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<figure><img src="image_1.png" /></figure>

FREE BOUNDARY PROBLEMS
FOR THE LAPLACE EQUATION :
A PRIORI ESTIMATES,
EXISTENCE THEOREMS,
ASYMPTOTIC BEHAVIOURS

Alain DERVIEUX
Hideo KAWARADA

Rapport de Recherche N° 364

Octobre 1979

48p1
FREE BOUNDARY PROBLEMS FOR THE LAPLACE EQUATION : A PRIORI ESTIMATES,

EXISTENCE THEOREMS, ASYMPTOTIC BEHAVIOURS

Alain DERVIEUX *

Hidéo KAWARADA **

**RESUME** : Ce travail est consacré à l'étude de Problèmes à Frontière Libre avec l'équation de Laplace Δu = f dans le domaine et les deux conditions u = 1 et |grad u| = λ = cte sur la frontière libre. Dans les problèmes modèles que nous envisageons, nous avons les trois cas suivants :

1) f = 0, λ est donné ≠ 0, 2) f = 0,λ est inconnu et la longueur de la frontière libre est donnée, 3) f ≠ 0,λ = 0 (Problème de l'Obstacle).

**SUMMARY** : This work is devoted to Free Boundary Problems with Laplace equation Δu = f in the domain and the two conditions on the Free Boundary u = 1 and |grad u| = λ = const. In the model problems which we study, three cases arise : 1) f = 0,λ is given, 2) f = 0,λ is unknown and the length of the Free Boundary is given, 3) f ≠ 0,λ = 0 (Obstacle Problem).

<footnote>* IRIA-LABORIA, Laboratoire de Recherche en Informatique et Automatique, B.P. 105, 78150 Le Chesnay (France)</footnote>

<footnote>** TOKYO UNIVERSITY, Department of Applied Physics. Faculty of Engineering, Bunkyo-Ku, Tokyo 113 (Japan)</footnote>
## INTRODUCTION

This work contains two parts

The first part concerns the two following Free Boundary Problems (F.B.P) :

Problem 1 : Find a function $u$ and a curve $\gamma$ such that

$$ 
\left\{ 
\begin{array}{l} 
u = 0 \text{ on a (given) boundary } \Gamma \\ 
u = 1 \text{ on the curve } \gamma \text{ (k is given)} \\ 
|\text{grad } u| = k \\ 
\Delta u = 0 \text{ in the domain between } \Gamma \text{ and } \gamma 
\end{array} 
\right. 
$$

Problem 2 : Find function $u$ and a curve $\gamma$ such that

$$ 
\left\{ 
\begin{array}{l} 
u = 0 \text{ on } \Gamma(\text{given}) \\ 
u = 1 \text{ on the curve } \gamma \\ 
|\text{grad } u| \text{ is constant} \\ 
\displaystyle\int_{\gamma} \frac{\partial u}{\partial \nu} \, d\sigma = C(\text{given}) \\ 
(\nu \text{ is the outward normal vector}) \\ 
\Delta u = 0 \text{ in the domain between } \Gamma \text{ and } \gamma. 
\end{array} 
\right. 
$$

Problem 1 has a hydrodynamical origin and is a generalization of a wave or jet problem. Problem 2, which can be interpreted as an inverse problem with respect to the first one, is connected with the Stefan problem and occurs in cryogenics experiments ; that kind of problem is also introduced to modelize plasma configurations in tokamak machines when the plasma is subject to skin effect (see for example A.S. DEMIDOV [1]).

It is interesting to note that these two problems have not been transformed into Variational Inequalities (nor Quasi-Variational Inequalities ; see C. BAIOCCHI [1]) so that methods developed for that family of problems are not, as far as we know, available for the above two problems.

Our main point of view is to regard the Free Boundary as the most important unknown variable.
A very simple a priori estimate is exhibited for the Free Boundary of
**Problem 1** and consequences are derived for some asymptotic behavior of this
problem.

From continuity and asymptotic properties of **Problem 1**, existence and uniqueness of a solution to **Problem 2** is showed, as well as an iterative process is constructed for a convergent approximation of the solution of **Problem 2** by solutions of **Problem 1**.

Many authors studied **Problem 1**: for instance A. BEURLING, D.E. TEPPER, G. WILDENBERG studied it when the Free Boundary is *outside* of the fixed one. The "inner case" has been considered by I.I. DANILJUK. Conformal mappings have been used by these authors whose results we use ; but, on the other hand, our methods do not use that kind of arguments, which allows us to hope that they could be widely generalized.

In the *second part* we study a particular *obstacle problem*

$\Delta u = f \text{ in } \Omega$

$u = 1$ on the curve AB

$\frac{\partial u}{\partial x} = 0$ on AD $\cup$ BC

$u = \frac{\partial u}{\partial \nu} = 0$ on the free boundary DC (fig.1)

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

The important point of this study is to obtain an a priori estimate for
the *slope* of the Free Boundary ;

Indeed this can be considered only if the geometrical data have a distin-
guished direction : the slope of CD is said to be zero if CD a *straight*
*segment* perpendicular to AD and BC.

It is interesting to note that this estimate may imply the regularity
(and even smoothness) of the free boundary.

Naturally this estimate has some classical-type *applications*, two examples
of which we give, concerning optimal control and asymptotic behaviour.
It is important to note with D.G.SCHAEFFER [2] that the regularity of the data does not generally imply the regularity of the free boundary ; as in D.KINDERLEHRER, G. STAMPACCHIA [1] (see also C. BAIOCCHI et al. [1]) we start from assumption concerning only the data.

The plan is following

# PART I: Jet-like problems

## 1. INNER PROBLEM

1.1 Formulation of the Free Boundary Problem

1.2 The circular case

1.3 An a priori estimate for the length of the Free Boundary

1.4 Consequences for asymptotic behaviour

## 2. OUTER PROBLEMS

2.1 Position of the problems

2.2 Some known properties

2.3 Main results

2.4 Proofs

# PART II: An obstacle problem

## 1. A REGULARITY THEOREM VIA AN A PRIORI ESTIMATE

1.1 Preliminaries

1.2 Main results : an a priori estimate for the Free Boundary

1.3 Proof

## 2. APPLICATION TO AN OPTIMAL CONTROL PROBLEM

2.1 Position of the general problem

2.2 Definition of our problem

2.3 Existence of an optimal control

2.4 Proof

## 3. APPLICATION TO AN ASYMPTOTIC STUDY

3.1 Formulation of the problem and main result

3.2 Proof
# FIRST PART : JET - LIKE PROBLEMS

## 1. INNER PROBLEM

### 1.1. Formulation of the Free Boundary Problem

$\Gamma$ is a smooth curve without self-intersection points which lies in the 2-dimensional space ; $Q(x,y)$ is a real-valued, non negative function defined in the region $\Omega$ bounded by the curve $\Gamma$ ($\Omega$ is bounded). The F.B.P. is

Find another simple closed sufficiently smooth curve $\gamma \subset \Omega$ which, together with $\Gamma$ forms the boundary of a region $\Omega_\gamma$ for which there exists a function $u$ such that

$$ (1) \quad \left\{ \begin{array}{lll} \Delta u = 0 & \text{in } \Omega_\gamma \\ u = 0 & \text{on } \Gamma \\ u = 1 & \\ |\text{grad } u| = Q & \text{on } \gamma \end{array} \right. $$

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

**Theorem 1:** (DANILJUK [1,2,3]) we suppose that $Q$ is a function of $C^\infty(\bar{\Omega})$ such that

$$ (2) \quad \left\{ \begin{array}{ll} \displaystyle \forall x,y \in \bar{\Omega} \quad Q(x,y) \ge Q > 0 & \\ \displaystyle \text{Inf}_{\gamma,z} \int_{\Omega_\gamma} \left\{ |\text{grad } z|^2 + Q^2 \right\} \, dx \, dy \le \int_{\Omega} Q^2 \, dx \, dy & \end{array} \right. $$

where the infimum is taken over every smooth curve $\gamma$ and every smooth function $z$ such that $z = 0$ on $\Gamma$ and $z = 1$ on $\gamma$.
Then problem $\rho_o$ admits at least one solution such that $\gamma$ is a $C^\infty$ curve.

The solution of Problem $\rho_o$ does not always exist, and is generally not unique; to illustrate this, we shall study a particular case which gives a good intuitive idea of what happens in the general case a well for the number of solutions as for their asymptotic behaviour.

## 1.2. The circular case

The supplementary assumptions are

$$ (3) \quad \left\{ \begin{array}{l} Q = k = \text{const.} \\ \Gamma \text{ is a circle with radius } R; \\ \text{We put } k_{\min} = \exp(1-\ln R) \end{array} \right. $$

**Proposition 1 (trivial) :** Under the assumptions (3),

(i) Problem $\rho_o$ admits

no symmetric solution for $k < k_{\min}$

one symmetric solution for $k = k_{\min}$

two symmetric solutions for $k > k_{\min}$

(k_{\min} corresponds to a turning point for the branch of symmetric solutions);

(ii) If k tends to infinity, a first symmetric free boundary converges to the center of $\Gamma$ while the second converges toward $\Gamma$.

## 1.3 An a priori estimate for the length of the free boundary

### 1.3.1. Estimate for the distance between the boundaries

**Proposition 2 :** For every pair of positive constants ($\epsilon_o, Q_o$), for any $\Gamma$ and $Q$ such that

$$ Q(x) < Q_o \quad \text{Vx s.t. } d(x, \Gamma) < \epsilon_o $$

there exists a positive constant $\epsilon_1 = \epsilon_1(\epsilon_o, Q_o, \Gamma)$

such that the corresponding Free Boundary $\gamma$ satisfies

$$ (4) \quad d(\gamma, \Gamma) > \epsilon_1 $$
**Proof:** Let us first consider the case where $\Gamma$ is a circle with the origin $O$ as center and $R>0$ as radius, and let us introduce the domain

$$ \Omega_o = \{ x ; d(x, \Gamma) \le \epsilon_o \} $$

and the function

$$ \varphi(x) = -K \ln\left(\frac{\|x\|}{R}\right) $$

where $K$ is chosen sufficiently large so that

$$ (5) \quad - \frac{\partial \varphi}{\partial r} > \Omega_o \quad \text{with} \quad r = \|x\|, $$

and

$$ (6) \quad \varphi(x) > 1 \quad \text{for} \quad \|x\| = R - \epsilon_o; $$

we shall show that

$$ (7) \quad \varphi \ge u \quad \text{in} \quad \Omega_o \cap \Omega_\gamma. $$

For this purpose, we consider the two following cases

(i) $d(\gamma, \Gamma) \ge \epsilon_o$; then (7) follows trivially from (6) and the Maximum Principle.

(ii) $d(\gamma, \Gamma) < \epsilon_o$; we use a contradiction argument :

if (7) does not hold, then $\max_{\gamma \cap \Omega_o} (u-\varphi)$ is positive and occurs on $\gamma \cap \Omega_o$; moreover, since $\varphi$ depends only on $\|x\|$ and since $u$ is constant on $\gamma$, this maximum occurs in point(s) $x_o$ such that $d(x_o, \Gamma) = d(\gamma, \Gamma)$ and $\gamma$ is tangent at $x_o$ to a circle with the origin as center, so that

$$ \frac{\partial}{\partial n}(u-\varphi)|_{x_o} = \left(\frac{\partial u}{\partial n} + \frac{\partial \varphi}{\partial r}\right)|_{x_o}; $$

but, using (5) and the assumption of Proposition 2 we get

$$ (8) \quad \frac{\partial}{\partial n}(u-\varphi)|_{x_o} < 0. $$

On the other hand, by the strong Maximum Principle, we have

$$ \frac{\partial}{\partial n}(u-\varphi)|_{x_o} > 0 $$

which is in contradiction with (8).
Thus (7) is proved.

