

where $v$ is to $\omega$ what $v_\epsilon$ is to $u_\epsilon$.

The reader will verify that (1.33)(1.34) is identical to (1.31), (1.32).

<figure><img src="image_4.png" /><figcaption>Figure 4</figcaption></figure>
## II - OTHER ELLIPTIC PROBLEMS.

### II.1) Higher order operators.

Let us show on an example that the method described in Section I can be adapted to higher order operators.

For example let $u_n$ be the solution of

$$ (2.1) \qquad \Delta^2 u_n = f_n \quad \text{in } \Omega_n = \mathcal{O}-F_n $$

$$ (2.2) \qquad u_n = \frac{\partial u}{\partial n} = 0 \quad \text{on } \partial\Omega_n $$

where $\Omega_n$, $F_n$, $f_n$ satisfy the hypotheses of section I.

**Theorem 2** : If H1-H2-H3 hold, then {$\hat{u}_n$} converges to $\hat{u}$ in $H^2(\mathcal{O})$ strongly, where $\hat{u} = 0$ in $F$ and where the restriction $u$ of $\hat{u}$ to $\mathcal{O}-F$ is the solution of :

$$ (2.3) \qquad \Delta^2 u = f \quad \text{in } \Omega. $$

$$ (2.4) \qquad u = \frac{\partial u}{\partial n} = 0 \quad \text{on } \partial\Omega $$

the $\hat{u}$ denotes the extension by zero in $H_o^2(\mathcal{O})$

**Proof** : A variational formulation of (2.1)-(2.2) is

$$ (2.5) \qquad \int_{\Omega_n} \Delta u_n \Delta v dx = \int_{\Omega_n} f_n v dx \quad \forall v \in H_o^2(\Omega_n) , u_n \in H_o^2(\Omega_n) $$

We proceed exactly as in Theorem 1. From (2.5) we obtain :

$$ \| \hat{u}_n \|_{H^2(\mathcal{O})} \le C $$

Therefore

$$ 0 = X_n \hat{u}_n \to \ell \hat{u} \quad \text{in } L^2(\mathcal{O}) \quad \text{weakly} $$

$$ 0 : X_n \nabla \hat{u}_n \to \ell \nabla \hat{u} \quad \text{in } (L^2(\mathcal{O}))^N \quad \text{weakly.} $$

Therefore $\hat{u}=0$ in $F$ and $\nabla \hat{u}=0$ in $F$. Since $\mathcal{D}(\mathcal{O})$ is dense in $H_o^2(\mathcal{O})$, one can show that
$\forall v \in H_o^2(\Omega) \quad \exists v_n \in H_o^2(\Omega_n)$ such that $\hat{v}_n \to \hat{v}$ in $H^2(\bar{\Omega})$ strongly.

The proof can be completed as in theorem 1. ■

## II.2) Stationary Navier-Stokes equation.

In this section $N=2$ or $3$.

With the notations of section I, let us suppose that there exists an open set of $\mathbb{R}^N$, $D$ such that

$$F_n \subset \bar{D} \subset \mathcal{O} \quad \forall n$$

Then let $(u_n, p_n)$ be the solution of

$$ (2.6) \quad \left\{ \begin{array}{l} v\Delta u_n - u_n \nabla u_n = \nabla p_n - f, \quad \nabla \cdot u_n = 0 \quad \text{in } \Omega_n = \mathcal{O} - F_n \\ u_n|_{\partial \mathcal{O}} = z|_{\partial \mathcal{O}}, \quad u_n|_{\partial F_n} = 0 \end{array} \right. $$

where

$z \in H^1(\mathcal{O})^N, f_n \in L^2(\mathcal{O})^N \text{ and } \nabla \cdot z = 0 \text{ and } z=0 \text{ in } D.$

We shall assume that $v, z, f_n$ and $\mathcal{O}$ are such that (2.6) has a unique solution (see Ladyzhenskaya [7]).

A variational formulation for (2.6) is

$$ (2.7) \quad \left\{ \begin{array}{l} \displaystyle \int_{\Omega_n} (v \frac{\partial u_n}{\partial x_i} - u_n \frac{\partial v}{\partial x_i}) \cdot u_n \, dx = \int_{\Omega_n} f_n v \, dx \quad \forall v \in V(\Omega_n) \\ \\ \qquad u_n - z \in V(\Omega_n) = \{v \in D(\Omega_n)^N \mid \nabla \cdot v = 0\} \quad \text{H}^1(\mathcal{O}) \\ \\ \displaystyle \int_{\Omega_n} \frac{\partial u_n}{\partial x_i} \frac{\partial v}{\partial x_i} - u_n \cdot u_n \frac{\partial v}{\partial x_i} \, dx = \int_{\Omega_n} f_n v \, dx \quad \forall v \in V(\Omega_n) \end{array} \right. $$

**Proposition 1** : If hypotheses H1-H3 are satisfied, then $\{u_n\}$ solutions of (2.6) (i.e. (2.7)) converge strongly in $H^1(\mathcal{O})^N$ to $\hat{u}$, the zero-extension, in $\mathcal{O}$, of the solution $u$ of (2.7) with $\Omega_n = \mathcal{O}-F$.

