# Hadamard's variational formula for a mixed problem and an application to a problem related to a signorini-like variational inequality

Bernadette Palmerio, Alain Dervieux

► To cite this version:

Bernadette Palmerio, Alain Dervieux. Hadamard's variational formula for a mixed problem and an application to a problem related to a signorini-like variational inequality. [Research Report] IRIA-RR-339, IRIA. 1979, pp.35. hal-04716605

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HADAMARD'S
VARIATIONAL FORMULA
FOR A MIXED PROBLEM AND
AN APPLICATION TO A PROBLEM
RELATED TO A SIGNORINI-LIKE
VARIATIONAL INEQUALITY

Bernadette PALMERIO
Alain DERVIEUX

Rapport de Recherche N° 339

Janvier 1979

<figure><img src="image_3.png" /></figure>
HADAMARD'S VARIATIONAL FORMULA FOR A MIXED PROBLEM AND
AN APPLICATION TO A PROBLEM RELATED TO A SIGNORINI-LIKE
VARIATIONAL INEQUALITY

Bernadette Palmerio*, Alain Dervieux**

Résumé :

Nous donnons une formule de variation par rapport au domaine géométrique (formule de Hadamard) de la solution d'un problème mêlé ; nous mettons en évidence les singularités qu'elle contient et nous en tirons une méthode d'approche du problème du contrôle optimal d'un problème en relation avec une inéquation variationnelle elliptique du type Signorini.

Abstract :

We give a variational formula for the solution of a Mixed Boundary Value Problem when the geo-
metrical domain is perturbed (Hadamard's Variational Formula) ; we show that this formula
contains singularities ; the result is used to construct a new approach to the optimal control pro-
blem for a problem related to a Signorini-like Elliptic Variational Inequality.

<footnote>* Université de Nice, I.M.S.P. - Parc Valrose - 06034 Nice Cédex (France)</footnote>

<footnote>**IRIA/LABORIA - B.P. 105 - 78150 Le Chesnay (France)</footnote>
The purpose of this work is the study of the *variation of the solution of an Elliptic Mixed Boundary Value Problem* when the domain is perturbed. *This first type of problem* would, at first glance, seem to be particularly far from concrete considerations; let us explain the physical contexts in which it can occur :

In *flow problems* with jets, wakes or cavities, it is useful to introduce mixed problems, with Dirichlet conditions along the walls and Neumann condition along the free boundary (cf. P.R. GARABEDIAN [1]); it is then necessary to study the behaviour of the corresponding solution when geome-trical data are perturbed.

In *inverse problems* : for example when the front part of a profile is optimized (cf. O. PIRONNEAU [1]).

The results which we got have made it possible to study a *second type of problem* concerning the perturbation of a problem related to a Signorini-like Elliptic Variational Inequality.

Our contribution to these two problems is the following :

For the *first kind*, we establish by means of a new method a *variational formula* of Hadamard-type; till now, as far as we know, Hadamard's formulae are computed either by differentiating integrals (J. HADAMARD [1]) or by a Stokes formula (P.R. GARABEDIAN [2]); for the two cases, the state function must be regular enough (in particular a Green formula is used); in this work we deal with a "model problem" which does not verify this condition; morevoer formal identification leads almost surely to false results; the methods proposed here gives a rigourous alternative computation.

For the *second kind* of problem, we establish *differentiability results* which bring information about the behaviour of a geometrical unknown similar to a free boundary.

The plan of the paper is the following :

1. Hadamard's formula

1.1. Geometrical parametrization

1.2. Formulation of the problem
1.3. An extended definition of trace

1.4. Main results

1.5. Proof of Theorem 1

2. Moving the Neumann Dirichlet point

2.1. Formulation of the problem

2.2. Regularity for a mixed problem

2.3. Main results

2.4. Proofs

3. Perturbation of a problem related to a Signorini-like inequality

3.1. Formulation of Problems $\rho_1$ and $\rho_2$

3.2. Singular points for $\rho_2$

3.3. Perturbation results for regular points of $\rho_2$

3.4. Relations between $\rho_1$ and $\rho_2$: some remarks and hints,
# 1. - HADAMARD'S FORMULA

## 1.1. Geometrical parametrization

Let B be an open ball of $\mathbb{R}^N$, $\mathcal{T}$ the set of one-to-one mappings $T$ from $\bar{B}$ to $\bar{B}$ such that $T$ and $T^{-1}$ are twice continuously differentiable, and that $T(x)=x$ for every point $x$ of $\partial B$; let $\mathcal{T}_{ad}$ be a subset of $\mathcal{T}$, and $\Omega$ be an open domain of $\mathbb{R}^n$ such that

$\bar{\Omega} \subset B$; $\partial\Omega$ is a $C^2$-manifold with dimension $N-1$ and $\Omega$ is locally on one side of its boundary.

Let $\Gamma$ be a measurable subset of $\partial\Omega$. The set $\mathcal{U}_{ad}$ of admissible geometrical variables $(\Omega_T, \Gamma_T)$ is defined from the pair $(\Omega, \Gamma)$ as follows

$$ \mathcal{U}_{ad} = \{ (\Omega_T, \Gamma_T) ; \Omega_T = T(\Omega), \Gamma_T = T(\Gamma) ; T \in \mathcal{T}_{ad} \} $$

## 1.2. Formulation of the problem

We are interested in solving the following problem

$$ (\mathcal{P}) \quad \left\{ \begin{array}{l} \text{to minimize in } \mathcal{T}_{ad} \text{ the functional} \\ j(T) = \frac{1}{2} \| z(T) - z_d \|^2_{L^2(\Omega_o)} \end{array} \right. $$

where $z(T)$ is the solution of the following mixed Neumann-Dirichlet boundary value problem

$$ (1.1) \quad \left\{ \begin{array}{l} z(T) \in H_o^1(\Omega_T, \Gamma_T) = \{ \phi \in H^1(\Omega_T) ; \phi|_{\Gamma_T} = 0 \}, \\ \forall \phi \in H_o^1(\Omega_T, \Gamma_T), \\ \displaystyle\int_{\Omega_T} \{ \langle \operatorname{grad} z(T), \operatorname{grad} \phi \rangle + z(T)\phi - f\phi \} \, dv = 0 \end{array} \right. $$

with the following data

$$ \left\{ \begin{array}{l} \Omega_o \text{ is a fixed open set such that (assumption about } \mathcal{T}_{ad}) \\ \forall T \in \mathcal{T}_{ad}, \bar{\Omega}_o \subset \Omega_T \\ z_d \in L^2(\Omega_o) \\ f \in H^1(B). \end{array} \right. $$
In this work, we shall be interested in the classical way to obtain necessary optimality conditions for this minimization problem : it is enough to compute the derivative of j.