From the definition of $\varphi$ and from (6), there exists a number $r_o$ belonging to $]R-\epsilon_o, R[$ such that

$$ (9) \quad \varphi(x) = 1 \quad \text{if} \quad \|x\| = r_o $$

therefore, if we chose

$$ \epsilon_1 = \min \{ \epsilon_o, R - r_o \}, $$

the conclusion follows from (7) and (9).

Let us now deal with the **general case** ($\Gamma$ is no more a circle). The region $\Omega$ is a regular simply connected domain; then there exists a (fixed) regular conformal mapping $T$ which maps $\Omega$ into the disk $\{\rho < 1\}$. Let us denote by $\tilde{u}$ the transported solution :

$$ \tilde{u} = u \circ T^{-1}; $$

by invariance of harmonicity by conformal mapping we have

$$ \Delta \tilde{u} = 0 \quad \text{in} \quad T(\Omega_\gamma) $$

on the other hand we have

$$ \tilde{u} = 0 \quad \text{on} \quad \{\rho = 1\} = T(\Gamma) $$

$$ \tilde{u} = 1 \quad \text{on} \quad \tilde{\gamma} = T(\gamma); $$

thus if we introduce

$$ \tilde{Q} = \frac{\partial \tilde{u}}{\partial \nu} \Big|_{\tilde{\gamma}}; $$

an $L^\infty$-estimate for $Q$ implies an $L^\infty$ estimate for $\tilde{Q}$ because the conformal mapping $T$ belongs to $W^{1,\infty}$ and its inverse too. Thus $\tilde{u}$ fulfills the conditions of the above particular case : we deduce an estimate for the distance $d(\tilde{\gamma}, \{\rho=1\})$ and therefore an estimate for $d(\gamma, \Gamma)$.
### 1.3.2. Estimate for the length of the Free Boundary

**Proposition 3:** Under the assumptions of Proposition 1, we have, for every solution $\gamma$ of Problem (1),

$$ (10) \quad \int_{\gamma} Q d\sigma \le K (\epsilon_0, Q_0, \Gamma) ; K \text{ indep. of } Q. $$

*Proof:* we have

$$ (11) \quad d(\gamma, \Gamma) > \epsilon_1 \quad \epsilon_1 \text{ fixed;} $$

Let $v$ be defined by

$$ (12) \quad \left\{ \begin{array}{ll} v \in H^2(\Omega) & \\ v(x) = 0 & \text{for } d(x, \Gamma) < \frac{\epsilon_1}{2} \\ v(x) = 1 & \text{for } d(x, \Gamma) > \epsilon_1. \end{array} \right. $$

Let us estimate the difference $z = u-v$, where $u$ is the solution of the F.B.P. for the datum $Q$; we have

$$ (13) \quad \left\{ \begin{array}{l} \Delta z = - \Delta v \\ z|_{\Gamma} = 0 \\ z|_{\gamma} = 0 \end{array} \right. $$

thus

$$ \begin{align*} \|z\|_{H_0^1(\Omega_\gamma)}^2 &= - \int_{\Omega_\gamma} z \Delta v \, dx \\ &\le \|z\|_{L^2(\Omega_\gamma)} \|\Delta v\|_{L^2(\Omega_\gamma)}. \end{align*} $$

Then (by Friedrichs-Poincaré inequality)

$$ (14) \quad \|z\|_{H_0^1(\Omega)} \le K(\Omega) \|\Delta v\|_{L^2(\Omega)} $$

where we use the same notation for the zero extension of $z$.
Let us also denote by $u$ the 1-extension of $u$ in $\Omega$; the statement (14) implies

$$ (15) \qquad \|u\|_{H_o^1(\Omega)} \le (K(\Omega) + 1) \|v\|_{H_o^2(\Omega)} \cdot $$

By Green's formula

$$ \begin{align*} \int_{\Gamma} v \frac{\partial u}{\partial v} d\sigma &= \int_{\Omega} \left[ v \Delta u + \langle \text{grad } v, \text{grad } u \rangle \right] dx dy - \int_{\Gamma} v \frac{\partial u}{\partial v} d\sigma \\ &= \int_{\Omega} \langle \text{grad } v, \text{grad } u \rangle dx dy \\ &\le \|u\|_{H_o^1(\Omega)} \|v\|_{H_o^2(\Omega)} \end{align*} $$

and by (15)

$$ \le (K(\Omega) + 1) \|v\|_{H^2(\Omega)}^2 $$

so that we have proved

$$ \int_{\gamma} Q d\sigma \le \text{const.} $$

### 1.4. Consequences for asymptotic behavior

The following result is a direct consequence of Proposition 3.

**Proposition 4; We suppose that**

$$ (i) \quad d(x, \Gamma) < \epsilon_o \quad \Rightarrow \quad Q_n(x) < Q_o $$

$$ (ii) \quad \forall x \in \bar{S} \text{ c h, } \quad Q_n(x) \ge k_n \xrightarrow{n \to +\infty} \infty $$

for a fixed domain $\bar{S}$; then

$$ \begin{gathered} \max_{\gamma} \ell(\gamma \cap \bar{S}) \to 0 \\ \gamma \end{gathered} $$

where the maximum is taken over the solution(s) of Problem $\rho_0$.
**Proof:** Indeed we have for every solution $\gamma$ of (1):

$$k_n \cdot \ell(\gamma \cap \bar{S}) \le \int_{\gamma \cap \bar{S}} Q d\sigma \le \int_{\gamma} Q d\sigma \le K(\epsilon_o, Q_o, \Gamma). \quad \blacksquare$$

Now we shall deduce that either the free boundary $\gamma_n$ goes out of $\bar{S}$, or shrinks to a point:

**Proposition 5 :** Under the hypothesis of Proposition 3 and if $\gamma_n$ stays completely in $\bar{S}$ for every $n$, then the length $\ell(\gamma_n)$ tends to zero when $n$ tends to infinity and there exist subsequences $\gamma_n'$ which tend to a point; in this case, the corresponding solution $u_n'$ converges weakly to zero in $H_o^1(\Omega)$.

**Proof :** We can choose a sequence of points $x_n$ belonging to $\gamma_n$ thus there exist subsequences $(x_{n'})$ such that

$$x_{n'} \to x^* \quad \text{as } n' \to \infty.$$

Then it is evident that

$$\underset{y \in \gamma_n'}{\operatorname{Max}} d(y, x^*) \le d(x_{n'}, x^*) + \underset{y \in \gamma_{n'}}{\operatorname{Max}} d(y, x_{n'})$$

so that

$$\underset{y \in \gamma_{n'}}{\operatorname{Max}} d(y, x^*) \to 0 \quad \text{or} \quad \gamma_{n'} \to x^*.$$

Let us deal with the convergence of $u_n$, which we still denote $u_n :$ from the proof of Proposition 2 we know that (after an extension by 1)

$$\|u_n\|_{H_o^1(\Omega)} < K \text{ (indep. of } n).$$

Then a subsequence $u_n$ converges weakly

$$(16) \quad u_n \to u^* \quad \text{in } H_o^1(\Omega) \quad \text{weakly.}$$

**Lemma 1 :** For any $x_o \in \Omega$, $x_o \ne x^*$ there exist $\epsilon > 0$ and $n_o \in \mathbb{N}$ such that $n > n_o \Rightarrow d(x_o, \gamma) > \epsilon$.
**Proof of the Lemma 1 :** we have

$$d(x_o, y_n) \geq d(x_o, x^*) - \max_{y \in \gamma_n} d(y, x^*)$$

but

$$\max_{y \in \gamma_n} d(y, x^*) < l(\gamma_n) + d(\gamma_n, x^*) \to 0$$

which implies the conclusion. ■

**End of the proof of Proposition 5 :** Using the above Lemma we deduce

$$V n > n_o,$$

$$B(x_o, \epsilon) = \{y; |x_o - y| < \epsilon\} \subset \Omega_{\gamma_n};$$

for any function $\varphi \in \mathcal{B}(B(x_o, \epsilon))$ we have, from the weak convergence (cf. (16))

$$\int_{\Omega} \text{grad } u_n, \text{grad } \varphi > dx \to \int_{\Omega} < \text{grad } u^*, \text{grad } \varphi > dx \text{ as } n \to \infty.$$

Now the first member is zero, thus

$$\Delta u^* = 0 \text{ in } B(x_o, \epsilon).$$

The point $x_o$ was arbitrary chosen, then

$$(17) \qquad \Delta u^* = 0 \text{ p.p. in } \Omega,$$

so that $u^*$ is the unique, vanishing (from (16) and (17)) limit. ■

## 2. OUTER PROBLEMS

### 2.1. Position of the problems

Let $\mathcal{B}$ be a doubly connected region limited by the point at infinity and a convex boundary¹ component $\Gamma$ which does not reduce to a point; if $k$ is a given positive number then it is known (A. BEURLING [1]) that there exists an unique annulus $\Omega \subset \mathcal{B}$ having $\Gamma$ as one boundary component and an other boundary component $\gamma$, the Free Boundary, such that there exists a function $u$ defined in $\Omega$ satisfying (cf. Fig.5)
$$
\begin{equation}
\begin{cases}
(18) & \Delta u = 0 \text{ on } \Omega \\
(19) & u = 0 \text{ on } \Gamma \\
(20) & u = 1 \text{ on } \gamma \\
(21) & |\text{grad } u| = k \text{ on } \gamma.
\end{cases}
\end{equation}
$$

<figure><img src="image_4.png" /><figcaption>Figure 3</figcaption></figure>

That problem will be called Problem $\mathcal{P}$ ; we shall also consider a
second problem, Problem $(\mathcal{P}')$, in which (21) is replaced by (21')
To find $(u,\gamma)$ such that (18) (19) (20) (21') hold, with
$(\mathcal{P}')$
$$ (21') \qquad |\text{grad } u| = \frac{C}{l(\gamma)} \text{ on } \gamma $$
where C is a given positive number and l(γ) is the length of γ.

Since |grad u| is constant on γ, the number C denotes the circulation of
this quantity along γ; from (18) (19) (20) and from Green's formula, it
represents too the total energy of the system

$$
C = \int_{\Omega} |\text{grad } u|^2 dx dy.
$$

2.2. Some known properties

After A. BEURLING, D.E.TEPPER [1,2] (cf. also more recently TEPPER,
WILDENBERG [1]) studied the qualitative properties of the Free Boundary
of Problem (P) ; we give here some results of those authors which will
be useful in the sequel.

**Proposition 6 (Regularity property):** The solution $\gamma$ of Problem (P) is an analytic closed convex curve.
**Proposition 7 (Monotonicity property):** If $k_1 > k_2 > 0$, then the corresponding domains verify

$$ \bar{\Omega}_{k_1} \subset \Omega_{k_2} . $$

**Proposition 8 (Surjection property):** For any point $z_o$ of $\mathcal{B}$, there exists some $k$ such that

$$ z_o \in \mathcal{E}_{\gamma_k} $$

where $\gamma_k$ denotes the Free Boundary corresponding to $k$, that is to say

$$ (22) \qquad \mathcal{B} = \bigcup_{k>0} \gamma_k . $$

**Proposition 9 (Asymptotic behaviour):** Let

$$ (23) \qquad z = f(w) = c w + \alpha_o + \sum_{n=1}^{\infty} \alpha_n w^{-n} $$

be a conformal mapping which transforms $\mathcal{B}$ into the exterior circle; the annulus $f^{-1}(\Omega_k^*)$ will be denoted by $\Omega_k^*$; then for $k>0$ there exists two positive numbers $\alpha_k, \beta_k$ such that

$$ \{w; 1 < |w| < \alpha_k\} \subset \Omega_k^* \subset \{w; 1 < |w| < \beta_k\} $$

and $(\beta_k - \alpha_k)$ tends to zero when $k$ tends to zéro.