**Proof** : From (2.7) it is shown in Ladyzhenskaya [7, p.121] that $\hat{u}_n$ is bounded in $H^1(\mathcal{O})$, furthermore the bound does not depend upon $\Omega_n$ (see [7]). Therefore, for a subsequence, $\hat{u}_n \to \hat{u}$ weakly in $H^1(\mathcal{O})^N$. The rest of the proof goes exactly as in theorem 1. There is no difficulty in taking the
limit in (2.7) with the non linear terms since $\hat{u}_n \to \hat{u}$ strongly in $L^2(\mathcal{O})^N$.
To show that the convergence is in the strong sense in $H^1$ let us take
$v = u_n - z$ in (2.7). Note that

$$ \int_{\Omega_n} u_n \cdot \nabla (u_n - z) (u_n - z) dx = 0, \text{ therefore } \| \nabla \hat{u}_n \|_{L^2(\mathcal{O})}^2 \to \| \nabla \hat{u} \|_{L^2(\mathcal{O})}^2 $$
which, in turn, implies that $\| \nabla (\hat{u}_n - \hat{u}) \|_{L^2(\mathcal{O})} \to 0$.

**Remark 5 : Neumann problems.**

The study of the asymptotic behavior of {$u_n$} where $u_n$ is the solution of (1.4) in $\mathcal{O}-F_n$ with an homogeneous Neumann boundary condition on $\partial F_n$ is an open problem. When $F_n$ is as in (1.11), the theory of homogenization seems to indicate that $u_n \to u$ with

$$ (2.8) \quad \frac{\partial}{\partial x_j} \left( (1-\ell)a_{ij} \frac{\partial u}{\partial x_i} \right) = f \quad \text{in } \mathcal{O} - \cap F_n $$

If $F_n$ is as in (1.16) with $g \in C^2[0,1]$ then (2.8) can give

$$ (2.9) \quad \frac{\partial u}{\partial v_A} = 0 \quad \text{on } \partial(\cup F_n - \cap F_n) $$

<figure><img src="image_5.png" /><figcaption>Figure 5</figcaption></figure>

### III - APPLICATIONS.

#### III.1 Ciliated micro-organisms

Some microorganisms are provided with a dense coat of flexible cilia which are used for swimming. Microorganisms like the Paramecia move their cilia in a synchronized motion in such a manner that it appears that periodic waves are propagated on the envelope of the cilia. It is well-known in fluid mechanics that in order to study such motions (called symplectic beat) one may replace the ciliated microorganism by the outer envelope of its cilia (see Blakes [4]). We shall now prove this result and also obtain a formula for the momentum of the microorganism due to the fluid.

Thus let $F_n(t)$ be the microorganism at time $t$ and $n$ the number of cilia; let $\mathcal{O}-F_n$ be the region occupied by the fluid and $u_n(x,t)$ the speed of the fluid at $x \in \mathcal{O}-F_n(t)$ at time $t$. Let $z(x,t)$ be the speed of the cilia passing through $x \in \partial F_n(t)$ at time $t$. Then $u_n$ is given by
$$
(3.1) \quad \Delta u_n = \nabla p, \quad \nabla \cdot u_n = 0 \quad \text{in } \mathcal{O} - F_n(t) \quad \forall t \in ]0, T[
$$

$$
(3.2) \quad u_n = z(t) \text{ on } \partial F_n(t), \quad u_n|_{\partial \mathcal{O}} = 0 \quad \forall t \in ]0, T[
$$

Note that if the cilia do not move in a synchronized beat or if the
cilia bend too much so that their tips are at some time inside the
cilia layer, then it is not possible to define z(t) ∈ H¹(Ω) when
n → ∞. This is why this analysis apply to simplectic beats only.

The equations of the motion of the organism are obtained by writing
Newton's law. The fluid exerts a net force on the organism which must
balance the drag ; the forces of inertia are neglectible. If the
microorganism moves at constant speed **m**, the drag is (assuming **O** is
large) :

$$
(3.3) \quad d_n^m = \int_{\mathcal{O} - F_n} 2 \sigma_{ij}^m \sigma_{ij}^m dx \quad \text{where} \quad \sigma_{ij}^m = \frac{1}{2} \left( \frac{\partial u_{nj}^m}{\partial x_i} + \frac{\partial u_{ni}^m}{\partial x_j} \right)
$$

and where $u_n^m$ is the solution of (3.1) (3.2) with $\vec{z} = \vec{m}$. The net force of the fluid on $F_n$ is

$$
(3.4) \quad \vec{f}_n = \int_{\partial F_n} (2\sigma \underline{z} p^z \mathbb{I}) d\vec{\partial F}_n \quad (\mathbb{I} = (\langle \delta_{ij} \rangle))
$$

and the net momentum is

$$
(3.5) \quad \vec{\tau}_n = \int_{\partial F_n} \left[ (2\sigma \underline{z} p^z \underline{I})(x) d(\partial \vec{F}_n)(x) \right] \times \vec{x}
$$

Newton's laws are : $f_n = \tau_n = 0$

**Proposition 2**

For simplectic motions, when the number of cilia, n, tends to infinity,
the drag d_n^m ; the net force, due to the fluid, f_n and the net momentum
τ' on the organism tends respectively to d_m, f and τ' = τ + τ", where
d^m, f, τ are the same quantity computed for F = U F_n (enveloppe of the
cilia) and where
$$
(3.6) \quad \tau'' = \int_F (1-\ell(x)) \operatorname{rot} z \, dx, \text{ and } \ell = \lim_{n \to \infty} \chi_F^n .
$$

In other words to study the rectilinear motions of the micro-
organism one may consider a body with the shape of the enveloppe of
the tips of the cilia having the speed of the tips of the cilia.
However for the momentum there is a corrective factor τ", which takes
into account that fact that in between the cilia there is a moving
fluid.