### 1.3. An extended definition of trace

**Definition 1** : We shall say that a matrix function $\mathcal{A} = (a_{ij})_{1 \le i,j \le N}$ belonging to $[L^1(\{x_N < 0\})]^N$ admits a normal trace $\sigma e_N|_{x_N=0}$ if, for any $V$ belonging to $[\mathcal{B}(\{x_N=0\})]^N$ ($1$), for any lift $\tilde{V}$ of $V$ in $[C^\infty(\{x_N \le 0\})]^N$ the following limit exists and is independent of the choice of the lift $\tilde{V}$ :

$$ \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{-\epsilon < x_N < 0} dx_N \int_{R^{N-1}} \langle \sigma e_N, \tilde{V} \rangle dx_1 \dots dx_{N-1} ; $$

this limit, if it exists, will be denoted by $(\sigma e_N, V)_*$.

This product is an extension of a $(N-1)$-dimensional surface integral. ■

**Proposition 1** :

(i) The above definition generalizes the classical notion of trace for a continuous matrix function.

(ii) The above normal trace, if it exists, belongs to the **algebrical** dual of $[\mathcal{B}(\{x_N=0\})]^N$.

**Proof** : (i) is evident. Because it is possible to choose $\tilde{V}$ independent of $x_N$, the linearity is immediately derived and (ii) follows. ■

In order to extend that definition to a regular domain of $R^N$, we shall prove the following result :

**Proposition 2** : For a given $C^2$ regular, bounded domain $\Omega$ there exist

$\alpha > 0, \beta > 0,$

$\gamma(\partial\Omega)$ a neighbourhood of $\partial\Omega$
and $\gamma$ a $C^1$-diffeomorphism ($\gamma$ and $\gamma^{-1}$ belong to $C^1$) which transforms $\mathcal{V}(\partial\Omega)$ in $\partial\Omega \times ]-\alpha, +\alpha[$. ■

**Sketch of the proof** : Let us consider, for a point $P$ of $\partial\Omega$, the outward normal vector $n_p$ from $P$ and the segment $S_p$ defined by

$$ (1.2) \qquad S_p = \{ M \in \mathbb{R}^N ; \vec{PM} = \lambda n_p , \lambda \in [-\alpha, +\alpha[ \}. $$

The domain $\Omega$ is $C^2$-regular, thus, for an $\alpha$ small enough, all the segments $S_p$ do not intersect one another ; then it is easy to see that

$$ (1.3) \qquad \left\{ \begin{array}{l} \gamma^{-1} : \partial\Omega \times ]-\alpha, +\alpha[ \to \bigcup_{P \in \partial\Omega} S_p = \mathcal{V}(\partial\Omega) \\[2ex] \gamma^{-1}(P, \lambda) = M \text{ defined by (1.2)} \end{array} \right. $$

defines a $C^1$-diffeomorphism. ■

**Definition 2.** - Under the assumptions of Proposition 2, we shall say that a matrix function $A = (a_{ij})_{1 \le i,j \le N}$ belonging to $[L^1(\Omega)]^{N^2}$ admits a normal trace $\sigma \cdot n|_{\partial\Omega}$ if for any $V$ of $[\mathcal{L}(\partial\Omega)]^N$ and for any lift $\tilde{V}$ of $V$ in $[C^\infty(\bar{\Omega})]^N$ the following limit exists and is independent of the choice of the lift $\tilde{V}$.

$$ (\sigma n, V)_* = \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{\partial\Omega} d\sigma \int_{-\epsilon}^{\epsilon} dx_N \left[ \frac{\langle \sigma^T \gamma' e_N, V \rangle}{|\det \gamma'|} \right] \circ \gamma^{-1} . \quad \blacksquare $$

**Proposition 3** :

(i) The above definition gives an extension of the classical trace of a continuous matrix function.

(ii) The above normal trace, if it exists, belongs to the *algebraic* dual of $[\mathcal{L}(\partial\Omega)]^N$. ■

**Proof** : The part (ii) can be showed as hinted in Proposition 1.

(i) Let us suppose that $A$ belongs to $[C^0(\bar{\Omega})]^N$; let $I$ be the limit

$$ I = \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{\partial \Omega} d\sigma \int_{-\epsilon}^{\epsilon} dx_N \left[ \frac{\langle \sigma^T \gamma' e_N, V \rangle}{|\det \gamma'|} \right] \circ \gamma^{-1} $$
thus

$$
I = \int_{\partial \Omega} \left[ \frac{\langle \tau^{\tau_{\gamma'}} e_{N}, V \rangle}{|\det \gamma'|} \right] \circ \gamma^{-1} d\sigma ;
$$

using $\gamma$ to do a <u>substitution in the surface integral</u>, we get

$$
I = \int_{\partial\Omega} \frac{\langle \tau^{\tau_{\gamma'}} e_{N}, V \rangle}{|\det \gamma'|} |\det \gamma'|' \| [\tau^{\tau_{\gamma'}}]^{-1} n \|_{\mathbb{R}^N} d\sigma .
$$

Now, we have

$$
t_{\gamma'} e_N = \frac{1}{\| [t_{\gamma'}]^{-1} n \|_{\mathbb{R}^N}} n
$$

therefore

$$
I = \int_{\partial \Omega} \langle \partial_n, V \rangle \, d\sigma . \blacksquare
$$

1.4. Main results.

**Theorem 1** : Under the assumptions of Sections 1.1, 1.2,

(i) the mapping $T \to z(T)|_{\Omega_o}$ is Fréchet differentiable on $\mathcal{C}_{ad}$ w.r.t. the
norms $[W^1, \infty(B)]^N \to H^1(\Omega_o)$ and its derivative is the restriction to $\Omega_o$
of the unique solution $\frac{\partial z}{\partial T}(T)$ of the following system

for $\theta \in [W^1, \infty(B)]^N$ such that $\theta|_{\partial B} = 0$,

$$
\frac{\partial z}{\partial T}(T) \cdot \theta \in L^2(\Omega_T),
$$

for $\phi \in L^2(\Omega_T)$, with $\overline{\text{supp}} \phi \subset \Omega$, let $p_\phi$ the (unique) solution of