### 2.3. Main results

**Theorem 2:** (Existence and uniqueness). Under the assumptions of Paragraph 2.1, and for every given positive number $C$, there exists a unique solution $(\bar{u}, \bar{\gamma})$ of Problem (P').

We need, for the statement of the next result, to introduce the following notions :

**Definition:** the polar parametrization of the regular boundary $\gamma$ of a starlike set is defined as a functions still denoted $\gamma$ (in the sequel this simplification of notation will be used systematically) such that
$$ (24) \quad \left\{ \begin{array}{l} \gamma \in C^\infty(0,2\pi), \text{ periodic with period } 2\pi \\ \text{the boundary } \gamma = \{(r \cos \theta, r \sin \theta), \theta \in [0,2\pi], r = \gamma(\theta)\}. \end{array} \right. \blacksquare $$

This parametrization is a fortiori possible for regular boundaries of convex sets : the origin of the axis is chosen inside Γ.

Now we shall show how the solution of Problem P' can be approximated by solutions of Problem P ; let us consider the following iterative process :

**Step 1 :** Choose $k_0$ such that $c_0 = k_0 \cdot l(\gamma_0) > C$

**Step 2 :** Define $k_{n+1} = \frac{C}{l(\gamma_n)}$,

$$ \gamma_{n+1} = \gamma(k_{n+1}) $$

$$ c_{n+1} = k_{n+1} \cdot l(\gamma_{n+1}) $$

where $\gamma(k)$ denotes the F.B. solution of Problem P corresponding to the datum k.

**Theorem 3 (Approximating process):** Under the assumptions of Theorem 2, the above algorithm converges monotonically to the unique solution of P' for the datum C :

$$ \gamma_n \to \gamma \quad \text{in } C^0(0,2\pi) \text{ (i.e. uniformly)} $$

$$ l(\gamma_n) \to l(\gamma) $$

$$ c_n \to C $$

$$ \tilde{u}_n \to \tilde{u} \quad \text{in } C^0(\bar{D}) \text{ and } H^1(\bar{D}). \blacksquare $$

**Remark 1:** In a numerical implementation, Problem P, whose solution is used in Step 2, can be solved for example by "trial methods" (see the complete survey of C.W. CRYER [1]).

## 2.4. Proofs

### 2.4.1. Monotonocity and convergence results for Problem P

Let us first recall the following classical result (see for example M. BERGER [1]).
**Lemma 2:** Let $\Omega_i$ ($i=1,2$) be two closed convex regions of $\mathbb{R}^2$ with smooth boundaries $\gamma_i = \partial\Omega_i$ such that

$$ \Omega_2 \subset \Omega_1, \Omega_2 \neq \Omega_1 $$

then

$$ \ell(\gamma_1) > \ell(\gamma_2). $$

<figure><img src="image_5.png" /></figure>

**Lemma 3:** Let $(k_n)$ ($n=1,2,3,...$) be a sequence of positive numbers such that

$$ (25) \quad \left\{ \begin{array}{@{}l@{}} k_1 > k_2 > \dots > k_n > k_{n+1} \rightarrow k_o \in ]0,\infty[ \\ \qquad \text{when } n \rightarrow +\infty; \end{array} \right. $$

Let us denote by $\gamma_n$ (resp. $\ell_n$, $u_n$) the corresponding Free Boundary (resp. its length, and the distributed solution) for problem P ($n = 0,1,2,...$).

Then, when $n$ tends to infinity we have

$$ (i) \quad \gamma_n \le \gamma_{n+1} \rightarrow \gamma_o \text{ in } C^0(0,2\pi) \text{ (see (24))} $$

$$ (ii) \quad \ell_n \rightarrow \ell_o, $$

Moreover if $\tilde{u}_n$ denotes the 1-extension of $u_n$ in the open set $\Omega$ whose boundaries are $\Gamma$ and $\gamma_o$ ($n=0,1,2,...$), then

$$ (iii) \quad \tilde{u}_n \rightarrow u_o \text{ in } C^0(\Omega) $$

$$ (iv) \quad \tilde{u}_n \rightarrow u_o \text{ in } H^1(\Omega). $$

<figure><img src="image_6.png" /></figure>

**Proof:** Using (24), then sequence of boundaries $\gamma_1, \dots, \gamma_n$ corresponds to a sequence of functions $\gamma_1, \dots, \gamma_n$; from Proposition 7, this sequence is strictly increasing and converges to a function $\gamma_\infty$:

$$ \gamma_n \nearrow \gamma_\infty \text{ everywhere in } [0,2\pi] $$

$$ \gamma_\infty \le \gamma_o. $$

Let us show that

$$ (26) \quad \gamma_\infty = \gamma_o; $$

we use a contradiction argument: if (26) does not hold, then, if we denote by $\Omega_\infty$ the bounded domain with $\gamma_\infty$ as boundary, there exists a point P such that
$$ (27) \quad P \in \Omega_o \setminus \bar{\Omega}_\infty; $$

but from Proposition 8 there exists a positive number $k^*$ such that $P$ belongs to the Free Boundary $\gamma(k^*)$ of Problem P for $k=k^*$; if $k_o \ge k^*$, then, by Proposition 7, we have $\gamma(k^*) \ge \gamma(k_o)$, which contradicts (27); thus $k^* > k_o$; therefore there exists some $k_n$ in the sequence $(k_n)$ such that $k^* > k_n$ and $\gamma_{k^*} < \gamma_{k_n}$, which contradicts (27); thus (26) is proved.

The limit of $\gamma_n$ is then smooth; we can use Dini's theorem so that (i) is proved.

(ii) From (i) we deduce

$$ (28) \quad \left\{ \begin{array}{l} \text{for every } \epsilon > 0 \text{ there exists } n_o \text{ such that} \\ \text{for } n > n_o, \quad (1-\epsilon) \gamma_o \le \gamma_{k_n} \le \gamma_o; \end{array} \right. $$

thus, from Lemma 2

$$ (29) \quad l[(1-\epsilon) \gamma_o] \le l(\gamma_{k_n}) = l_n \le l(\gamma_o); $$

the length is a homogeneous function of the parametrization; therefore

(29) becomes

$$ (1-\epsilon) \ell_o \le \ell_n \le \ell_o \quad \text{or} \quad \epsilon \le \ell_o - \ell_n \le \epsilon \ell_o $$

from which (ii) is deduced.

(iii) From the Maximum Principle, we have

$$ (30) \quad |\tilde{u}_n - u_o|_{C^0(\Omega)} \le |u_n - u_o|_{C^0(\gamma_n)}; $$

but $u_n = 1$ on $\gamma_n$, $u_o = 1$ on $\gamma_o$ and $u_o$ is smooth so that (i) implies

$$ (31) \quad |u_n - u_o|_{C^0(\gamma_n)} \to 0 \text{ when } n \to \infty; $$

which, jointly (30) implies (iii).
(iv) **First Step**: weak convergence: we shall show the following result:

**Lemma 4**: The sequence $\tilde{u}_n$ is bounded in $H^1(\Omega)$.

**Proof**: we see easily by Green's formula that

$$ 
\begin{aligned}
\int_{\Omega} |\operatorname{grad} \tilde{u}_n|^2 dx &= \int_{\Omega_n} |\operatorname{grad} \tilde{u}_n|^2 dx = \int_{\partial \Omega_n} \tilde{u}_n \frac{\partial \tilde{u}_n}{\partial n} d\sigma \\
&= k \cdot l_n ;
\end{aligned}
$$

from (ii) that integral is bounded; this, jointly with Poincaré's inequality, implies the conclusion of Lemma 4. ■

Now, statement (iii) implies the strong convergence of $\tilde{u}_n$ toward $u_o$ in $L^2(\Omega)$, therefore

$$ \tilde{u}_n \to u_o \text{ weakly in } H^1(\Omega) \text{ as } n \to \infty. $$

**Second Step**: Strong convergence: we easily see that

$$ 
\begin{aligned}
\int_{\Omega} |\operatorname{grad}(\tilde{u}_n - u_o)|^2 dx = & -\int_{\Omega} \langle \operatorname{grad} u_o, \operatorname{grad}(\tilde{u}_n - u_o) \rangle dx \\
& + \int_{\Omega_n} \langle \operatorname{grad} \tilde{u}_n, \operatorname{grad}(\tilde{u}_n - u_o) \rangle dx ;
\end{aligned}
$$

now, using Green's formula and the conditions satisfied by $\tilde{u}_n$, this becomes

$$ 
\begin{aligned}
\int_{\Omega} |\operatorname{grad}(\tilde{u}_n - u_o)|^2 dx = & -\int_{\Omega} \langle \operatorname{grad} u_o, \operatorname{grad}(\tilde{u}_n - u_o) \rangle dx \\
& + k_n + \int_{\partial \Omega_n} (\tilde{u}_n - u_o) d\sigma ;
\end{aligned}
$$

from the weak convergence, the first integral of the right hand side converges to zero; so does the second integral from the uniform convergence; then the conclusion of Lemma 3 follows. ■

The following result is trivially deduced from Lemma 3:

**Lemma 5**: The mapping $k \to C_k = k \cdot l(\gamma_k)$ is a continuous function for $0 < k < \infty$. ■
**Lemma 6 (k + o):** Let $k_n$ be a sequence of positive numbers such that
$$k_1 > k_2 > \dots > k_n > k_{n+1} \quad \rightarrow_{o} \quad n \rightarrow \infty$$
then

$$ (32) \qquad \gamma_1 < \gamma_2 < \dots < \gamma_n < \gamma_{n+1} \quad \rightarrow_{+\infty} \quad \text{every where,} $$

$$ (33) \qquad \tilde{u}_1 \ge \tilde{u}_2 \ge \dots \ge \tilde{u}_n \ge \tilde{u}_{n+1} \quad \rightarrow_{o} \quad 0 $$

weakly in $H^1(\Omega)$ for every open bounded set $\Omega$ included in $\mathcal{B}$ and uniformly
in every compact subset of $\mathcal{B}$.

Moreover, for every compact subset $K'$ of $\Omega_{n+1}$, we have

$$ (34) \qquad \tilde{u}_n > \tilde{u}_{n+1} \quad \text{on} \quad K'. $$

**Proof :** The convergence of $\gamma_n$ to infinity is already asserted in Proposition 9, and the inequalities between the $\gamma_n$ in Proposition 7, thus (32) holds.

Inequalities for $\tilde{u}_n$ are easily derived from (32) : indeed, from the Maximum Principle we have

$$ o \le \tilde{u}_1 - \tilde{u}_2 \le \operatorname{Max}_{\gamma_1} (u_1 - u_2) \text{ in } \mathcal{B} $$

with strict inequalities in $K'$.

Let us finally show the convergence of $u_n$ :
we denote by $\tilde{u}_n$ the extension of $u_n$ by 1 in $\mathcal{B}$ ; as in the proof of
Lemma 3 (iv) and using (i) of that lemma, we have

$$ (35) \qquad \|\tilde{u}_n\|_{H^1(\Omega)} \le \text{const.} $$

Thus we can extract $H^1(\Omega)$ weakly converging subsequences ; the limit will
be unique and zero if we show the uniform convergence to zero on $K$ ;
for this purpose, we use a supersolution process :

Let $r_s$ be a positive number such that $r_s < \text{Min } \Gamma$ ; from Proposition 9,
for every positive number $k$ there exists a circle of radius $R_s(k)$ such
that

(36) 	$R_s(k) < \gamma_k$ everywhere,

(37) 	$R_s(k) \to \infty$ when $k \to 0$.