Proof : The fact that d_n^m → d^m is due to proposition 1 (adapted to (3.1)). By integrations by parts one can show that for any constant vector v_o of ℝ^N → ℝ

$$
(3.7) \quad \vec{f}_n \cdot \vec{v}_o = \int_{\mathcal{O}-F_n} 2 \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \left( \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} \right) dx
$$

where v is any extension of v_o in H_o^1(\mathcal{O})

Therefore, from proposition 1 and since v = v_o in F,

$$
(3.8) \quad f_n \cdot v_o \rightarrow \int_{\mathcal{O}-F} 2 \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \left( \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} \right) dx = \int_{\partial F} (2\mathcal{G}-p\mathbb{I})\vec{v}_o \cdot d\vec{\partial F} = f \cdot v_o
$$

similarly, by integrations by parts, one can show that

$$
(3.9) \quad \vec{\tau}_n \cdot \vec{v}_o = \int_{\mathcal{O}-F_n} 2 \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \left( \frac{\partial (\vec{x} \times \vec{v})_i}{\partial x_j} + \frac{\partial (\vec{x} \times \vec{v})_j}{\partial x_i} \right) dx
$$

Therefore

$$
\begin{align*}
\tau_n \cdot v_o \rightarrow & \int_{\mathcal{O}} 2(1-\ell) \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) \left( \frac{\partial (\vec{x} \times \vec{v})_i}{\partial x_j} + \frac{\partial (\vec{x} \times \vec{v})_j}{\partial x_i} \right) dx \\
& = \int_{\partial F} (2\mathcal{G}-p\mathbb{I}) d\vec{\partial F} \times \vec{x} \cdot \vec{v}_o + \int_{F} (1-\ell)v_o \operatorname{rot} z \, dx
\end{align*}
$$

since $u_n \to u = z$ in F, and since $v = v_o$ in F. ■
### III.2) Applications to optimum design problems.

Let $u_{\Omega}$ be the solution of a P.D.E. in $\Omega$, $\Omega \subset \mathcal{O}$ ($\mathcal{O}$ bounded). Let $J(\Omega, u_{\Omega})$ be a functional. Problems of the type

$$ (3.10) \quad \text{Min} \quad \Omega \in \mathcal{S} \quad J(\Omega, u_{\Omega}) $$

are called optimum design problems. $\mathcal{S}$ is referred as the class of admissible domains. Such problems can be solved by iterative methods (PIRONNEAU [16]) which generates a sequence $\{\Omega_n\}$ such that $J(\Omega_n, u_{\Omega_n})$ is decreasing. Such studies show that it is not necessary to include in $\mathcal{S}$ constraints of the type $\partial\Omega \in \mathcal{C}^2$ to obtain accumulation points for $\{\Omega_n\}$.

We can also extend the existence theorems of GARABEDIAN [6], LIONS [2], MURAT-SIMON [14]. For notational convenience, let $u_{\Omega}$ be the solution of

$$ (3.11) \quad \text{Au} = f \text{ in } \Omega, u \in H_o^1(\Omega) $$

where $A$ is as in (1.5), and let

$$ (3.12) \quad J(\Omega, u_{\Omega}) = \|u-z_d\|^2_{L^2(\Omega)}, \quad z_d \text{ given in } L^2(\mathcal{O}) $$

#### Definition

Let $\theta>0, \varepsilon>0$; $B(x,\varepsilon)$ be the open ball of center $x$ and radius $\varepsilon$ and $C(x,\theta)$ be an open half cone of vertex $x$ and angle $\theta$. The set $\Omega$ satisfies the *uniform cone property* if for any $x \in \Omega$ one can insert $K = C(x,\theta) \cap B(x,\varepsilon)$ in $\Omega$.

The following result is essentially in CHENAIS [18], the previous technique enables us to give a simpler proof.

**Theorem 4** : If there exists $\theta>0, \varepsilon>0$ such that $\mathcal{S} = \{\Omega | \Omega \text{ open } \subset \mathcal{O}, \Omega \text{ or } \mathcal{O}'-\Omega\}$ satisfies the uniform cone property with parameters $\theta, \varepsilon$ then (3.10)(3.12) has at least one solution.

*Proof* : Let $\{\Omega_n\}$ be a minimizing sequence of $J$; let us extract a subsequence (denoted $\{\Omega_n\}$ also) such that
$\bar{\Omega}_n \to \Omega'$ Hausdorff

$F_n = \bar{\Omega} - \bar{\Omega}_n \to F \quad \text{''}$

$\partial\Omega_n \to A \quad \text{''}$

1) $\mathring{A} \neq \emptyset$ - Otherwise there exists $B(x, \varepsilon/2) \subset A$. Let $\{x_1, x_2, \dots, x_p\}$ be $p$ points on $\partial B(x, \varepsilon/2)$ such that $\forall K$ of vertex $x$ there exists $x_i$, $i \le p$, in $K$. By definition of $A$ there exists $y, y_1, \dots, y_p$ in $\partial\Omega_n$ arbitrarily close to $x, x_1, \dots, x_p$ respectively; this contradicts the fact that $\Omega_n$ or $\mathring{O} - \bar{\Omega}_n$ satisfies the cone property.

2) $\partial F = A = \partial \bar{\Omega}'$ - Since $\partial F_n \subset F_n$ we have $A \subset F$, and also $\partial F_n \subset \bar{\Omega}_n$ implies $A \subset \bar{\Omega}'$. Conversely let $x \in F \cap \bar{\Omega}'$; then there exists $\{x_n\}$, $x_n \in F_n$ and $\{y_n\}$, $y_n \in \bar{\Omega}_n$ such that $d(x, x_n) \le \frac{1}{n}$ and $d(x, y_n) \le \frac{1}{n}$. Therefore $x \in A$. Altogether this shows that $A = F \cap \bar{\Omega}'$.