$$
(1.4) \quad \left\{
\begin{array}{ll}
\displaystyle P_{\phi} \in H_{o}^{1}(\Omega_{T}, \Gamma_{T}) \\
\displaystyle \int_{\Omega_{T}} \{\langle \text{grad } p_{\phi}, \text{grad } \psi \rangle + p_{\phi}\psi - \phi\psi\} \, dv = 0 & \text{for every } \psi \in H_{o}^{1}(\Omega_{T}, \Gamma_{T})
\end{array}
\right.
$$

then the matrix

$$
(1.5) \quad \mathcal{A} = \left( \frac{\partial p_{\phi}}{\partial x_j} \frac{\partial z(T)}{\partial x_i} + \frac{\partial p_{\phi}}{\partial x_i} \frac{\partial z(T)}{\partial x_j} - \delta_{ij} [\langle \text{grad } z(T), \text{grad } p_{\phi} \rangle + z p_{\phi}^{-fp_{\phi}}] \right)_{1 \le i,j \le N}
$$

admits a normal trace in the sense of Definition 2; this trace belongs to
$[W^1, \infty(\partial \Omega_T)^N]'$ and we have
$$
(1.6) \quad \int_{\Omega_T} \phi \frac{\partial z}{\partial T}(T) \cdot \theta \, dv = (\mathcal{A}n, \theta_0 T^{-1})_*
$$

(ii) the functional $j$ is Fréchet differentiable on $\mathcal{C}_{\text{ad}}$ with respect to the norm $[W^1, \infty(B)]^N$ and its derivative is given by

$$
(1.7) \quad \left\{
\begin{array}{@{}l@{}}
  \forall \theta \in [W^1, \infty(B)]^N \text{ such that } \theta|_{\partial B} = 0, \\
  \frac{\partial j}{\partial T}(T) \cdot \theta = (\mathcal{A}n, \theta_0 T^{-1})_*
\end{array}
\right.
$$

where $\mathcal{A}$ is given by (1.4) (1.5), putting

$$
(1.8) \qquad \phi = \chi_{\Omega_o}[z(T) - z_d]
$$

$(\chi_{\Omega_o}$ is the characteristic function of $\Omega_o$). $\blacksquare$

**Remark 1** : The statement of the system (1.4) (1.5) (1.6) whose $\frac{\partial z}{\partial T}(T) \cdot \theta$ is the solution could be compared to "very weak" $L^2$-posed systems with elliptic second order operators (see J.L. LIONS, E. MAGENES [1] or for a simple explanation J.L. LIONS [1] Chapter 2, paragraph 4.2) ; but we do not know how to directly study this system for the following reasons :

- when $\phi$ in a $L^2(\Omega)$-or even smooth- function, the solution $p_\phi$ of (1.4) does not generally belong to $H^2(\Omega)$ so that the classical transposition argument does not work.

- To say that the right-hand side of (1.6) is definite, is equivalent to establishing a non standard regularity result : the only proof which we know is the most important part of the next section and uses explicitly the domain dependence techniques. ■

**Remark 2** : Choosing $p_\phi$ with a compact support in $\Omega_T$, (this choice is possible)
we obtain

$$
\int_{\Omega_T} \left[ \frac{\partial z}{\partial T}(T) \cdot \theta \right] (\Delta - 1) p_{\phi} \, dv = 0
$$

so that

$$
(\Delta - 1) \left[ \frac{\partial z}{\partial T}(T) \cdot \theta \right] = 0 \text{ in } \mathcal{B}'(\Omega_T) ;
$$

the singularities of $\frac{\partial z}{\partial T}(T) \cdot \theta$ lie only on the boundary. ■
**Remark 3** : In the particular cases

$$ \Gamma_T = \emptyset, \Gamma_T = \partial\Omega_T, $$

using the interpretations of the corresponding problems (Neumann problem, Dirichlet problem), we obtain the classical results for regular systems, i.e.,

$$ \frac{\partial j}{\partial T}(T) \cdot \theta = \int_{\partial \Omega_T} \langle G_i, \theta \cdot T^{-1} \rangle d\sigma $$

with

$$ G_1 = -\{\langle \mathrm{grad} \ z(T), \mathrm{grad} \ p \rangle + z(T)p - fp \} \ n $$

for the Neumann problem (respectively

$$ G_2 = \langle \mathrm{grad} \ z(T), \mathrm{grad} \ p \rangle \ n $$

for the Dirichlet problem), where $G_i$ belongs to $L^1(\partial\Omega_T)$, since $z(T)$ belongs to $H^2(\Omega_T)$ in those cases.

In the general case, such a transform is still available on every open subset of $\partial\Omega_T - \Gamma_T$ (resp. $\Gamma_T$) but not on the whole set $\partial\Omega_T$: we shall specify singularities which arise in a more particular context in Section 2. ■

### 1.5. *Proof of Theorem 1*

Let us prove first the part (ii).

**Step 1** : We apply a classical process in domain dependence studies (see P.R. GARABEDIAN [2], and also F. MURAT, J. SIMON [1,2], B. PALMERIO, A. DERVIEUX [1], O. PIRONNEAU [2]) which consists in considering a new mixed problem in a fixed domain by transporting the first one in the domain $\Omega$ by means of the diffeomorphism $T$; the so-called "transported state" $\tilde{z}(T) = z(T) \cdot T$ is the solution of the following system (obtained by transporting (1.1))

$$ (1.9) \quad \left\{ \begin{array}{l} \tilde{z}(T) \in H_0^1(\Omega, \Gamma), \quad \forall \psi \in H_0^1(\Omega, \Gamma) \\ \displaystyle\int_{\Omega} \{\langle [T_T']^{-1} \operatorname{grad} \tilde{z}(T), [T_{T'}']^{-1} \operatorname{grad} \psi \rangle + \tilde{z}(T)\psi - (f \cdot T)\psi\} |\det T'| \, dv = 0, \end{array} \right. $$

and the functional is given by

$$ (1.10) \quad j(T) = \frac{1}{2} \int_{\Omega_0} [\tilde{z}(T) \cdot T^{-1} - z_d]^2 dv. $$
**Lemma 1** : The diffeomorphisms of the sets $\mathcal{T}$ and $\mathcal{T}_{ad}$, instead of being supposed to belong to $[C^2(\bar{B})]^N$, are only supposed to belong to $[W^{1,\infty}(B)]^N$ ($^2$).