If we denote by $u_s(k)$ the solution of the Dirichlet problem:

(38) 	$\left\{
\begin{array}{ll}
\Delta u_s(k) = 0 & \text{for } r_s < \sqrt{x^2+y^2} < R_s(k) \\
u_s(k) = 0 & \text{for } \sqrt{x^2+y^2} = r_s \\
u_s(k) = 1 & \text{for } \sqrt{x^2+y^2} = R_s,
\end{array}
\right.$

then, by the Maximum Principle, we get

(39) 	$0 \le u_k \le u_s(k) \quad \text{in } \Omega_k \cap \left\{ \sqrt{x^2+y^2} < R_s(k) \right\}$

where $\Omega_k$ denotes the bounded domain limited by $\mathbb{P}$ and $\gamma_k$.

But $u_s(k)$ is given by

(40) 	$u_s(k) = 1 - \frac{\log \sqrt{x^2+y^2} - \log R_s(k)}{\log \gamma_s - \log R_s(k)}$

which converges uniformly in $K$ to zero when $R_s(k)$ tends to infinity, which implies, jointly with (39) and (40) the uniform convergence of $u_k$ on $K$ to zero.

**Lemma 7** ($k \to \infty$) : Let $k_n$ be a sequence of positive numbers such that

$k_1 < k_2 < \dots < k_n < k_{n+1} \xrightarrow{n \to \infty} \infty \text{ as } n \to \infty,$

then

(41) 	$\gamma_n \to \Gamma$ uniformly, i.e. in $C^\alpha(0,2\pi)$

and

(42) 	$\tilde{u}_n \to 1$ in $L^2(\mathbb{R})$.
**Proof**: Let us note first that (42) is a trivial consequence of (41) since meas $(\tilde{u}_n \neq 1) = \text{meas} (\tilde{u}_n \in [0,1] \rightarrow 0 \ (\text{in } \mathcal{B}))$ when $d(\gamma_n, \Gamma)$ tends to zero.

Statement (41) follows from monotonicity and surjection properties (Proposition 7 and 8), via Dini's theorem as for Lemma 3. ■

**Lemma 8**: The mapping $k \rightarrow C_k$ defined in Lemma 5 have the following properties

(i) If $k_1 < k_2$ then $C_{k_1} < C_{k_2}$

(ii) $C_k \rightarrow 0$ when $k \rightarrow 0$

(iii) $C_k \rightarrow \infty$ when $k \rightarrow \infty$.

**Proof**: Using Green's formula, we get

$$ (43) \qquad C_k = k \int_{\Gamma_k} d\sigma = \int_{\Gamma} |\text{grad } u_k| \, d\sigma. $$

If $k_1 < k_2$, then, from (33) (34), and by applying the **strong** Maximum Principle to $u_1 - u_2$ in $\Omega_2$,

we obtain

$$ \frac{\partial}{\partial \nu} (u_1 - u_2) > 0 \text{ on } \Gamma; $$

therefore

$$ |\text{grad } u_1| < |\text{grad } u_2| \text{ on } \Gamma, $$

which, together with (43), implies (i).

(ii) Let $v$ be a smooth function defined on $\mathcal{B}$ such that

$$ \begin{align*} v &= -1 \text{ on } \Gamma \\ v &= 0 \text{ for } \sqrt{x^2+y^2} > \text{Max } \Gamma(\theta) + 1; \tag{0,2\Pi} \end{align*} $$

then, by Green's formula,

$$ C_k = \int_{\Gamma} |\text{grad } u_k| \, d\sigma = \int_{\Gamma} v \frac{\partial u_k}{\partial v} \, d\sigma = \int_{\mathcal{B}} \langle \text{grad } u_k, \text{grad } v \rangle \, dx dy $$

then, using the $H^1(\Omega)$-weak convergence of $u_k$ toward zero (cf. (33)), the part (ii) is proved. ■
(iii) Since $l(\gamma_k)$ is greater than $l(\Gamma)$ (Lemma 1) which is positive, we have

$$k \cdot \ell_k \to +\infty .$$

### 2.4.2. Proof of Theorem 2

Taking into account the above Lemmas, we need only to note that $k \mapsto C_k$ is a continuous, monotone and strictly increasing function defined on $[0, +\infty[$, with range $[0, +\infty[$:

thus for any given $c$ in $[0, +\infty[$ there exists a unique corresponding $k$ in $[0, +\infty[$ such that $c = C_k$, and Theorem 2 is proved.

### 2.4.3. Proof of Theorem 3

Let us note first that Step 1 is always possible (see Paragraph 2.4.2); monotonicity properties are explicit in the following lemma

**Lemma 9 :** The sequence $\{k_n, \Omega_n, c_n\}$ verify

(i) $k_0 > k_1 > k_2 > \dots > k_n > k_{n+1} > \dots$

(ii) $\Omega_0 \subset \Omega_1 \subset \Omega_2 \subset \dots \subset \Omega_n \subset \Omega_{n+1} \subset \dots$

(iii) $c_0 > c_1 > c_2 > \dots > c_n > c_{n+1} > \dots > C > 0.$

**Proof :** By induction; from $c_n > C$ and $k_n \cdot l(\gamma_n) = c_n$, $k_{n+1} \cdot l(\gamma_n) = C$, we derive $k_{n+1} < k_n$ which implies $\Omega_{n+1} \supset \Omega_n$; thus, using Lemma 2, we get $l(\gamma_{n+1}) > l(\gamma_n)$; then

$$c_{n+1} = k_{n+1} \cdot l(\gamma_{n+1}) > k_{n+1} \cdot l(\gamma_n) = C.$$

**Lemma 10 :** The sequence $(k_n)$ converges to a positive constant $k_\infty$.

**Proof :** If $k_\infty = 0$, then, by Lemma 6, $c_n \to 0$, which contradicts Lemma 9 (iii).

To complete the proof of Theorem 3, we use Lemma 3 :
$$ \gamma_n \to \gamma = \gamma(k_\infty) \quad \text{uniformly} $$

$$ \tilde{u}_n \to u = u(k_\infty) \quad \text{in } H_0^1(\mathcal{D}) \\ \text{loc} $$

$$ \ell_n \to \ell(\gamma) $$

and indeed

$$ c_n = k_n \cdot \ell_n \to k \cdot \ell(\gamma) = \lim_{n \to +\infty} k_{n+1} \cdot \ell(\gamma_n) = C. $$

<figure><img src="image_7.png" /></figure>
SECOND PART : AN OBSTACLE PROBLEM

# 1. A REGULARITY THEOREM VIA AN A PRIORI ESTIMATE

## 1.1 Preliminaries

The data are the following

$$
\begin{align*}
& 0 < \alpha < \beta < +\infty \\
& 0 < \delta < +\infty \\
& 0 < M < +\infty \\
& 0 < a,b \ ; \ b \le \sqrt{\frac{2}{\alpha}} - a ;
\end{align*}
$$

We consider the set

$$
\Sigma_o = \{ \sigma \in W^{1,\infty}_{(0,1)}, -a \le \sigma \le b \}
$$

and the following Free Boundary Problem

F.B.P. : For function f of L²(ℝ²), for a σ given in the set Σ₀,
find a pair (u,γ) such that (see the figure)

$$
(1) \quad \gamma \in W^{1,\infty}(0,1)
$$

$$
(2) \quad \Omega_{\sigma, \gamma} = \{ (x,y), x \in [0,1], \sigma(x) < y < \gamma(x) \}
$$

$$
(3) \quad \left\{
\begin{array}{ll}
u \in H^1(\Omega_{\sigma, \gamma}) & \\
\Delta u = f \quad \text{in } \Omega_{\sigma, \gamma} & \\
u = 1 \quad \text{on } \sigma = \{(x,y), x \in [0,1], 1 = \sigma(x)\} & \\
\frac{\partial u}{\partial \nu} = 0 \quad \text{on } \text{AD} \cup \text{U} \quad \text{BC} & \\
u = \frac{\partial u}{\partial \nu} = 0 \quad \text{on } \gamma = \{(x,y), x \in [0,1], y = \gamma(x)\}.
\end{array}
\right.
$$

<figure><img src="image_8.png" /><figcaption>Figure 4</figcaption></figure>
This system is related to an Obstacle-type Ellyptic Variational Inequality which we can write as $(\text{Max } \sigma(x) < r < \infty) \qquad x \in [0,1]$

$$ (4) \quad \left\{ \begin{array}{l} u \in K_r, \quad V_{\phi \in K_r} \\ \int_{\Omega_{\sigma,r}} \{ \langle \text{grad } u, \text{grad}(\phi-u) \rangle + f(\phi-u) \} \,dx \,dy \ge 0 \end{array} \right. $$

with

$$ \Omega_{\sigma,r} = \{ (x,y), \ o < x < 1, \ \sigma(x) < y < r \} $$

$$ K_r = \{ \phi \in H^1(\Omega_{\sigma,r}), \phi \ge 0, \phi|_{\sigma} = 1, \phi|_{\{y=r\}} = 0 \}. $$

More precisely, the above two problems are connected by the following (classical :see for instance D. KINDERLEHRER, G. STAMPACCHIA [1]). results :

**Proposition 1.** (i) If a solution $u$ of (4) is such that the set

$$ (5) \quad \Omega_{+}(u) = \{ (x,y), \ o < x < 1, \ \sigma(x) < y, \ u(x,y) > o \} $$

can be expressed by

$$ (6) \quad \Omega_{+}(u) = \{ (x,y), \ o < x < 1, \ \sigma(x) < y < \gamma(x) \} $$

with

$$ \gamma \in W^{1,\infty}(0,1), \quad \sigma(x) < \gamma(x) < r \quad \forall x \in [0,1] $$

then solution $u$ is a solution of (1) (2) (3).

(ii) Let us assume that for an $r_o > 0$ we have

$$ f|_{\Omega_{r_o}} > 0 $$

then every solution $(u,\gamma)$ of (1) (2) (3) such that

$$ \left\{ \begin{array}{l} u \ge o \quad \text{in } \Omega_{r_o} \\ \gamma(x) \le r_o \quad \forall x \in [0,1] \end{array} \right. $$

is also a solution of (4) for $r \ge r_o$.