3) $\mathring{\Omega}' \in \mathcal{S}$: Suppose that $\bar{\Omega}'$ and $F$ do not satisfy the uniform cone property, then: $\exists x, x_1, \dots, x_p \in \partial F$ such that $\forall K$ of vertex $x$, $\exists i$ such that $K \cap \partial F \supset \{x_i\}$. Again $(x, x_1, \dots, x_p)$ is arbitrarily close to $(y, y_1, \dots, y_p)$ in $\partial F_n$ and the above statement says that neither $\Omega_n$ nor $\mathring{O} - \bar{\Omega}_n$ satisfies the cone property.

Therefore the assumptions H1-H3 are satisfied and the theorem follows from theorem 1.

<figure><img src="image_6.png" /></figure>
In some special cases, one need not restrain $\mathcal{S}$ at all.

**Proposition 3**: If $u_{\Omega}$ is bounded in $C^{0}(\mathcal{O})$ uniformly and uniformly lip-
schitz for all $\Omega \in \mathcal{S}$, then (3.10)-(3.12) has at least one solution

**Proof**: We proceed as in the proof of theorem 4. Therefore it suffices
to show that if $F_n \to F$ (Hausdorff), $\hat{u}_{\Omega_n} = \hat{u}_n \to \hat{u} = \hat{u}_{\Omega}$ in $H^1(\mathcal{O})$ weakly,
$\|\hat{u}_n\|_{C^0(\mathcal{O})} \le C \quad \forall n$, then $\hat{u} = 0$ in $F$ and $u \in H^1_0(\Omega)$. Let $x \in F$ then, by
definition, there exists $\{x_n\}$, $x_n \in F_n$, $x_n \to x$. We have :

$$
|\hat{u}_n(x_n) - \hat{u}(x)| \le |\hat{u}_n(x_n) - \hat{u}(x_n)| + |\hat{u}(x_n) - \hat{u}(x)| \to 0
$$

and : $\hat{u}_n(x_n) = 0$, therefore $\hat{u}(x) = 0 \quad \forall x \in F$

and $\hat{u} \in C^0(\mathcal{O}) \cap H_0^1(\mathcal{O})$

Now let

$$
\hat{u} = (\hat{u} - \epsilon)^{+} + (\hat{u} + \epsilon)^{-}
$$

The continuity of $\hat{u}$ implies that :

$$
\hat{u}_{\epsilon} \in H_{0}^{1}(\mathcal{O} - \mathcal{F})
$$

The fact that $\hat{u}_\epsilon \to \hat{u}$ in $H^1(\mathcal{O})$, yields. $\hat{u} \in H_0^1(\mathcal{O} - \mathcal{F})$. ■

**Corollary 3**: The optimum design problem :

$$
(3.13) \quad \text{Min}_{\Omega \in \mathcal{S}} \left\{ \|u-z_d\|_{L^2(\Omega)}^2 \quad \middle| \quad \begin{array}{l} -\displaystyle \frac{\partial}{\partial x_i} \left( \left( \left| \frac{\partial u}{\partial x_i} \right|^{p-2} \frac{\partial u}{\partial x_i} \right) \right) = f \\ \qquad u \Big|_{\partial \Omega} = 0 \end{array} \right\}
$$

has at least one solution : provided that $f \in W^{-1,p'}$ with $p' < 2$, and
that $S = \{\Omega | \Omega \subset \mathcal{O}\}$, $\mathcal{O}$ open bounded in $\mathbb{R}^2$, $\mathcal{A}$ regular.

**Proof**: From LIONS [8; 2.2], the P.D.E. in (3.13) has a unique solution
in $W_0^{1,p}(\Omega)(\frac{1}{p} + \frac{1}{p'} = 1)$ and $\|u_\Omega\|_{W_0^{1,p}(\Omega)} \le K \quad \forall \Omega \in \mathcal{S}$.

Now $W_0^{1,p}(\mathcal{O}) \subset C^0(\mathcal{O})$ with continuous injection, and
$|u(x)-u(y)| < K \|u\|_{W_0^{1,p}} |x-y|^{\alpha}$ for some $\alpha > 0$.
Therefore the result is a consequence of Proposition 3.■

### III.3) Applications to variational inequalities.

As usual we shall consider an example. Let $\mathcal{O}$ be a bounded open set of $\mathbb{R}^2$ with regular boundary $\Gamma_1 \cup \Gamma_2$ (see figure 6)

<figure><img src="image_7.png" /><figcaption>Figure 6</figcaption></figure>

Consider the variational inequality :

$$ (3.14) \quad a(u,\phi-u) \ge (f,\phi-u) + \int_{\Gamma_1} v(\phi-u)d\Gamma \quad \forall \phi \in K $$

$$ (3.15) \quad K = \{ \phi | \phi \in H^1(\mathcal{O}) , \phi \ge 0 \text{ a.e., } \phi|_{\Gamma_2} = 0 \} $$

with

$$ (3.16) \quad a(u,\phi) = \int_{\mathcal{O}} (\nabla u \cdot \nabla \phi + u \phi) dx $$

We assume that :

$$ \begin{cases} f \in L^2(\mathcal{O}) & f \le \alpha < 0 \\ v \in H^{1/2}(\Gamma_1) , & v \ge 0 \end{cases} $$