Then the mappings $T \to \tilde{z}(T)$ (defined by (1.9)) and $T \to j(T)$ are Fréchet-differentiable from $\mathcal{T}_{ad}$ equipped with the $W^{1,\infty}(B)^N$ norm, respectively in $H^1(\Omega)$ and $\mathbb{R}$; the derivative of $j$ is given by

$$ (1.11) \quad \left\{ \begin{array}{@{}l@{}} \displaystyle \frac{\partial j}{\partial T}(T) \cdot \theta = \int_{\Omega_T} \langle [V']^{\cdot T} V' \rangle \operatorname{grad} z(T) \, , \operatorname{grad} p > \, dv \\[2ex] \displaystyle - \int_{\Omega_T} \{ \langle \operatorname{grad} z(T), \operatorname{grad} p \rangle + zp-fp \} \operatorname{div} V \, dv \\[2ex] \displaystyle + \int_{\Omega_T} \{ -p \langle \operatorname{grad} f, V \rangle + \chi_{\Omega_o} (z(T)-z_d) \langle \operatorname{grad} z(T), V \rangle \} \, dv \end{array} \right. $$

with $V = \theta_o T^{-1}$ and where $p$ is the solution of (1.4) with
$\Phi = \chi_{\Omega_o} [z(T)-z_d]$. ■

**Proof** : Considering (1.9), we can see that we are led now to a perturbation problem in which only the coefficients and right-hand side are perturbed (instead of the domain) : following G. CHAVENT [1], we introduce a mapping $\Phi$ representing the (new) state equation :

$$ (1.12) \quad \left\{ \begin{array}{@{}l@{}} \Phi : \mathcal{T}_{ad} \times H_o^1(\Omega, \Gamma) \to [H_o^1(\Omega, \Gamma)]' \\[2ex] \qquad (T, \alpha) \to \Phi(T, \alpha) \\[2ex] \qquad \begin{array}{@{}l@{}} \text{such that } \forall \psi \in H_o^1(\Omega, \Gamma) \\[2ex] (\Phi(T, \alpha), \psi) = [H_o^1(\Omega, \Gamma)]' \times H_o^1(\Omega, \Gamma) \\[2ex] \displaystyle \int_{\Omega} \{ \langle [\cdot T']^{-1} \operatorname{grad} \alpha, [\cdot T']^{-1} \operatorname{grad} \psi \rangle + \alpha \psi - (f_o T) \psi \} |\det T'| \, dv. \end{array} \end{array} \right. $$

It is evident that

$$ \alpha = \tilde{z}(T) \iff \Phi(T, \alpha) = 0; $$
now it can be easily shown that $\Phi$ is continuously differentiable for the topologies : $[W^1, \infty(\mathbf{B})]^N \times H_0^1(\Omega, \Gamma) \to [H_0^1(\Omega, \Gamma)]'$, and that, for $T$ belonging to $\mathcal{C}_{ad}'$,

$$ \frac{\partial \Phi}{\partial \alpha} (T, \alpha) \in \text{Isom} \{ H_0^1(\Omega, \Gamma) \to [H_0^1(\Omega, \Gamma)]' \} ; $$

then, applying the Implicit Function Theorem to $\Phi$, we derive the differentiability of $T \to \tilde{z}(T)$, and therefore (using (1.10)) of $T \to j(T)$.

Let us compute the derivative of $j :$ by the substitution of $\Omega, \theta$ by $T(\Omega), \theta \cdot T^{-1}$, we see that it is enough to restrict ourselves to the case $T \equiv \text{Identity}$:

In that case, by differentiating equality (1.9), we derive the equation of the derivative $\frac{\partial \tilde{z}}{\partial T}$ of $\tilde{z}$:

$$ (1.13) \quad \left\{ \begin{array}{ll} \displaystyle \frac{\partial \tilde{z}}{\partial T}(I) \cdot \theta \in H_0^1(\Omega, \Gamma) & , \forall \psi \in H_0^1(\Omega, \Gamma) \\[2ex] \displaystyle \int_{\Omega} [<\text{grad } (\frac{\partial \tilde{z}}{\partial T}(I) \cdot \theta), \text{grad } \psi> + \psi \frac{\partial \tilde{z}}{\partial T}(I) \cdot \theta] \, \text{d}v = \\[2ex] \displaystyle \int_{\Omega} \{<(\theta'+t^\theta)\text{grad } z(I), \text{grad } \psi> \\[2ex] \displaystyle \quad - [<\text{grad } z(I), \text{grad } \psi> + z(I)\psi - f\psi]\, \text{div } \theta - \psi<\text{grad } f, \theta>\} \, \text{d}v. \end{array} \right. $$

The differentiation of (1.10) gives

$$ \frac{\partial j}{\partial T}(I) = \int_{\Omega_0} (\tilde{z}(I) - z_d) \left[ \frac{\partial \tilde{z}}{\partial T}(I) + \langle \text{grad } \tilde{z}(I), \theta \rangle \right] \text{d}v. $$

From this, from the adjoint state equation ((1.4) with $\phi = \chi_{\Omega_0} [z(I)-z_d]$) and from (1.13) with $\psi = p$ we get (1.11) with $V = \theta$, and $\Omega_T = \Omega$.

Returning to the general case, we have only to substitute $\Omega, \theta$ by $\Omega_T, \theta \cdot T^{-1}$; then we get (1.11). ■

**Step 2** : We shall now use a family $(\theta_\epsilon)$ of increments which are equivalent to a given $\theta$ in the following way $(\epsilon, t$ are positive, sufficiently small real numbers)

$$ (1.14) \quad \begin{cases} [T+t\theta_\epsilon] (\Omega) = [T+t\theta] (\Omega) \\ [T+t\theta_\epsilon] (\Gamma) = [T+t\theta] (\Gamma). \end{cases} $$
For this purpose, we introduce the mapping ℓ_ε

$$ (1.15) \qquad \left\{ \begin{array}{ll} \ell_{\epsilon} : \mathbb{R}^N \to \mathbb{R} \\ \ell_{\epsilon}(x_1, x_2, \dots, x_N) = & \begin{cases} 0 & \text{if } x_N \le -\epsilon \\ \frac{1}{\epsilon}(\epsilon+x_N) & \text{if } -\epsilon \le x_N \le 0 \\ \frac{1}{\epsilon}(\epsilon-x_N) & \text{if } 0 \le x_N \le \epsilon \\ 0 & \text{if } x_N \ge \epsilon, \end{cases} \\ . & \\ & \qquad 0 \text{ if } x_N \ge \epsilon, \end{array} \right. $$

and we consider

$$ (1.16) \qquad \left\{ \begin{array}{ll} \theta_{\epsilon} = V_{\epsilon} \circ T & \text{with} \\ V_{\epsilon} = & \begin{cases} (\ell_{\epsilon} \circ \gamma) \, V & \text{on } \gamma(\partial \Omega_{T}) \\ 0 & \text{elsewhere} \end{cases} \\ V = \theta_{\epsilon} \circ T^{-1} & ; \end{array} \right. $$

where $\gamma$ is the diffeomorphism related to $\Omega_T$ defined in Proposition 2;
the statement (1.14) holds by construction; thus follows trivially the