We have also the proposition
**Proposition 2:(i)** (see J.L. LIONS, G. STAMPACCHIA [1]) for every
$r > \operatorname{Max} \sigma(x)$ the inequality (4) has an unique solution.
$x \in [0,1]$

(ii) Moreover, if $\alpha < f < \beta$, $\sigma$ belong to $\Sigma_0$ and $r > \sqrt{\frac{2}{\beta}}$, then the solution of (4) verifies

$$
(7) \quad [0,1[ \times ]b, \sqrt{\frac{2}{\beta}} - a [c \le \Omega_+(u) \le ]0,1[ \times ] - a, \sqrt{\frac{2}{\alpha}} + b [ \\
\qquad (\Omega_+(u) \text{ is defined in } (6)].
$$

**Proof of (ii):** Let us introduce the following functions

$\underline{u} = \frac{1}{2} \beta \left( y + a - \sqrt{\frac{2}{\beta}} \right)^2$ if $y \le \sqrt{\frac{2}{\beta}} - a$, $0$ otherwise

$\bar{u} = \frac{1}{2} \alpha (y - b - \sqrt{\frac{2}{\alpha}})^2$ if $y \le \sqrt{\frac{2}{\alpha}} + b$, $0$ otherwise;

then it will be enough to show

$$
(8) \quad \underline{u} \le u \le \bar{u}
$$

Since it will imply

$$
\operatorname{supp} \underline{u} \subset \operatorname{supp} u \subset \operatorname{supp} \bar{u} .
$$

Statement (8) is a property of supersolutions (see H. LEWY, G. STAMPACCHIA
[1] ; let us give an outline of its proof :

**Step 1** : Let $\zeta = \operatorname{Inf}(u, \bar{u})$ then $\zeta$ belongs to $K_r$ and we have

$$
\int_{\Omega_{\sigma}, r} \left[ \langle \operatorname{grad} u, \operatorname{grad}(\zeta-u) \rangle + f(\zeta-u) \right] dx dy \geq 0.
$$
**Step 2** : From $\zeta - u \le 0$, we derive, by a Green formula

$$ \int_{\Omega_{\sigma,r}} \left[ \langle \operatorname{grad} \bar{u}, \operatorname{grad}(\zeta-u) \rangle + \alpha (\zeta-u) \right] dx dy \le 0; $$

using again $\zeta \le u$, this can be also written as

$$ \int_{\Omega_{\sigma,r}} \cap \{\zeta < u\} \left[ \langle \operatorname{grad} \zeta, \operatorname{grad}(\zeta-u) \rangle + \alpha(\zeta-u) \right] dx dy \le 0 $$

**Step 3** : Computing the difference between the conclusions of Step 1 and Step 2 we get

$$ \int_{\Omega_{\sigma,r}} \left[ \langle \operatorname{grad}(\zeta-u), \operatorname{grad}(\zeta-u) \rangle + (\alpha-f)(\zeta-u) \right] dx dy \le 0 $$

Which implies $\zeta = u$ and $u \le \bar{u}$.

By a similar argument we get $u \le u$.

These properties will lead us to special common solutions of (1) (2) (3) and (4); let us assume that $\sigma \equiv o$ and that the function $f_o$ is taken equal to a function $f_o$ such that

$$ (9) \quad \left\{ \begin{array}{l} f_o \text{ is infinitely differentiable, } \alpha \le f_o \le \beta, \\ \frac{\partial f}{\partial x} \bigg|_o = 0, \quad \frac{\partial f}{\partial y} \bigg|_o \ge \delta. \end{array} \right. $$

Then the problem is reduced to a *one-dimensional* Variational Inequality;
from Proposition 2, the Free Boundary exists and consists of *one horizontal* segment; in particular, this Free Boundary is *smooth* and so is the solution $u$; the purpose of next section is to show that this smoothness still holds when $\sigma$ and $f$ are in a neighborhood (to precise later) of those special data $\sigma \equiv o$ and $f = f_o(y)$.

## 1.2. Main result : a priori estimates for the Free Boundary
**Proposition 3; Let be**

$$r > \sqrt{\frac{2}{\alpha}} + b$$

$$ \mathcal{J} = \{ f \in L^2([0,1[ \times [\alpha, \beta], o \le \alpha \le f \le \beta, \\ \frac{\partial f}{\partial y} \ge \delta ; \left| \frac{\partial f}{\partial x} \right| \le M \} $$

$$ \Sigma = \{ \sigma \in W^{1,\infty}(0,1), -a \le \sigma \le b, \left| \frac{d\sigma}{dx} \right| \le \frac{M}{\delta} \}; $$

then, for $(f, \sigma)$ belonging to $\mathcal{J} \times \Sigma$, the corresponding Free Boundary $\gamma$, solution of (1) (2) (3) and (4) belongs to $W^{1,\infty}(0,1)$ and verifies the estimates

$$(10) \qquad \sqrt{\frac{2}{\beta}} - a \le \gamma \le \sqrt{\frac{2}{\alpha}} + b$$

$$(11) \qquad \left| \frac{d\gamma}{dx} \right| \le \frac{M}{\delta}.$$

**Corollary 1:** Under the hypothesis of Proposition 3, assuming moreover that $f$ belongs to $C^{\ell+\alpha}$ ($[0,1] \times [-a,r]$) with $\ell \ge 1$ and $y \in \mathbb{R}$, $f(0,y) = f(1,y)$ and $V_m \in \mathbb{N}$, $m \le \ell$, $\forall y \in [-a,r]$ $\frac{\partial^m f}{\partial x^m}(0,y) = \frac{\partial^m f}{\partial x^m}(1,y) = 0$;

then the corresponding Free Boundary $\gamma$ is $C^{\ell+\alpha+1}$.

### 1.3. Proofs

**Proof of Proposition 3**

**Step 1:** Let us note first that the Maximum Principle applied to the solution $u$ of (4) corresponding to the data $f$ and $\sigma$ gives

$$(12) \qquad u \le 1$$

$$(13) \qquad \frac{\partial u}{\partial v} < 0 \text{ on } \sigma.$$

For every $\varepsilon$ belonging to the interval $[-\frac{\delta}{M}, \frac{\delta}{M}]$, we shall apply the Maximum Principle to the function $w_\varepsilon = \frac{\partial u}{\partial y} + \varepsilon \frac{\partial u}{\partial x}$
If $\varepsilon = 0$, $w_0 = \frac{\partial u}{\partial y}$ and

$$(14) \quad \left\{ \begin{array}{l} \Delta w_o = \frac{\partial f}{\partial y} \ge \delta > 0 \text{ on } \Omega_+ = \{(x,y) \in \Omega, r; u(x,y) > 0\} \\ w_o < 0 \text{ on } \sigma \text{ from (13) and } \frac{\partial u}{\partial \tau} = 0, \text{ where} \\ \tau \text{ denotes an unit tangential vector} \end{array} \right.$$
thus the maximum of $u$ on $\Omega_+$ occurs on the boundary $\partial\Omega_+$; by a symmetry argument, we see that it does not occur on a vertical side so that from (14) and from

$$w_o = 0 \text{ on } \Omega_+, r \in \Omega_+$$

we derive

$$ (15) \qquad \begin{array}{l} w_o \le 0 \text{ on } \Omega_+; \\ \text{For an arbitrary } \epsilon, \text{ we have} \end{array} $$

$$ (16) \quad \left\{ \begin{array}{l} \Delta w_{\epsilon} = \displaystyle\frac{\partial f}{\partial y} + \epsilon \displaystyle\frac{\partial f}{\partial x} \ge \delta - \frac{\partial}{M} M \ge 0 \\ w_{\epsilon} = w_o \le 0 \text{ on vertical sides (using (15))} \\ w_{\epsilon} = 0 \text{ on } \Omega_{\sigma}, r \in \Omega_{+}; \end{array} \right. $$

on $\sigma$ we have

$$w_{\epsilon} = \langle (\epsilon,1), v \rangle > \frac{\partial u}{\partial v} + \langle (\epsilon,1), \tau \rangle > \frac{\partial u}{\partial \tau};$$

the second term of the sum is identically zero; from the bound of $\frac{d\sigma}{dx}$ (see the definition of $\Sigma$) we derive

$$\langle (\epsilon,1), v \rangle = \langle (\epsilon,1), \left( 1 + \left[ \frac{d\sigma}{dx} \right]^2 \right)^{-\frac{1}{2}} \left( -\frac{d\sigma}{dx}, 1 \right) \rangle \ge 0$$

(because $|\epsilon \frac{d\sigma}{dx}| \le 1$) so that, using (13), we get

$$w_{\epsilon} \le 0 \text{ on } \sigma.$$

Thus

$$ (17) \qquad w_{\epsilon} < 0 \text{ on } \Omega_{+}. $$

**Step 2:** Statement (17) implies that $u$ is non increasing along the direction $(\epsilon,1)$; let us introduce the following function

$$\forall x \in [0,1], \gamma(x) = \sup \{y \in [\sigma, r], (x,y) \in \Omega_+\}.$$

For every point $(x_o, \gamma(x_o))$ of $\partial\Omega_+$ we have $u(x_o, \gamma(x_o)) = 0$; since $u$ is a non negative function and is non increasing in the direction $(\epsilon,1)$, $u$ vanishes also on the half lines $\{(x_o \pm \lambda\epsilon, \gamma(x_o) +\lambda), \lambda \ge 0\}$ with $(x_o, \gamma(x_o))$ as origin; choosing every point of those half-lines as a new origin, we derive that $u$ vanishes in the set $\{(x_o + \mu\epsilon, \gamma(x_o) +\lambda), \lambda \ge 0, \mu \le \lambda\}$ so that (17) can be interpreted as follows
$$ \begin{array}{l} \Omega_+ \subset \{(x,y), y \le \frac{1}{\epsilon}(x-x_0) + \gamma(x_0)\} \\ \text{or } y \le -\frac{1}{\epsilon}(x-x_0) + \gamma(x_0) \}. \end{array} $$

and similarly

$$ \begin{array}{l} \Omega_{\sigma, r} \cap \Omega_+ \subset \{(x, y), y \ge \frac{1}{\epsilon}(x-x_0) + \gamma(x_0)\} \\ \text{or } y \ge -\frac{1}{\epsilon}(x-x_0) + \gamma(x_0) \}. \end{array} $$

Now, those statements imply that $\gamma$ is a Lipschitz continuous function satisfying

$$ (18) \quad \operatorname{Sup}_{x \in ]0,1[} \left| \frac{d\gamma}{dx} \right| \le \frac{1}{\epsilon} $$

and also, using (15), that

$$ \begin{align*} \partial\Omega_+ &= \{(x, \gamma(x)), x \in [0,1]\} \cup \{0\} \times [\sigma(0), \gamma(0)] \\ &\cup \{1\} \times [\sigma(1), \gamma(1)] \cup \sigma; \end{align*} $$

thus the conclusion of Proposition 3 follows.

<figure><img src="image_9.png" /></figure>

**Remark 1**: Similarly the estimate (18) also holds for every curve $u = \text{const.}$ i.e. if we denote by $\gamma_k$ the following set $(1>k>0)$

$$ \gamma_k = \{(x,y) \in \partial\Omega_+, u(x,y) = k\} $$

then there exists a parametrization, still denoted $\gamma_k$, such that

$$ (19) \quad \gamma_k \in W^{1,\infty}(0,1), \quad \operatorname{Sup}_{x \in ]0,1[} \left| \frac{d\gamma_k}{dx} \right| \le \frac{M}{\delta} $$

and

$$ (20) \quad \text{the set } \gamma_k = \{(x, \gamma_k(x)), x \in [1,0]\}; $$

moreover it is easy to see that $\gamma_k$ converges uniformly to $\gamma$ when $k$ tends to infinity.

<figure><img src="image_10.png" /></figure>

**Remark 2**: The conclusions of Proposition 3 and Remark 1 still hold if the set $\gamma$ is replaced by
$$
\begin{gather*}
\mathfrak{g}^* = \{ f \in L^2 \left( ]0,1[ \times ]0,r[ \right) : a, r \in \mathfrak{L} \, , \ o < \alpha \le f \le \beta \, , \\
\frac{\partial f}{\partial y} + \epsilon \frac{\partial f}{\partial x} \ge 0 \quad \forall \epsilon \in [-\frac{\delta}{M}, +\frac{\delta}{M}] \} .
\end{gather*}
$$

<figure><img src="image_11.png" /></figure>

Proof of Corollary 1

From H. BREZIS, D. KINDERLEHRER [1], we have

$$
(21) \quad u \in C^{1,1} \left( [0,1] \times [0,r] \right);
$$

now, from the Lipschitz property of $\gamma$, we deduce that this free boundary
has no zero-density point with respect to the support of $u$; this jointly
with (21), is sufficient to apply the theorem of L. CAFFARELLI [1] firs-
tly, and then those of D. KINDERLEHRER, L. NIRENBERG [1], from which the
smoothness of $\gamma$ is derived (the corners are dealt with by symmetry argu-
ments).