By considering a penalized problem, it is classical to show that the unique solution of the inequality (3.14) is in $H^2(\mathcal{O})$ and that

$$ (3.17) \quad \|u\|_{H^2(\mathcal{O})} \le K(|f|_{L^2(\mathcal{O})} + |v|_{H^{1/2}(\Gamma_1)}) $$

(see LIONS [23])
**Theorem 5** : The set $F = \{x \in \mathcal{O} | u(x)=0\}$ depends continuously on the data $f$ and $v$ for the Hausdorff topology, up to a null set : i.e.

$$ (f_n, v_n) \to (f, v) \text{ weakly in } L^2(\mathcal{O}) \times H^{1/2}(\Gamma_1) \underset{\mathcal{N}=\emptyset}{\Rightarrow} F_n \to F \subset \mathcal{U} \in \text{Hausdorff} $$

**Proof :**

Let $(f_n, v_n)$ be a family of functions of $L^2(\mathcal{O}) \times H^{1/2}(\Gamma_1)$ such that

$$ (3.18) \qquad (f_n, v_n) \to (f, v) \text{ in } L^2(\mathcal{O}) \times H^{1/2}(\Gamma_1) \text{ weakly.} $$

From (3.17)

$$ (3.19) \qquad u_n \to u \text{ in } H^2(\mathcal{O}) \text{ weakly} $$

Let $F_n = \{x \in \mathcal{O} | u_n(x) = 0\}$, $F_n$ be compact we can find a subsequence and $F'$ such that :

$$ (3.20) \qquad F_n \to F' \text{ (Hausdorff)} $$

We note that $u_n$ satisfies

$$ (3.21) \qquad \left\{ \begin{array}{l} - \Delta u_n + u_n = f_n \text{ in } \mathcal{O}-F_n, u_n = 0 \text{ in } F_n \\ \\ \frac{\partial u_n}{\partial n}|_{\Gamma_1} = v_n \\ \\ u_n|_{\partial F_n} = 0 \end{array} \right. $$

From Proposition 3 and (3.18)-(3.21) $u$ satisfies

$$ \left\{ \begin{array}{l} - \Delta u + u = f \text{ in } \mathcal{O}-F', u=0 \text{ in } F' \\ \\ \frac{\partial v}{\partial n}|_{\Gamma_1} = v \\ \\ u|_{\partial F'} = 0 \end{array} \right. $$
Therefore $F' \subset F$

Suppose that $\tilde{F}' - \tilde{F} \neq \emptyset$. Let $D$ be an open set with regular boundary in $F-F'$.
Then $u$ satisfies

$$ - \Delta u + u = f \text{ in } D \ ; \ u|_{\partial D} = 0 $$

By hypotheses $f < 0$, therefore $u < 0$ in $D$ which is not allowed. ■

In many optimal control problems, it is the free boundary which is observed
therefore one can choose the following cost functions

(3.22) 	$J(v) = (d(\tilde{F}(v), F_d))$ (d = Hausdorff's distance, up to null sets)

**Corollary 4 :** The optimal control problem :

$\text{Min } J(v), \ V \text{ bounded in } H^{1/2}(\Gamma_l)$
$\nu \in V$

where $J$ is given by (3.16)-(3.22) has a solution.

**Remark 6** : In practice $J(v)$ is not convenient. However such an analysis
shows that functionals like

$J(v) = \|u-z_d\|_{L^2(\Omega)}^2$, $z_d$ properly chosen

are feasible to control the free boundary.

# IV - PARABOLIC AND HYPERBOLIC PROBLEMS.

## IV.1) Parabolic problems :

Let :

• $\mathcal{O}$ a bounded open set in $\mathbb{R}^N$, $Q = \mathcal{O} \times ]0,T[$

• $\Omega_n = \mathcal{O} - F_n$ as in I.1)

• $V_n = H_0^1(\Omega_n)$, $V = H_0^1(\Omega)$

• $a(t;\phi,\psi) = (A(t)\phi,\psi)$ a bilinear continuous form on $V \times V$ such that :

$$ -|a(t;\phi,\psi)| \le K \|\phi\|_V \|\psi\|_V \quad \forall \phi, \psi \in V \quad \forall \Omega $$

- There exists $\lambda$ such that :

$$ a(t;\phi,\phi) + \lambda |\phi|^2_{L^2(\Omega)} \ge \alpha \|\phi\|^2_V, \quad \forall \phi \in V $$
Without loss of generality, we can take $\lambda=0$

Let

$$
(4.1) \quad W(0,T;\Omega_n) = \{\phi \mid \phi \in L^2(0,T;V_n) , \frac{d\phi}{dt} \in L^2(0,T;V'_n)\}
$$

Let $u_n \in W_n(0,T)$ be the solution of :

$$
(4.2) \quad \left\{
\begin{array}{ll}
\displaystyle \frac{\partial u_n}{\partial t} + A(t)u_n = f_n & \text{in } \Omega_n \times ]0,T[ \\
\\
u_n(x,0) = 0
\end{array}
\right.
$$

As usual $u_n$ is defined from its extension $\hat{u}_n$ in $Q$ by zero which is the
solution of

$$
(4.3) \quad \begin{gather*}
\int_0^T [-\langle \hat{u}_n, \frac{\partial \hat{w}}{\partial t} \rangle + a(t; \hat{u}_n, \hat{w})] dt = \int_0^T (f_n, \hat{w}) dt \\
\forall \hat{w} \text{ such that } w \in W_o(0,T; \Omega_n) = \{\phi \mid \phi \in W_n(0,T; \Omega_n), \phi(x,T) = 0\}
\end{gather*}
$$