**Lemma 2** : For every $\epsilon$ positive, small enough, we have

$$ (1.17) \qquad j'(T) \cdot \theta = j'(T) \cdot \theta_{\epsilon} . \qquad \blacksquare $$

From the construction of $V_\epsilon$ and from (1.11), we obtain the differential
$j'(T) \cdot \theta$ as an integral on the set $\Delta_\epsilon$

$$ \Delta_{\epsilon} = \gamma^{-1}\{\partial\Omega_{T} \times ]-\epsilon, 0[\} . $$

Making $\epsilon$ tend to zero, we shall obtain as limit a new expression of this
differential as an integral along the boundary $\partial\Omega_T$ in the sense of Definitions
1, 2.

To compute this limit, we shall need the following lemma
**Lemma 3**

(i) If $u$ and $w$ are two functions of $H^1(\Omega_T)$, then

$$ \int_{\Delta_{\varepsilon}} \langle V'_{\varepsilon} \operatorname{grad} u, \operatorname{grad} w \rangle \, dv - \int_{\Delta_{\varepsilon}} \langle \operatorname{grad} u, \operatorname{grad} (l_{\varepsilon} \circ \gamma) \rangle \langle \operatorname{grad} w, V \rangle \, dv \to 0 \\ \text{when } \varepsilon \to 0. $$

(ii) If $a$ is a function of $L^1(\Omega_T)$ then

$$ \int_{\Delta_{\varepsilon}} a \, \operatorname{div} V_{\varepsilon} \, dv - \int_{\Delta_{\varepsilon}} a \, \langle \operatorname{grad} (l_{\varepsilon} \circ \gamma), V \rangle \, dv \to 0 \\ \text{when } \varepsilon \to 0. \blacksquare $$

**Proof** : This lemma is a direct consequence of Lebesgue's theorem and of (1.16). ■

Let us consider the expression (1.11), putting $V=V_{\varepsilon}$; the lemma 3 is used to transform the first and second integrals; the third integral tends to zero because it contains no derivative of $V_{\varepsilon}$; thus we can deduce the following lemma :

**Lemma 4.** - The following expression has a limit equal to $\frac{\partial j}{\partial T}(T) \cdot \theta$ when $\varepsilon$ tends to zero

$$ \left\{ \begin{aligned} I_{\varepsilon} &= \int_{\Delta_{\varepsilon}} \{ \langle \operatorname{grad} (l_{\varepsilon} \circ \gamma), \operatorname{grad} z \rangle - \langle \operatorname{grad} p, V \rangle \\ &\quad + \langle \operatorname{grad} (l_{\varepsilon} \circ \gamma), \operatorname{grad} p \rangle - \langle \operatorname{grad} z, V \rangle \\ &\quad - [\langle \operatorname{grad} z, \operatorname{grad} p \rangle + zp - fp] \langle \operatorname{grad} (l_{\varepsilon} \circ \gamma), V \rangle \} \, dv \\ &\quad - [\langle \operatorname{grad} z, \operatorname{grad} p \rangle + zp - fp] \langle \operatorname{grad} (l_{\varepsilon} \circ \gamma), V \rangle \end{aligned} \right. \\ \text{with } z = z(T), \quad p \text{ defined as in Lemma 1, and } V=\theta \circ T^{-1}. \blacksquare $$

Now a simple computation gives

$$ \operatorname{grad} (l_{\varepsilon} \circ \gamma) = \frac{1}{\varepsilon} t_{\gamma'} e_N $$

so that we have, using the notation (1.5) and the transport of $\Delta_{\varepsilon}$ on $\partial\Omega_T \times ]-\varepsilon, 0[$ by $\gamma^{-1}$,
$$
\begin{align*}
I_{\varepsilon} &= \frac{1}{\varepsilon} \int_{\Delta_{\varepsilon}} \langle V, \nabla^{\mathrm{t}\gamma'} e_{N} \rangle \, dv \\
&= \frac{1}{\varepsilon} \int_{\partial\Omega_{T}} d\sigma \int_{-\varepsilon}^{0} dx_{N} \left[ \frac{\langle \nabla^{\mathrm{t}\gamma'} e_{N}, V \rangle}{|\det \gamma'|} \right] \circ \gamma^{-1}
\end{align*}
$$

Now we recognize the *trace* of $\nabla$ in the sense of Definition 2.

From Lemma 4, the limit of $I_ε$ when $ε$ tends to zero exists for every increment $tV$ small enough in $[W^1, \infty(\mathbf{B})]^N$. Let us see that this limit does not depend on the lift : since when $\theta_1 = \theta_2$ on $\partial\Omega$ (or $V_1=V_2$ on $\partial\Omega_T$) we have $(T+t\theta_1)(\Omega) = (T+t\theta_2)(\Omega)$ and $(T+t\theta_1)(\Gamma) = T+t\theta_2(\Gamma)$, we conclude, as in Lemma 2, that the limit of $I_ε$ for $V=V_1$ is identical to the limit for $V=V_2$.

Then the conditions of Definition 2 are fulfilled. It is also interesting to notice that, from (1.11) this trace does not even depend on the choice of $\gamma$. Let us now show that this trace belongs to $[W^1, \infty(\partial\Omega_T)]^N$: from (1.11) it belongs to $[W^1, \infty(\mathbf{B})]^N$; but it does not depend on the lift $\tilde{V}$ of $V$; thus, using the canonical lift defined by choosing

$$ \tilde{V}_{\circ \gamma^{-1}} \text{ independent of } x_N, $$

this implies that the above trace is an element of $[W^1, \infty(\partial\Omega_T)]^N$.

**Proof of (i) : First step.**

We need the following lemma :

**Lemma 5** : For $\epsilon$ small enough we have

$$ V_{\epsilon}|_{\Omega_{o}} = 0. $$

**Proof** : the thickness of $\Delta_g$ tends to zero, and $\bar{\Omega}_o \subset \Omega_T$. ■

Replacing every small enough increment $\theta$ by $\theta_\epsilon$ defined by (1.16), we have, from (1.14)

$$ z(T+\theta) = z(T+\theta_{\epsilon}) = \tilde{z}(T+\theta_{\epsilon}) \circ (T+\theta_{\epsilon})^{-1} ; $$

using Lemma 5, we get, since $T+\theta_\epsilon = (I+V_\epsilon) \circ T$

$$ z(T+\theta)|_{\Omega_o} = \tilde{z}(T+\theta_{\epsilon}) \circ T^{-1}|_{\Omega_o} $$
thus the differentiability follows from Lemma 1 (with respect to $\theta_{\epsilon}$, thus to $\theta$).