<figure><img src="image_12.png" /></figure>

2. APPLICATION TO AN OPTIMAL CONTROL PROBLEM

2.1. Position of the general problem

Let us consider a Free Boundary Problem in which a datum (the control
variable) is supposed to vary : to an arbitrary value of this datum cor-
responds an unique free boundary and an unique value of a cost functional
(to minimize) expressed with respect to this Free Boundary (the F.B. is
"observed"). The existence of a control value minimizing the cost (an op-
timal control) is generally an open problem.

The arguments to deal with this problem need schematically two steps :

(i) <u>A priori estimates</u> for the Free Boundary.

(ii) Continuity of the F.B with respect to the control variable.

Step (i) is strongly related with the close problem of controlability,
i.e. looking for data such that the corresponding F.B is identical (or
close) to a given one ; for that point see for instance D.G. SCHAEFFER
[3].

The second step deals with dependence with respect to data : with argu-
ments analogous to those of D.G. SCHAEFFER [1], some differentiability
results with respect to data are proved in DERVIEUX [1] for c^m norms (with
m large enough).
For the two steps, see O. PIRONNEAU, C. SAGUEZ [1] [2]: the cost function introduced by the authors is the Hausdorff distance between the free domain and a "desired" one.

Now the two topologies above ($C^m$ norms, Hausdorff distance) make the results difficult to apply to very concrete cases (for instance numerical experiments) ; in fact, it seems necessary to use a reasonable intermediary topology : this paragraph is an attempt in this direction for a particular problem, with the hope of a possibility of generalization.

## 2.2. Definition of our problem

The notations and assumptions are identical to the ones in Paragraph 1 ; we consider the following subset of $\mathcal{F}$ ($M'$ is a strictly positive constant) :

$$ \mathcal{F}_{ad} = \{ f \in W^1,^\infty ( ]0,1[ \times ]0,r[, f \in \mathcal{F}, |\frac{\partial f}{\partial y}| < M' \} $$

From the above section if the pair of data $(f, \sigma)$ belongs to $\mathcal{F}_{ad} \times \Sigma$, then the corresponding Free Boundaries will belong to the set

$$ \Gamma_{ad} = \{ \gamma \in W^1,^\infty (0,1) ; \sqrt{\frac{2}{\beta}} - a < \gamma < \sqrt{\frac{2}{\alpha}} + b ; \text{Max} |\gamma'| < \frac{M}{\delta} \}. $$

Let $J$ be a functional defined on $\Gamma_{ad}$:

$$ J : \Gamma_{ad} \longrightarrow \mathbb{R}_{+} ; $$

we assume that $J$ is lower semi-continuous on $\Gamma_{ad}$ equipped with the strong topology of $C^T(0,1)$ with $\sigma < \tau < 1$.

Let us consider the functional

$$ j : \mathcal{F}_{ad} \times \Sigma \longrightarrow \mathbb{R}_{+} $$

$$ V(f, \sigma) \in \mathcal{F}_{ad} \times \Sigma, j(f, \sigma) = J(\gamma(f, \sigma)) $$

where $\gamma(f, \sigma)$ is the parametrization of the Free Boundary which corresponds to the data $f$ and $\sigma$ in Problem (1) (2) (3).

We consider the following problem

**Problem (P):** To find a pair $(f_{opt}, \sigma_{opt})$ belonging to $\mathcal{F}_{ad} \times \Sigma$ such that

$$ j(f_{opt}, \sigma_{opt}) = \min_{f \in \mathcal{F}_{ad}, \sigma \in \Sigma} j(f, \sigma). $$
**Example**: if we want the free boundary to be as close as possible to a so-called "desired" curve the parametrization of which is $y = \gamma_d(x)$, we may choose

$$J(\gamma) = \int_{0}^{1} [\gamma(x) - \gamma_d(x)]^2 dx.$$

### 2.3. Existence of an optimal control

**Proposition 4**: Problem 1 admits at least one solution.

**Remark 3**: It is also possible to consider, instead of J, a functional $\bar{J}$ depending both on the control variables $(f$ and $\sigma)$ and on the states functions $(u(f,\sigma), \gamma(f,\sigma))$ solutions of (1) (2) (3)(4). Then Proposition 8 still holds if the mapping $(f,\sigma, \gamma, \tilde{u}) \to \bar{J}(f,\sigma, \gamma, \tilde{u})$ is continuous from $\mathcal{F}_{ad} \times \Sigma \times \Gamma_{ad} \times H^1(]0,1[ \times ]0,r[)$ equipped with the norm of $L^2(]0,1[ \times [C^T(0,1)]^2 \times H^1(]0,1[ \times ]0,r[)$ into $\mathbb{R}$ $(u(f,\sigma)$ is extended by 1 and 0).

**Remark 4**: In accordance to what is said in Section 2.1 we have also the following non-controllability result: if the desired boundary $\gamma_d$ in the above example does not verify (26) (27), then there does not exist a pair $(f_{opt}, \sigma_{opt})$ in $\mathcal{F}_{ad} \times \Sigma$ such that $j(f_{opt}, \sigma_{opt}) = 0$.

**Remark 5**: The case where $f$ is fixed (the control variable is only $\sigma$) is studied in DERVIEUX [1] where are given some further informations concerning physical motivations and necessary optimality conditions, and also numerical results.

**Remark 6**: Using Remark 1, we can get a result similar to Proposition 4 with the following functional :

$$j_k(f, \sigma) = J(\gamma_k(f, \sigma))$$

where $\gamma_k$ is the isovalue curve $\{(x,y), u(x,y) = k\}$.
## 2.4. Proof

We shall use the following lemma

**Lemma 1:** Let us assume that $f_n$, for $n$ belonging $\mathbb{N}$, and $f$ belong to $L^2(D)$ with

$$D = ]0,T[ \times ]-a,r[, \quad r > \frac{\sqrt{2}}{\alpha} + b$$

and

$$f(x) > 0 \text{ a.e. in } D,$$

that $f_n$ converges to $f$ in $L^2(D)$ when $n$ tends to infinity that $\sigma_n$, for belonging to $\mathbb{N}$, and $\sigma$ belongs to $C^0(0,1)$, verify

$$-a < \sigma_n, \sigma < b,$$

that $\sigma_n$ converges to $\sigma$ in $C^0(0,1)$ when $n$ tends to infinity, and that the Free Boundaries $\gamma_n$ corresponding to $(f_n, \sigma_n)$ as solution of (1) (2) (3)

(4) exist and verify

$$(22) \quad \left\{ \begin{array}{l} b < \gamma_n < r \\ \left| -\frac{d\gamma_n}{dx} \right| \le C, \quad \text{where } C \text{ is a constant not depending on } n. \end{array} \right.$$

Then the solutions $(u_n, \gamma_n)$ of (1) (2) (3) (4) converge to the solution $(u, \gamma)$ corresponding to $(f, \sigma)$ in the following ways:

(i) The extensions $\tilde{u}_n$ of $u_n$ by 1 for $y \le \sigma_n(x)$ converge to the extension $\tilde{u}$ of $u$ strongly in $H^1(D)$.

(ii) the free boundaries $\gamma_n$ converge to $\gamma$ strongly in $C^T(0,1)$ for $0 \le \tau < 1$ their first derivatives converge in $L^\infty(0,1)$ weakly star to $\gamma'$.

## Proof of Lemma 1

(i) : Convergence of $u_n$; We put

$$\Omega_{\sigma_n, \gamma_n} = \{(x,y), 0 < x < 1, \sigma_n(x) < y < \gamma_n(x)\};$$

then

$$\|\tilde{u}_n\|_{H^1(D)} \le \sqrt{\text{meas}(D)} + \|u_n\|_{H^1(\Omega_{\sigma_n, \gamma_n})};$$
the following estimate is classical (and trivial : put $\varphi = 0$ in the variational inequality) :

$$
\|u_n\|_{H^1(\Omega_{\sigma_n, \gamma_n})} \leq |f_n|_{L^2(\Omega_{\sigma_n, \gamma_n})}
$$

$$
\text{thus} \quad \|u_n\|_{H^1(D)} \le \text{const.} (1 + |f_n|_{L^2(D)}).
$$

Extracting a subsequence still denoted $\tilde{u}_n$, we get

$$
(23) \quad \tilde{u} \to u^* \text{ weakly in } H^1(D), \text{ strongly in } L^2(D).
$$

For an arbitrary $\epsilon$, we consider a function $\varphi$ belonging to $C^\infty(\bar{D})$ such that

$$
\varphi \ge 0 ; \varphi(x,y) = 1 \text{ for } (x,y) \in D, y < \sigma(x) + \epsilon ;
$$

then for $n$ sufficiently large, since $\sigma_n$ converges uniformly to $\sigma$, the support of $\varphi - 1$ is included in the set $\{(x,y) \in D, y > \sigma_n(x)\}$; thus we can substitute in $\varphi$ the Variational Inequality verified by $u_n$:

$$
(24) \quad \int_{\Omega_{\sigma_n}} \langle \operatorname{grad} u_n, \operatorname{grad}(\varphi - u_n) \rangle \, dx \, dy \geq \int_{\Omega_{\sigma_n}} f_n (\varphi - u_n) \, dx \, dy
$$

with

$$
\Omega_{\sigma_n} = \{(x, y) \in D, y > \sigma_n(x)\}.
$$

We want now *to pass to limit* :from the definition of the extension $\tilde{u}_n$,

(24) can be written as :

$$
(25) \quad \left\{
\begin{aligned}
& \int_D \langle \operatorname{grad} \tilde{u}_n, \operatorname{grad}(\varphi - \tilde{u}_n) \rangle \, dx \, dy \ge \\
& \quad - \int_D f_n (\varphi - \tilde{u}_n) \, dx \, dy - \int_D \bar{\Omega}_{\sigma_n} f_n \tilde{u}_n \, dx \, dy .
\end{aligned}
\right.
$$

Now, $\tilde{u}_n$ is bounded in $H^1(D)$, thus (see for example J. NEČAS [1]) also in $L^4(D)$, thus, by Hölder inequality
$$
\left| \int_{D \setminus \bar{\Omega}_{\sigma_n}} f_n \tilde{u}_n \, dx \, dy - \int_{D \setminus \bar{\Omega}_{\sigma}} f_n \tilde{u}_n \, dx \, dy \right| \le \\
\le |f_n|_{L^2(D)} |\tilde{u}_n|_{L^4(D)} |1|_{L^4((\Omega_{\sigma_n} \setminus \bar{\Omega}_{\sigma}) \cup (\Omega_{\sigma} \setminus \Omega_{\sigma_n}))}.
$$

where $\Omega_{\sigma} = \{(x,y) \ D,y > \sigma(x)\}$; from the convergence of $\sigma_n$ to $\sigma$, the right-
hand side tends to zero as $n$ tends to infinity, and, using the conver-
gences of $f_n$ and $\tilde{u}_n$ (see (23)), we get

$$
\int_{D \setminus \bar{\Omega}_{\sigma_n}} f_n \tilde{u}_n \, dx \, dy \longrightarrow \int_{D \setminus \bar{\Omega}_{\sigma}} f u^* \, dx \, dy.
$$

Let us now consider the remaining terms of (25) : using the H¹(D) weak
lower semi-continuity of the H<sub>O</sub><sup>1</sup>(D) canonical semi-norm, we obtain

$$
(26) \quad \int_{\Omega_{\sigma}} \langle \operatorname{grad} u^*, \operatorname{grad} (\varphi - u^*) \rangle_{dx dy} \geq - \int_{\Omega_{\sigma}} f (\varphi - u^*)_{dx dy} ;
$$

making $\epsilon$ tend to zero, and by a density argument with respect to $\varphi$, the
inequality (26) still holds for every nonnegative function $\varphi$ of $H^1(\Omega_\sigma)$
equal to 1 on $\sigma$.