In order to study {$u_n$}, we shall assume the following :

H4 \quad $f_n \to f$ in $L^2(Q)$ weakly

$$
H5 \quad i) F_n \to F \text{ Hausdorff.}
$$

$$
\begin{align*}
\text{ii) } F \text{ closed and such that : } & \phi \in W(0,T; \mathcal{O}) \\
& \text{, } \phi=0 \text{ a.e. in } F \times ]0,T[ \\
& \Rightarrow \phi \in W(0,T; \mathcal{O} - F)
\end{align*}
$$

$$
H6 \quad \begin{array}{l} i) X_n = X_{F_n \times ]0,T[} \to \ell \text{ in } L^\infty(Q) \text{ weakstar.} \\ ii) \{(x,t) | \ell(x,t) > 0\} = F \times ]0,T[ \text{ a.e.} \end{array}
$$

$$
ii) \{ (x,t) | \ell(x,t) > 0 \} = F \times ]0,T[ \text{ a.e.}
$$

Here and in the sequel we shall need the following result :

**Lemma 2** : Let $Q(\Omega) = \{(\Omega(t),t) | t \in ]0,T[\}, \Omega(t)$ open in

$$
\mathcal{D}_1(\Omega; 0, T) = \{\phi \in C^\infty(\bar{Q}) , \text{supp } \phi \subset \{(\Omega(t),t), t \in [0, T[\} \}
$$

$$
\text{Let } \overline{\mathcal{D}}_1(\Omega; 0, T) = V(\Omega), \text{ if } \bar{Q} - Q(\Omega_n) \to \bar{Q} - Q(\Omega) \text{ (Hausdorff)}
$$

then

$$
\forall \phi \in V(\Omega) \ \exists \{\phi_n\} \ \phi_n \in V(\Omega_n) : \phi_n \to \phi \text{ strongly in } V
$$
**Proof** : see PIRONNEAU [16], lemma I.3.1 . ■

**Theorem 6** : If H4-H5-H6 hold, the sequence {$\hat{u}_n$} converges strongly in $L^2(0,T;V)$ to $\hat{u}$, with $\hat{u}=0$ in $F \times ]0,T[$ and $\hat{u} = u \in W(0,T;\Omega)$ in $\Omega \times ]0,T[$ $= (\mathcal{O}-F) \times ]0,T[$ solution of

$$
(4.4) \quad \left\{
\begin{array}{l}
\displaystyle \frac{\partial u}{\partial t} + A(t)u = f \quad \text{in } \Omega \times ]0,T[ \\[2ex]
u(x,o) = 0
\end{array}
\right.
$$

**Proof** : From (4.3) one can show that :

$$
(4.5) \quad \| \hat{u}_n \|_{L^2(0,T;H_o^1(\mathcal{O})) \cap L^\infty(0,T;L^2(\mathcal{O}))} \le C
$$

$$
(4.6) \quad \| \hat{u}_n(T) \|_{L^2(\mathcal{O})} \le C
$$

From the continuity of A(·) :

$$
(4.7) \quad \| A(t) \hat{u}_n \|_{H^{-1}(\mathcal{O})} \le K \| \hat{u}_n \|_{H^1(\mathcal{O})}
$$

The Fourier transform of (4.2) multiplied by $w \in H_o^1(\Omega_n)$ is

$$
(4.8) \quad i\tau(\tilde{u}_n, \tilde{w}) + \langle \tilde{A}(t)\tilde{u}_n, \tilde{w} \rangle = (\tilde{f}_n, \tilde{w}) - (\hat{u}_n(T), \hat{w}) \exp(-i\tau T)
$$

where the $\sim$ denotes the Fourier transform.

Taking $\hat{w} = \tilde{u}_n$ in (4.8) yields :

$$
(4.9) \quad \left\{
\begin{aligned}
|\tau| |\tilde{u}_n(\tau)|^2 &\le ( \|\tilde{A}(t)\tilde{u}_n\|_{H^{-1}(\mathcal{O})} + \|\tilde{f}_n\|_{L^2(\mathcal{O})} ) \|\tilde{u}_n(\tau)\|_{H_o^1(\mathcal{O})} \\
&\quad + |\hat{u}_n(T)|_{L^2(\mathcal{O})} |\tilde{u}_n(\tau)|_{L^2(\mathcal{O})}^2
\end{aligned}
\right.
$$

From (4.5)-(4.6)-(4.7) we obtain

$$
(4.10) \quad |\tau| |\tilde{u}_n(\tau)|^2_{L^2(\mathcal{O})} \le C_1 \|\tilde{u}_n(\tau)\|_{H_o^1(\mathcal{O})} + C_2 |\tilde{u}_n(\tau)|^2_{L^2(\mathcal{O})}
$$
then

$$
\int_{-\infty}^{+\infty} \frac{|\tau| |\tilde{u}_n|^2}{1+\tau^\alpha} d\tau < +\infty \quad \text{if } \alpha < \frac{1}{2}.
$$

Therefore

$$
(4.11) \quad \| D^\gamma \hat{u}_n \|_{L^2(0,T; L^2(\Omega))} \le C \text{ for } 0 < \gamma < \frac{1}{4}
$$

Hence for a subsequence,

$$
(4.12) \quad \hat{u}_n \to \hat{u} \text{ in } L^2(0,T,H_o^1(\Omega)) \text{ weakly.}
$$

$$
(4.13) \quad D^\gamma \hat{u}_n \to D^\gamma \hat{u} \text{ in } L^2(0,T, L^2(\Omega)) \text{ weakly.}
$$

$$
(4.14) \quad \hat{u}_n \to \hat{u} \text{ in } L^2(0,T; L^2(\Omega)) \text{ strongly.}
$$