When $\bar{\Omega}_o$ describes the family of compact subsets of $\Omega_T$, the differential $\frac{\partial}{\partial T}(z(T)|_{\Omega}) \cdot \theta$ defines an unique function of $L_{\text{loc}}^2(\Omega_T)$ which we denote formally by $\frac{\partial z}{\partial T}(T) \cdot \theta$.

We shall show now that $\frac{\partial z}{\partial T}(T) \cdot \theta$ belongs to $L^2(\Omega_T)$:

Let $\rho_{\Omega}$ be a continuous linear extension operator from $H^1(\Omega)$ to $H^1(B)$:

$$ \forall \phi \in H^1(\Omega), \rho_{\Omega}\phi \in H^1(B) \text{ and } \rho_{\Omega}\phi|_{\Omega} = \phi. $$

Such an operator exists: see for instance J.L. LIONS [3].

From Lemma 1, $T \to \rho_{\Omega} \tilde{z}(T)$ is Fréchet differentiable in $H^1(B)$ so that $^{(3)}$ $T \to [\rho_{\Omega} \tilde{z}(T)] \cdot T^{-1}$ is Fréchet differentiable in $L^2(B)$; now

$$ [\rho_{\Omega} \tilde{z}(T)] \cdot T^{-1} = z(T) \text{ on } \Omega_T $$

thus

$$ \frac{\partial}{\partial T} \{[\rho_{\Omega} \tilde{z}(T)] \cdot T^{-1}\} \cdot \theta|_{\Omega_o} = \frac{\partial}{\partial T} [z(T)|_{\Omega_o}] \cdot \theta $$

for every $\Omega_o$ such that $\bar{\Omega}_o \subset \Omega_T$, so that

$$ \frac{\partial}{\partial T} \{[\rho_{\Omega} \tilde{z}(T)] \cdot T^{-1}\} \cdot \theta|_{\Omega_T} = \frac{\partial z}{\partial T}(T) \cdot \theta \in L^2(\Omega_T). $$

**Second step:** To compute $\frac{\partial z}{\partial T}(T)$, let us introduce an arbitrary function $\phi$ of $L^2(\Omega_T)$ such that $\text{supp } \phi \subset \bar{\Omega}_o$, and the functional

$$ j_{\phi}(T) = \int_{\Omega_o} \phi z(T) dv. $$

To know $\frac{\partial j_{\phi}}{\partial T}(T) \cdot \theta = \int_{\Omega_o} \phi \frac{\partial z}{\partial T}(T) \cdot \theta \, dv$ for every $\theta$, every $\phi$, every $\Omega_o$ is equivalent to know $\frac{\partial j_{\phi}}{\partial T}(T)$. Now the computation of $\frac{\partial j_{\phi}}{\partial T}(T)$ is identical to the proof of (i): the adjoint state is the solution $p_{\phi}$ of (1.4) and we get, as above, the existence of the expression (1.6).

■
## 2. - MOVING THE NEUMANN-DIRICHLET POINT

The results of this section are, with respect to some considerations, particular cases of those of Paragraph 1; they constitute a way to exhibit the singularities contained in the Hadamard's formula obtained above. In this new context, the mixed boundary value problem is posed in a fixed domain $\Omega$ in the plane $\mathbb{R}^2$, and the control variable consists of the position on $\partial\Omega$ of a point where the boundary condition is shifted (i.e. Neumann on one side, Dirichlet on the other side).

### 2.1. Formulation of the problem

$\Omega$ is a $C^2$-regular not necessarily bounded domain of $\mathbb{R}^2$ such that $\partial\Omega$ contains a straight segment AB; we choose a Cartesian coordinates system Oxy as follows (Fig. 1): O is the middle of AB, Ox is parallel to AB. The boundary condition if shifted in an inner point P of segment AB; this point is supposed to vary in the sequel ; let h be the abscissa of P.

We use as well the following data :

$$ egin{cases} 
\Omega_o \text{ is an open subset of } \Omega \text{ with } \bar{\Omega}_o \subset \Omega, \\ 
z_d \in L^2(\Omega_o), \\ 
f \in H^1(\Omega), \\ 
\alpha = \frac{1}{2} \text{ length of AB.} 
\end{cases} $$

We study the problem

$$ (\hat{\rho}) \quad \begin{cases} 
\text{Minimize in } [-\alpha, \alpha] \text{ the functional} \\ 
\hat{j}(h) = \frac{1}{2} \|z(h^t) - z_d\|_{L^2(\Omega_o)}^2 
\end{cases} $$

where $z(h)$ is the solution of the mixed problem (1.1) in which the subset $\Gamma_T = \Gamma_h$ is supposed to depend on h as explained above.

Just as in Section 1, we are interested in differentiability results and computation of the derivative of the functional to be minimized.
<figure><img src="image_4.png" /><figcaption>Figure 1</figcaption></figure>

## 2.2. Regularity for a mixed problem

The state function $z = z(h)$ admits generally singularities in the neighbourhood of the point P where the boundary condition is shifted. More precisely, in a neighbourhood of P, we have the following expansion (see P. GRISVARD [1,2], M. MERIGOT [1]), the polar coordinates $(r, \theta)$ being defined w.r.t. P

$$ (2.1) \qquad \left\{ \begin{array}{l} z(h+r \cos \theta, r \sin \theta) = z_{\frac{1}{2}} r^{\frac{1}{2}} \sin \frac{\theta}{2} + R \\ \text{with } z_{\frac{1}{2}} \in \mathbb{R} \text{ and } R \in H^{2+\varepsilon}(\Omega \cap \mathcal{V}(P)), \ 0 \le \varepsilon < \frac{1}{2} \\ \text{for a neighbourhood } \mathcal{V}(P) \text{ of P.} \end{array} \right. $$

In the sequel, $\phi_{\frac{1}{2}}$ will denote the singular term coefficient of such an expansion for the solution $\phi$ of a mixed problem.
## 2.3. Main results.