Trivially we have

$$
(27) \quad u^* \geq o \text{ on } D.
$$

On the other hand, we can easily verify that

$$
(28) \quad u^*|_{D \setminus \bar{\Omega}_{\sigma}} = 1.
$$

From (26) (27) (28) we deduce $u^* = u$ and the whole sequence $u_n$ converges
weakly to $u$; the strong convergence is shown later.

(ii) : *Convergence of* $\gamma_n$. From (22), after having extracted a subsequence still denoted $\gamma_n$, there exists a function $\gamma^*$ such that

$$
\gamma_n \to \gamma^* \text{ strongly in } C^T (0,1),
$$

$$
\gamma_n' \to (\gamma^*)' \text{ weakly star in } L^\infty(0,1);
$$
in particular, $\gamma_n$ converges uniformly to $\gamma^*$; by an analogous argument as in the proof of (i), we can deduce that the solutions $u_n$ of the Dirichlet problems in $\Omega_{\sigma_n, \gamma_n}$:

$$ egin{cases} 
\Delta u_n = f_n \text{ in } \Omega_{\sigma_n, \gamma_n} \\ 
\frac{\partial u_n}{\partial \nu} = 0 \text{ for } x = 0, \text{ and } x = 1 \\ 
u_n = 0 \text{ on } \gamma_n \\ 
u_n = 1 \text{ on } \sigma_n 
\end{cases} 
$$

have extensions $\tilde{u}_n$ (by $1$ for $y<\sigma(x)$, by $0$ for $y > \gamma_n(x)$), which converge weakly in $H^1(D)$ to the extension $\tilde{u}^{**}$ of the solution $u^{**}$ of the following Dirichlet problem:

$$ egin{cases} 
\Delta u^{**} = f \text{ in } \Omega_{\sigma, \gamma^*} \\ 
\frac{\partial u^{**}}{\partial \nu} = 0 \text{ for } x = 0 \text{ and } x = 1 \\ 
u^{**} = 0 \text{ on } \gamma^* \\ 
u^{**} = 1 \text{ on } \sigma. 
\end{cases} 
$$

Now, from Step 1, we derive $u^{**}=u$; from the Maximum Principle, we have

$$ \Omega_{\sigma, \gamma} = \text{Supp } u $$

$$ \Omega_{\sigma, \gamma^*} = \text{Supp } u^{**} $$

then $\Omega_{\sigma, \gamma} = \Omega_{\sigma, \gamma^*}$, and $\gamma = \gamma^*$ is the unique limit of the whole sequence $\gamma_n$, and (ii) is proved.

Using (mutatis mutandis) the Lemma 4 of Part I we deduce of (ii) the **strong convergence** of $\tilde{u}_n$ toward $u$.

**End of the proof of Proposition 4**

From Proposition 3 and Lemma 1, we derive that the mapping $(f,\sigma) \to \gamma$ $(f,\sigma)$ is sequentially continuous from $\mathcal{J}_{ad} \times \Sigma$, equipped with the $L^2(]0,1[ \times ]0,r[) \times C^0(0,1)$ norm, into $C^1(0,1)$; on the other hand, it is easy to see that the set $\mathcal{J}_{ad} \times \Sigma$ is compact in $L^2(]0,1[ \times ]0,r[) \times C^0(0,1)$; thus every minimizing sequence for problem $P$ converges to a minimum.
### 3. APPLICATION TO AN ASYMPTOTIC STUDY.

The a priori estimates established in the above section can be used for a lot of asymptotical studies; one interest of the example presented here is the special importance of these estimates for the derivation of convergence of the free boundary. There are quite few papers concerning the asymptotic behaviour of the Free Boundary for the Obstacle Problem; see as a recent work, W. ECKAUS, H.J.K. MOET [1].

#### 3.1. Formulation of the problem and main result

For $f$ belonging to $\mathfrak{F}$ (see Proposition 3) we consider the following Variational Inequality indexed by $\varepsilon > 0$.

$$ (29) \quad \left\{ \begin{array}{ll} D = ]0,1[ \times ]0,r[, r > \sqrt{\frac{2}{\alpha}} + b & \\ K_r = \{ \varphi \in H^1(D), \varphi(x,0) = 1, \varphi \ge 0 \}. & \\ \displaystyle\int_D \left\{ \begin{aligned} & \frac{1}{\varepsilon} \frac{\partial u}{\partial x} \quad \frac{\partial}{\partial x} (\varphi - u_\varepsilon) \\ & + \frac{\partial u_\varepsilon}{\partial y} \quad \frac{\partial}{\partial y} (\varphi - u_\varepsilon) + f(\varphi - u_\varepsilon) \end{aligned} \right\} dx dy \ge 0. & \end{array} \right. $$

The above methods apply (directly or with the substitution $\tilde{x} = \sqrt{\varepsilon} x$) so that we have the following result.

**Proposition 5.** *Under the above assumptions, for every $\varepsilon$ positive, we have the estimates*

$$ (30) \quad \left\{ \begin{array}{l} 0 \le u_{\varepsilon} \le 1 \\ \sqrt{\frac{2}{\beta}} - a \le \gamma_{\varepsilon} \le \sqrt{\frac{2}{\alpha}} + b \\ \left| -\frac{d\gamma_{\varepsilon}}{dx} \right| < \frac{M}{\delta} \end{array} \right. $$

where $\gamma_\varepsilon$ is the free boundary corresponding to $u_\varepsilon$.
We shall consider here the following problem :

**Problem :** What are the behaviours of $u_\epsilon$ and $\gamma_\epsilon$ when $\epsilon$ tends to zero ?

A first answer is the following result :

**Proposition 6 :** Under the above assumptions, when $\epsilon$ tends to zero we have

$$ (31) \quad \left\{ \begin{array}{ll} (a) & u_{\epsilon} \to u \quad \underset{\text{strongly in}}{H^1(D)} \\ (b) & \frac{1}{\sqrt{\epsilon}} \frac{\partial u_{\epsilon}}{\partial x} \to 0 \quad \underset{\text{strongly in}}{L^2(D)} \\ (c) & \begin{gathered} \gamma_{\epsilon} \to \gamma \quad \underset{\text{strongly in}}{C^T(0,1)}, \quad 0 < \tau < 1, \\ \gamma'_{\epsilon} + \gamma' \quad \underset{\text{weakly star in}}{L^{\infty}(0,1)}, \end{gathered} \end{array} \right. $$

where $u$ is the solution and $\gamma$ the Free Boundary of the one-dimensional following inequality :

$$ K_1 = \{ \varphi \in H^1(0,r), \varphi(0) = 1, \varphi \ge 0 \} $$

$$ (32) \quad \left\{ \begin{array}{ll} u \in K_1, & \forall \varphi \in K_1 \\ \int_0^r \left\{ \frac{\partial u}{\partial y} \cdot \frac{\partial}{\partial y} (\varphi - u) + \bar{f} (\varphi - u) \right\} dy \ge 0 & \end{array} \right. $$

with the datum

$$ (33) \quad \bar{f}(y) = \int_0^1 f(x,y) dx, \quad \forall y \in ]0,r[, $$

and where $\gamma$ is the ("horizontal") free boundary corresponding to $u$.

### 3.2 Proof

**Step 1 :** Estimates for $u_\epsilon$. Since $\gamma_\epsilon < r$, we can put $\varphi_\epsilon = 1$ in (29) so that
$$
\begin{align*}
\int_D \left\{ \frac{1}{\epsilon} \left| \frac{\partial u}{\partial x} \right|^2 + \left| \frac{\partial u}{\partial y} \right|^2 \right\} \mathrm{d}x \mathrm{d}y &\le - \int_D f (u_{\epsilon} - 1) \mathrm{d}x \mathrm{d}y \\
&\le |f|_{L^2(D)}^2 |u_{\epsilon} - 1|_{L^2(D)}^2 ;
\end{align*}
$$

Now, from a Poincaré inequality we have

$$
(34) \quad |u_{\epsilon-1}|_{L^2(D)} \le 2r |\frac{\partial u}{\partial y}|_{L^2(D)}^2
$$

then we get

$$
(35) \quad \left\{
\begin{array}{ll}
\displaystyle \left| \frac{\partial u}{\partial y} \right|_{L^2(D)}^2 & \le 2r |f|_{L^2(D)}^2, \\[1em]
\displaystyle \left| \frac{\partial u}{\partial x} \right|_{L^2(D)}^2 & \le 2r \sqrt{\epsilon} |f|_{L^2(D)}^2, \\[1em]
\displaystyle \left| u_{\epsilon} \right|_{L^2(D)}^2 & \le 4r^2 |f|_{L^2(D)}^2.
\end{array}
\right.
$$

Step 2: Weak convergence of $u_\epsilon$. Where $\epsilon$ tends to zero, from
any sequence $\epsilon_n$ we extract a subsequence still denotes $\epsilon_n$ such that

$$
(36) \quad \left\{
\begin{array}{ll}
u_{\epsilon_n} \to u^* \text{ weakly in } H^1(D), \text{ strongly in } L^2(D) \\
\\
\gamma_{\epsilon_n} \to \gamma^* \text{ strongly in } C^T(0,1) \text{ with } \tau \text{ fixed } \tau < 1.
\end{array}
\right.
$$

Let us choose in (29) φ belonging to K₁:

Since φ does not depend on x, we obtain

$$
(37) \quad \int_D \left\{ \frac{\partial u_{\epsilon_n}}{\partial y} \frac{\partial}{\partial y} (\varphi - u_{\epsilon_n}) + f(\varphi - u_{\epsilon_n}) \right\} \mathrm{d}x \mathrm{d}y \ge \frac{1}{\epsilon} \int_D \left| \frac{\partial u_{\epsilon_n}}{\partial x} \right|^2 \mathrm{d}x \mathrm{d}y \ge 0
$$

From (35) we derive

$$
\liminf_{n \to \infty} \int_D \left| \frac{\partial u_{\epsilon_n}}{\partial y} \right|^2 \mathrm{d}x \mathrm{d}y \ge \int_D \left| \frac{\partial u^*}{\partial y} \right|^2 \mathrm{d}x \mathrm{d}y
$$

$$
\lim_{n \to \infty} \int_D \left\{ \frac{\partial u_{\epsilon_n}}{\partial y} \frac{\partial \varphi}{\partial y} - f u_{\epsilon_n} \right\} \mathrm{d}x \mathrm{d}y = \int_D \left\{ \frac{\partial u^*}{\partial y} \frac{\partial \varphi}{\partial y} - f u^* \right\} \mathrm{d}x \mathrm{d}y
$$
thus

$$ \int_D \frac{\partial u^*}{\partial \bar{y}} \frac{\partial}{\partial y} (\varphi - u^*) \, dx \, dy \geq - \int_D f(\varphi - u^*) \, dx \, dy ; $$

Since $\varphi - u^*$ does not depend on $x$, the right-hand side is also equal to $\int_D \bar{f}(\varphi - u^*) \, dx \, dy$, therefore $u^*$ is equal to $u$ and all the sequence $u_{\epsilon_n}$ converge to $u$.