Therefore

$$
(4.15) \quad 0 = X_n \hat{u}_n \to \ell \hat{u} \text{ in } L^2(0,T; L^2(\Omega)) \text{ weakly}
$$

which implies

$$
(4.16) \quad \hat{u} = 0 \text{ in } \mathcal{F} \times [0,T]
$$

From (4.3) ; H5 ii) and lemma 2, we obtain that u is solution of (4.4).
To show the strong convergence, let us multiply (4.2) by u_n and integrate
the result over ]0,T[ ; by proceeding similarly with (4.4) we find that :

$$
(4.17) \quad \frac{1}{2} |u_n(T)|^2 + \int_0^T a(u_n, u_n) dt \to \frac{1}{2} |u(T)|^2 + \int_0^T a(u, u) dt
$$

which, together with the coercivity of a, yield the result. ■
IV.2) Parabolic problems in non cylindrical domains.

Let {$F_n$} be a family of closed sets of $Q = \emptyset \times ]0,T[$
Let $K_n = Q - F_n$ such that $K_n \cap \{t=T\} \neq \emptyset$

Following LIONS [ 8 ] , we define :

$$
(4.18) \quad H^1,^o(Q) = \{u \mid u \in L^2(Q) , D_x u \in L^2(Q)\}
$$

$$
(4.19) \quad H_o^1,^o(Q) = \overline{D(Q)} H^1,^o(Q)
$$

$$
(4.20) \quad \mathcal{B}(Q) = \{u \in H_o^1,^o(Q) \mid \frac{du}{dt} \in (H_o^1,^o(\mathcal{O}))'\}
$$

Let $u_n$ be the solution of

$$
(4.21) \quad \frac{\partial u}{\partial t} + Au_n = f_n \quad \text{in } K_n ; \quad u_n \in \mathcal{B}(K_n) , \quad u_n(x,o) = 0
$$

In addition to H4-H6, we assume :

$$
H7 \quad \left\{
\begin{array}{ll}
u=0 & \text{in } F_n \\
u \in H^1,^o(Q-F_n) &
\end{array}
\right.
\quad \Rightarrow \quad
u \in H^1,^o(Q-F_n) \quad \forall n
$$

**Theorem 7** : If H4-H7 holds, $\hat{u}_n \to \hat{u}$ in $H^1,^o(Q)$ strongly, where $\hat{u} = 0$ in $F$
and $u$ is solution of (4.21) in $K$ (with $f_n=f$).

**Proof** : The proof follows the proof of theorem 6, except that for the a priori estimate (4.11). Thus, following LIONS [8,II.8] we note that the solution $u_n^\epsilon$ of the penalized problem

$$
(4.22) \quad \left\{
\begin{array}{ll}
\displaystyle \frac{du_n^\epsilon}{dt} + Au_n^\epsilon + \frac{1}{\epsilon} X_n u_n^\epsilon = f_n & \text{in } Q \\
\\
u_n^\epsilon|_{\partial \Omega} = 0 \\
\\
u_n^\epsilon(x,o) = 0
\end{array}
\right.
$$

converges to $\hat{u}_n$ solution of 4.21 when $\epsilon \to 0$. From (4.22) we have
$$
(4.23) \quad \left\{
\begin{array}{l}
\|u_n^\epsilon\|_{L^2(0,T;H_0^1(\Omega))} \le C, \quad \|\frac{du_n^\epsilon}{dt}\|_{L^2(0,T;H^{-1}(\Omega))} \le C \\
\\
\displaystyle\frac{1}{\sqrt{\epsilon}} \|X_\epsilon u_n^\epsilon\| \le C. \quad (C \text{ independent of } \epsilon \text{ and } n)
\end{array}
\right.
$$

Therefore these bounds remain true for $\hat{u}_n$.

Using Theorem I,5.1 of [8], we have (4.14). The rest of the proof proceeds
as in Theorem 6, with a similar density result with {$w \in \mathcal{B}(Q), w(x,T) = 0$}
according to lemma 2. ■

IV.3) Second order hyperbolic problems.

We keep the notation of paragraph IV.1 except for :

$$
(4.24) \quad W(0,T;\Omega) = \{\phi \mid \phi \in L^2(0,T;V) \; ; \; \frac{d\phi}{dt} \in L^2(0,T;L^2(\Omega))\}
$$

Now $u_n \in W(O,T;\Omega_n)$ is the solution of :

$$
(4.25) \quad \left\{
\begin{array}{l}
\displaystyle \frac{d^2 u_n}{dt^2} + A(t)u_n = f_n \quad \text{in } \Omega_n \times ]0,T[ \\[1em]
u_n(0) = 0 \quad \text{in } \Omega_n \\[1em]
\displaystyle \frac{du_n}{dt}(0) = 0 \quad \text{in } \Omega_n.
\end{array}
\right.
$$

with the following additional assumption on A

$$
(4.26) \quad \forall \phi, \psi \in V, a(\cdot; \phi, \psi) \in C^1(0,T)
$$

We assume $\Omega_n$ and $\Omega$ sufficiently regular to define solutions of (4.25).