**Theorem 2**: (i) The mappings $h \to z(h)$ and $h \to z(h)|_{\Omega}$ are differentiable respectively in $L^2(\Omega)$ and $H^1(\Omega_o)$; their derivatives$^o$ for $h=0$$^{(4)}$ can be defined by the following systems

$\forall \phi \in L^2(\Omega)$ such that $\text{supp } \phi$ is compact in $\Omega$, we denote by $p_\phi$ the solution of system (2.3)

$$ (2.3) \qquad \begin{cases} p_{\phi} \in H_o^1(\Omega, \Gamma_o) , \forall \psi \in H_o^1(\Omega, \Gamma_o) \\ \int_{\Omega} \{\langle \text{grad } p_{\phi}, \text{grad } \psi \rangle + p_{\phi} \psi - \phi \psi\} \, dx \, dy = 0 \end{cases} $$

then we have

$$ (2.4) \qquad \left\{ \begin{aligned} \int_{\Omega} \phi \frac{\partial z}{\partial h} \, dx \, dy &= \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{D_{\epsilon}} \frac{x}{\sqrt{x^2+y^2}} \left( \frac{\partial z}{\partial y} \frac{\partial p_{\phi}}{\partial y} - \frac{\partial z}{\partial x} \frac{\partial p_{\phi}}{\partial x} \right) \\ &\quad - \frac{y}{\sqrt{x^2+y^2}} \left( \frac{\partial z}{\partial y} \frac{\partial p_{\phi}}{\partial x} + \frac{\partial z}{\partial x} \frac{\partial p_{\phi}}{\partial y} \right) \end{aligned} \right\} \, dx \, dy $$

with $z=z(o)$ and $D_{\epsilon} = \{(x,y) \in \Omega, \sqrt{x^2+y^2} < \epsilon\}$; we also have

$$ (2.5) \qquad \int_{\Omega} \phi \frac{\partial z}{\partial h} \, dx \, dy = \frac{\pi}{4} z_{\frac{1}{2}} p_{\frac{1}{2}} $$

where $z_{\frac{1}{2}}$ and $p_{\frac{1}{2}}$ are the singular term coefficients of $z(h)$ for $h=0$ and $p_{\phi}$.

(ii) The functional $\hat{j}$ is differentiable w.r.t. $h$, and its derivative for $h=0$ can be written as the right hand sides of (2.4), (2.5) where we have

$$ (2.6) \qquad \phi = \chi_{\Omega_o} [z(h) - z_d] . \quad \blacksquare $$
In the next section we shall also use the following results :

**Proposition 4** : (i) The functional

$$ \bar{j}(h) = \int_{\Omega} g z(h) dx dy, $$

with g belonging to $L^2(\Omega)$, is differentiable w.r.t. h and its derivative for h=0 can be written as in Theorem 2 (ii) with $\phi=g$.

(ii) If we consider the functional

$$ (2.7) \qquad \tilde{j}(h) = \frac{1}{2} \int_{\Omega} \{ |\operatorname{grad} z(h)|^2 + z(h)^2 \} \, dx \, dy $$

then $\tilde{j}$ is differentiable w.r.t. h and

$$ (2.8) \qquad \tilde{j}(0) = \frac{\pi}{4} (z_{\frac{1}{2}}(0))^2 $$

(iii) $\hat{j}, \bar{j}$, and $\tilde{j}$ are $C^k$ if f belongs to $H^k(\Omega)$.

**Remark 4** : It is interesting to note (and easy to show) that the derivative $\frac{\partial z}{\partial h}(h)$ is a non identically zero singular solution in $L^2(\Omega)$ of

$$ \left\{ \begin{array}{l} (\Delta - 1) \frac{\partial z}{\partial h}(h) = 0 \text{ on } \Omega \\ \\ \frac{\partial z}{\partial h}(h) = 0 \text{ on } \Gamma_h \\ \\ \frac{\partial}{\partial n} \frac{\partial z}{\partial h}(h) = 0 \text{ on } \partial\Omega \setminus \Gamma_h \end{array} \right. $$

where $n$ is the outward normal unit vector. ■

**Remark 5** : A first consequence of Theorem 2 is that the derivative $\hat{j}'(h)$ will vanish if and only if one (at least) of the two state functions $z$ and $p$ is regular ; the stationary points of $\tilde{j}$ arise only when $z(h)$ is "regular enough", i.e. when $z_{1/2}$ vanishes; moreover, this functional is monotone, increasing as $h$ increases ; its minimum is attained when the state equation is a Neumann problem on the whole segment $[-\alpha, +\alpha]$ ; this functional will be useful in the sequel. ■
## 2.4. Proofs

**Proof of Theorem 2 (ii) :**

**Step 1.** Let $\epsilon$ be a real, positive, small enough number, and $T_\epsilon(h)$ the
diffeomorphism defined by

$$ T_{\epsilon}(h) : \bar{\Omega} \rightarrow \bar{\Omega} \\ \left( \begin{matrix} x \\ y \end{matrix} \right) \rightarrow \left\{ \begin{matrix} \left( x + \frac{\epsilon - \sqrt{x^2 + y^2}}{\epsilon} \right) h & \text{for } \sqrt{x^2 + y^2} \le \epsilon \\ y & \text{otherwise.} \end{matrix} \right. $$

Thus Problem $\mathcal{P}$ (cf. Paragraph 1) and Problem $\hat{\mathcal{P}}$ are connected by

$$ \hat{j}(h) = j(T_{\epsilon}(h)), \forall \epsilon. $$

Though the domain is no longer necessarily bounded, Lemma 1 still applies and from a composite differentiation, formula (1.11) and Lemma 2 give an $\epsilon$-indexed family of expressions for the derivative $\hat{j}'$ under the form of integrals on the disk $D_\epsilon$ (we denote $z(h)$ by $z$ and by $p$ the adjoint state)

$$ \hat{j}'(0) = \int_{D_{\epsilon}} \{ \langle [\theta'_{\epsilon} + t_{\epsilon} \theta'] \rangle \operatorname{grad} z, \operatorname{grad} p \rangle \\ - \langle \operatorname{grad} z, \operatorname{grad} p \rangle + zp-fp \rangle \operatorname{div} \theta_{\epsilon} \} \, dx \, dy \\ - \int_{D_{\epsilon}} \{ p \langle \operatorname{grad} f, \theta'_{\epsilon} \rangle - \chi_{\Omega_0} [z-z_d] \langle \operatorname{grad} z, \theta'_{\epsilon} \rangle \} \, dx \, dy $$

with

$$ \theta_{\epsilon} = \left( \frac{\epsilon - \sqrt{x^2 + y^2}}{\epsilon} \right) \text{ on } D_{\epsilon} = \{(x, y) | \sqrt{x^2 + y^2} < \epsilon\} $$