**Step 3 :** *Strong convergence of $u_{\epsilon_n}$.* We first notice that $u_{\epsilon_n}$ belongs to $H^2(D)$ : for the neighbourhood of the Free Boundary, see for example J.L. LIONS [1] ; to see that there is no problem in the neighbourhood of the corners, we can consider the symmetry with respect to A D and B C (see the figure in section 1.1). Thus we can choose in (32)

$$ \varphi = u_{\epsilon_n}(x, \cdot) \in H^{3/2}(o, r) ; $$

by integration we obtain

$$ (38) \quad \int_D \frac{\partial u}{\partial \bar{y}} \frac{\partial}{\partial y} (u - u_{\epsilon_n}) \, dx \, dy \geq - \int_D \bar{f}(u_{\epsilon_n} - u) \, dx \, dy. $$

On the other hand, putting $\varphi = u$ in (29) we get

$$ (39) \quad \int_D \left\{ \frac{\partial u_{\epsilon_n}}{\partial x} \frac{\partial}{\partial x} (u - u_{\epsilon_n}) + \frac{\partial u}{\partial y} \frac{\partial}{\partial y} (u - u_{\epsilon_n}) + f(u - u_{\epsilon_n}) \right\} dx \, dy \geq 0. $$

by addition of (38) and (39) and using $\frac{\partial u}{\partial x} \equiv 0$ we have

$$ (40) \quad \left\{ \begin{aligned} & \int_D \frac{1}{\epsilon_n} \left| \frac{\partial}{\partial x} (u - u_{\epsilon_n}) \right|^2 + \left| \frac{\partial}{\partial y} (u - u_{\epsilon_n}) \right|^2 \, dx \, dy \\ & \leq \int_D (f - \bar{f}) (u - u_{\epsilon_n}) \, dx \, dy. \end{aligned} \right. $$

From (36) the right-hand side tends to zero thus (31a) and (31b) follow.

**Step 4 :** *Convergence of the free boundary.*

From (36), the free boundary $\gamma_{\epsilon_n}$ admits a limit $\gamma^*$ ; let us consider

$$ \Omega_* = \{ (x, y) \in D, y < \gamma^*(x) \} $$

$$ \Omega_+ = \{ (x, y) \in D, u(y) > 0 \}. $$
It remains to show that $\Omega_* = \Omega_+$.

Let $\mathcal{H}$ be a compact subset of $D\setminus\bar{\Omega}_*$; from the uniform convergence (1)
(see (36)), for $n$ sufficiently large ($n > n_o$) we have

$$
\mathcal{H} \subset D\setminus\bar{\Omega}_{\gamma\epsilon_n} = \{(x,y) \in D, y > \gamma\epsilon_n(x)\}
$$

thus

$$
u_{\epsilon_n} = 0
$$

and passing to the limit

$$
u|_{\mathcal{H}} = 0
$$

thus $\mathcal{H}$ is included into $D\setminus\bar{\Omega}_+$ ; since it holds for any $\mathcal{H}$ included into $D\setminus\bar{\Omega}_*$ we can deduce

$$
D\setminus\bar{\Omega}_* \subset D\setminus\bar{\Omega}_+
$$

thus

$\Omega_+ \subset \Omega_*.$

To prove the inverse inclusion, let us introduce a function $\varphi$ such that

$$
(41) \quad \left\{
\begin{array}{l}
\varphi \in C^\infty(\bar{D}), \\
\varphi(x,0) = 0 \quad \forall x \in [0,1];
\end{array}
\right.
$$

passing to the limit in the following equality

$$
\begin{align*}
& \int_D \varphi \left[ \frac{\partial^2 u_{\epsilon_n}}{\epsilon_n \partial x^2} + \frac{\partial^2 u_{\epsilon_n}}{\partial y^2} \right] dx dy \\
&= \int_0^1 dx \int_0^{\gamma_{\epsilon_n}(x)} f(x,y) \varphi(x,y) dy,
\end{align*}
$$

we derive (using Lebesgue's theorem and (36))

<footnote>(1) we see here (and also later) the importance of the a priori estimate
for the free boundary in this study.</footnote>
$$
(42) \quad \lim_{n \to \infty} \int_D \varphi \left[ \frac{1}{\epsilon_n} \frac{\partial^2 u_\epsilon}{\partial x^2} + \frac{\partial^2 u_\epsilon}{\partial y^2} \right] dx dy = \int_{\Omega_*} f_\varphi dx dy.
$$

Let us consider now, using (41),

$$
\begin{align*}
\int_D \frac{\partial^2 u_\epsilon}{\partial y^2} dx dy &= - \int_0^1 dx \int_0^{\gamma_\epsilon(x)} \frac{\partial u_\epsilon}{\partial y} \frac{\partial \varphi}{\partial y} dy \\
&= - \int_D \frac{\partial u_\epsilon}{\partial y} \frac{\partial \varphi}{\partial y} dx dy;
\end{align*}
$$

using again (36) we get:

$$
(43) \quad \left\{
\begin{aligned}
\lim_{n \to \infty} \int_D \varphi \frac{\partial^2 u_\epsilon}{\partial y^2} dx dy &= - \int_D \frac{\partial u_\epsilon}{\partial y} \frac{\partial \varphi}{\partial y} dx dy \\
&= \int_\Omega_+ \frac{\partial^2 u_\epsilon}{\partial y^2} dx dy \int_\Omega_+ \varphi \bar{f} dx dy.
\end{aligned}
\right.
$$

The difference of (42) and (43) we get

$$
(44) \quad \lim_{n \to \infty} \frac{1}{\epsilon} \int_D \varphi \frac{\partial^2 u_\epsilon}{\partial x^2} dx dy = \int_{\Omega_*} f_\varphi dx dy - \int_{\Omega_+} \bar{f}_\varphi dx dy .
$$

Let us suppose now that φ does not depend on x ; then

$$
\int_D \varphi \frac{\partial^2 u_\epsilon}{\partial x^2} dx dy = - \int_D \frac{\partial \varphi}{\partial x} \frac{\partial u_\epsilon}{\partial x} dx dy \\
+ \int_O \left[ \varphi \frac{\partial u_\epsilon}{\partial x} \right]_O^1 dy = 0,
$$

and (44) becomes

$$
O = \int_{\Omega_*} f \varphi \, dx \, dy - \int_{\Omega_+} \bar{f} \varphi \, dx \, dy ;
$$

since Ω + ⊂ Ω ∗, this implies

$$
(45) \quad \int_{\Omega_* \setminus \Omega_+} f \varphi \, dx \, dy + \int_{\Omega_+} (f - \bar{f}) \varphi \, dx \, dy = 0 .
$$
Using again that $\varphi$ does not depend on $x$, we see that the second integral vanishes; the function $f$ is larger than $\alpha$; we also may choose $\varphi$ larger than $\alpha$ for $y > \sqrt{\frac{2}{\alpha}} - b$ so that $\varphi > \alpha$ of $\Omega_+$; then (45) implies

$$ \alpha^2 \text{meas}(\Omega_* - \Omega_+) = 0; $$

this implies (jointly with $\Omega_+ \subset \Omega_*$)

$$ \gamma^* = \gamma. $$

<figure><img src="image_13.png" /></figure>
REFERENCES
==============

C. BAIOCCHI

[1] Problèmes à frontière libre et inéquations variationnelles, C.R. Académie Sciences Paris, t.283, Série A, p. 29-32 (12 Juillet 1976).

C. BAIOCCHI, V. COMINCIOLI, E. MAGENES, G. POZZI,

[1] Free Boundary Problems in the Theory of Fluid Flows through porous media : Existence and Uniqueness Theorems, Ann. di Mat. Pura ed Appl., IV, 97 (1973) p. 1-82

M. BERGER

[1] Géométrie. Cédic, Fernand Nathan, Paris (1977)

A. BEURLING

[1] Free Boundary problems for the Laplace equation. Institute for advanced Study, Seminar, Princeton, N.J., I (1957) p. 248-263

H. BREZIS, D. KINDERLEHRER,

[1] The smoothness of solutions to non linear variational inequalities, Indiana U. Math. J., 23, 9 (1974), p. 831-844

L.A. CAFFARELLI

[1] The regularity of Elliptic and Parabolic Free Boundaries, Bull. A.M.S., 82, N°4, July (1976), p. 616-618

C.W. CRYER

[1] Trial Methods for Free Boundary problems. Technical Summary Report. Mathematic Research Center, University of Wisconsin, Madison, Wisconsin (1977).

I.I. DANILJUK

[1] Existence theorem for a non linear Free Boundary Problem, U.M.J., 20, N°1 (1968) p. 2-29

[2] Study of a class of functional defined by integrals with variable domain, D.A.N. U.R.S.R. Série A, N°9 (1969) p. 783
[3] Sur une classe de fonctionnelles intégrales à domaine variable d'intégration, Actes du Congrès International Math., Nice, (1970), t. 2, p. 703-715

A.S. DEMIDOV

[1] The form of a steady plasma subject to the skin effect in a tokamak with non circular cross-section, Nuclear Fusion 15 (1975) 765-768

A. DERVIEUX

[1] Etude de la perturbation du Problème de l'Obstacle ; composition de la Frontière libre et application au contrôle optimal, LABORIA Report (to appear)

W. ECKHAUS, H.J. MOET

[1] Asymptotic solutions in Free Boundary Problems of Singularly perturbed Elliptic Variational Inequalities. University of Utrecht. December 1977 (To appear)

D. KINDERLEHRER, L. NIRENBERG

[1] Regularity in Free Boundary Problems, Annali della Scuela Norm. Sup. Pisa, Ser. IV (1977), p. 373-391

D. KINDERLEHRER, G. STAMPACCHIA

[1] A Free Boundary Value Problem in Potential Theory, Annales Inst. Fourier, Grenoble, 25, N°3 et 4 (1975), p.323-344

H. LEWY, G. STAMPACCHIA

[1] On the regularity of the solution of a variational inequality, Comm. Pure and Applied math., 22 (1969) p:153-188

J.L. LIONS

[1] Quelques méthodes de résolution des problèmes aux limites non-linéaires, Dunod, Paris 1969.

J.L. LIONS, G. STAMPACCHIA

[1] Variational Inequalities, Comm. Pure and Applied Math., 20 (1967), P.493-519
O. PIRONNEAU, C. SAGUEZ

[1] Comportement asymptotique des solutions d'équations aux dérivées partielles par rapport aux domaines, Comptes Rendus Académie Sciences Paris, **283** (1977)

[2] Asymptotic behaviour with respect to the domain of partial differential equations, LABORIA Report N°218 (1977)

D.G. SCHAEFFER

[1] A stability Theorem for the obstacle Problem, Advances in Math., **16** (1975), p.1-14

[2] Some Examples of Singularities in a Free Boundary, Annali della Scuola Norm. Sup. Pisa, **4**, N°1 (1977), p.133-144

[3] One-sided Estimates for the Curvature of the Free Boundary in the Obstacle Problem, Advances in Mathematics, **24**, (1977) p.78-98

D.E. TEPPER

[1] Free Boundary Problem, SIAM J. Math. Anal., **5** (1974) P. 841-846

[2] On a Free Boundary problem, the obstacle Case, SIAM J. Math. Anal., **6**, N°3 (1975), p. 503-505

D.E. TEPPER, G. WILDENBERG

[1] Some Infinite Free Boundary Problem (to appear).