Theorem 8 : If H4-H6 holds, $\hat{u}_n$ converges weakly in $W(0,T;\mathcal{O})$ to $\hat{u}$, with
$\hat{u} = 0$ in $F \times ]0,T[$ and $u \in W(0,T;\Omega)$ solution of.

$$
(4.27) \quad \left\{
\begin{array}{l}
\displaystyle \frac{d^2 u}{dt^2} + A(t)u = f. \\[1em]
u(0) = 0 \\[1em]
\displaystyle \frac{du}{dt}(0) = 0.
\end{array}
\right.
$$
Proof : From LIONS-MAGENES [12, Th. 3.8.1] ; $u_n \in L^2(0,T;V_n)$,
$u'_n \in L^2(0,T,L^2(\Omega_n))$. From [12 ; Lemma 3.8.2] ;

$$
(4.28) \quad \| u_n(t) \|_{L^\infty(0,T;V_n)} \le C \quad \forall n
$$

$$
(4.29) \quad \| u'_n(t) \|_{L^\infty(0,T;L^2(\Omega_n))} \le C
$$

If $\hat{w}$ denotes the zero extension, then, at least for a subsequence,

$$
(4.30) \quad \hat{u}_n \to \hat{u} \text{ in } L^2(0,T; H_o^1(\O)) \text{ weakly.}
$$

$$
(4.31) \quad \hat{u}'_n \to \hat{u}' \text{ in } L^2(0,T; L^2(\mathcal{O})) \text{ weakly.}
$$

$$
(4.32) \quad \hat{u}_n \to \hat{u} \text{ in } L^2(0,T; L^2(\mathcal{O})) \text{ strongly.}
$$

From lemma 2, as in Theorem 6 ;

$$
\forall w \in W_o(0,T;\Omega) = \{\phi \mid \phi \in W(0,T;\Omega), \phi(x,T) = 0\}
$$

$$
\exists w_n \in W_o(0,T;\Omega_n) \text{ such that } \hat{w}_n \to \hat{w} \text{ in } W(0,T;\mathcal{O}) \text{ strongly.}
$$

It is shown in [ , formula 3.8.23] that

$$
(4.33) \quad \int_o^T \left( -\left( \frac{d\hat{u}}{dt} \right)^n + \frac{d\hat{w}}{dt} \right) + a(t; \hat{u}, \hat{w}) dt = \int_Q f \hat{w} dx dt
$$

which implies

$$
(4.34) \quad \int_o^T \left( -\left( \frac{d\hat{u}}{dt} , \frac{d\hat{w}}{dt} \right) + a(t, \hat{u}, \hat{w}) \right) dt = \int_Q f \hat{w} dx dt \quad \forall w \in W_o(O,T,\Omega)
$$

As in theorem 6, $o=X_n \hat{u}_n \to \ell \hat{u}$ in $L^2(Q)$ weakly, which completes the proof.■

**Corollary 5** : If $f_n$ converges strongly to $f$, and $a(t,\phi,\psi)$ is independent of $t$ then $\hat{u}_n$ converges to $\hat{u}$ strongly in $L^2(0,T;H_o^1(\mathcal{O}))$.

Proof :Straightforward from [12, lemma 3.8.3].
**Remark 7**: Applications similar to these of III, can be found in the time dependent case. For examples :

1) turbulent flow over very irregular surfaces

2) Free boundary problems and variational inequalities with time dependent operators

3) Acoustic problems in irregular domains.

**Acknowledgement** : We are thankful to Professeur J.L. LIONS for his extremely helpful suggestions and also to Professeur R. GLOWINSKI and Dr. P. MORICE for their contributions.
BIBLIOGRAPHY

[1] ADAMS R., Sobolev Spaces, Academic Press, 1975.

[2] BEER G., The Hausdorff metric and convergence in measure, Michigan Math. J., 21, 1974.

[3] BENSOUSSAN A., LIONS J.L., PAPANICOLAU G., C.R.A.S. t.282, 21 Juillet 1975, 8 Septembre 1975, 14 Juin 1976.

[4] BLAKE J., A model for the micro-structure in ciliated organisms, (J. Fluid Mech. 1972, Vol. 55, pp. 1-23).

[5] DELLACHERIE C., Ensembles analytiques, capacités, mesures de Hausdorff. (Lectures Notes in Mathematics 295, Springer-Verlag).

[6] GARABEDIAN P., Partial differential equations, Wiley 1964.

[7] LADYZHENSKAYA O., Mathematical theory of viscous incompressible flow. Gordon & Breach, 1963.

[8] LIONS J.L., Quelques méthodes de résolution de problèmes aux limites non linéaires (Dunod-Gauthier-Villars, 1969).

[9] LIONS J.L., Some aspects of the optimal control of distributed parameter systems (SIAM Monograph 6., 1972).

[10] LIONS J.L., Cours Collège de France (1975-1976)

[11] LIONS J.L., Private communication.

[12] LIONS J.L., MAGENES E., Problèmes aux limites non homogènes et applications (Dunod 1968).

[13] MORICE Ph., Private Communication.

[14] MURAT R., SIMON J., Quelques résultats sur le contrôle par un domaine géométrique. Report 74003, Lab. Num. Analysis, University of Paris 6, 1974.

[15] NECAS J., La méthode directe en théorie des équations aux dérivées partielles, Masson 1967.

[16] PIRONNEAU O., Sur les problèmes d'optimisation de structure en mécanique des fluides (Thèse d'Etat, Paris VI, 1976).

[17] SAGUEZ C., Contrôle optimal de systèmes gouvernés par des inéquations variationnelles. Application à des problèmes de frontière libre (LABORIA report N° 191, 1976).

[18] CHENAIS D., Un résultat d'existence dans un problème d'identification Journal of Math. Anal. and Applications. Nov. 1975.