Now we make $\epsilon$ tend to zero : $\theta_{\epsilon}$ is bounded ; using Lebesgue's theorem, we see that the last integral vanishes as $\epsilon$ tends to zero. Replacing $\theta_{\epsilon}$ we have
$$
(2.9) \quad \left\{
\begin{aligned}
\hat{J}'(0) &= \lim_{\epsilon \to 0} \left\{ \int_{D_\epsilon} \frac{1}{\epsilon} \left\{ \frac{-2x}{\sqrt{x^2+y^2}} \frac{\partial z}{\partial x} \frac{\partial p}{\partial x} - \frac{y}{\sqrt{x^2+y^2}} \frac{\partial z}{\partial y} \frac{\partial p}{\partial x} \right. \right. \\
&\quad \left. - \frac{y}{\sqrt{x^2+y^2}} \frac{\partial z}{\partial x} \frac{\partial p}{\partial y} + \frac{x}{\sqrt{x^2+y^2}} \langle \operatorname{grad} z, \operatorname{grad} p \rangle \right\} dx dy \\
&\quad + \left. \int_{D_\epsilon} \frac{1}{\epsilon} \frac{x}{\sqrt{x^2+y^2}} [zp-fp] dx dy \right\} .
\end{aligned}
\right.
$$

From the expansion (2.1) of z, we derive

$$
\begin{gather*}
\max_{D_\epsilon} |z| \le K \sqrt{\epsilon} \\
\max_{D_\epsilon} |p| \le K \sqrt{\epsilon} \quad (K \text{ independent of } \epsilon) ;
\end{gather*}
$$

$$
\max_{D_{\epsilon}} |p| \le K \sqrt{\epsilon} \quad (K \text{ independent of } \epsilon) ;
$$

this implies (using Cauchy-Schwartz's inequality) that the last integral in (2.9)
vanishes as $\epsilon$ tends to zero ; thus (2.4) holds.

**Step 2** : Let us now look for the limit of each term of the remaining sum ;
when $r$ tends to zero, from (2.1), we have

$$
(2.10) \quad \left\{
\begin{array}{ll}
\displaystyle \frac{\partial z}{\partial x} - \frac{1}{2} z_{\frac{1}{2}} r^{-\frac{1}{2}} & \sin \frac{\theta}{2} \in H^{1+\epsilon}(\Omega \cap \mathcal{U}(P)) \quad 0 \le \epsilon < \frac{1}{2} \\
\\
\displaystyle \frac{\partial z}{\partial y} - \frac{1}{2} z_{\frac{1}{2}} r^{-\frac{1}{2}} & \cos \frac{\theta}{2} \in H^{1+\epsilon}(\Omega \cap \mathcal{U}(P)) \quad 0 \le \epsilon < \frac{1}{2} .
\end{array}
\right.
$$

From this and from analogous relations for p, we deduce

$$
\hat{J}'(0) = \lim_{\epsilon \to 0} \int_0^\pi \int_0^\epsilon \left\{ \frac{z_{\frac{1}{2}}}{4\epsilon r} \frac{P_{\frac{1}{2}}}{\sin^2 \frac{\theta}{2}} \left[ -\cos \theta \sin^2 \frac{\theta}{2} + \cos \theta \cos^2 \frac{\theta}{2} \right. \right. \\
\left. \left. + 2 \sin \theta \cos \frac{\theta}{2} \sin \frac{\theta}{2} \right] + \frac{1}{\epsilon} r^{-1/2} \mathbf{R} \right\} r dr d\theta ,
$$

with $R \in L^\infty(\Omega \cap \mathcal{U}(P))$ ; and then

$$
\hat{J}'(0) = \frac{\pi}{4} z_{\frac{1}{2}} P_{\frac{1}{2}} .
$$
**Proof of Theorem 2 (i) :**

Transposing exactly the proof of (i) in Theorem 1, we can derive most
of (i) from (ii) ; it remains to show the differentiability of $h \to z(h)$ in
$L^2(\Omega)$ : from Lemma 1, $h \to \tilde{z}(h) = z(h) \circ T_\varepsilon(h)$ is differentiable in $H^1(\Omega)$ ;
then $h \to z(h) = \tilde{z}(h) \circ T_\varepsilon^{-1}(h)$ is differentiable in $L^2(\Omega)$.■

**Proof of Proposition 4 : Proofs of (i) and (ii) :**

They are similar to the proof of (ii) in Theorem 2, only the adjoint states
are different. For $\tilde{j}$, it is the solution of the following system :

$$
(2.11) \quad \left\{
\begin{array}{ll}
\tilde{p} \in H_o^1(\Omega, \Gamma_h) , & \forall \psi \in H_o^1(\Omega, \Gamma_h) \\
\displaystyle\int_{\Omega} \{\langle \mathrm{grad} \, \tilde{p}, \mathrm{grad} \, \psi \rangle + \tilde{p}\psi \} \, dx \, dy = \displaystyle\int_{\Omega} \{\langle \mathrm{grad} \, z, \mathrm{grad} \, \psi \rangle + z\psi \} \, dx \, dy & \\
\text{so that } \tilde{p} \equiv z \text{ and (2.5) becomes (2.8).} &
\end{array}
\right.
$$

(iii) It is enough to show that (for a fixed ε) the Ck regularity of the
mapping h → Tε(h) and the belonging of f to Hk(Ω) implies the regularity
w.r.t. h of coefficients and right-hand side (f∘Tε)|det Tε'| in the trans-
ported state equation respectively in L∞(Ω) and [H01(Ω,Γ)]' (see (1.9) and
Lemma 1) (then the Implicit Function Theorem gives the conclusion); the
regularity of the coefficients is trivial because they are expressed by
means of rational functions of h; the regularity of (f∘Tε)|det Tε'| can be
shown by induction (see for instance F. MURAT, J. SIMON [2] paragraph 4.1.3.).

3. PERTURBATION OF A PROBLEM RELATED WITH A SIGNORINI-LIKE INEQUALITY

This part deals with a particular Signorini-like Elliptic Variational
Inequality (S.E.V.I.) ; we shall briefly write it as follows :

$$
\begin{equation}
\begin{aligned}
& (\rho_1) \quad \left\{
  \begin{array}{@{}l@{}}
    K = \{v \in H^1(\Omega) ; v \ge 0 \text{ a.e. on } \partial\Omega\} ; \\
    \quad \text{to find } u \in K \text{ s.t. } \forall v \in K \\
    \quad \displaystyle\int_{\Omega} \{\langle \mathrm{grad}\, u, \mathrm{grad}\, (v-u) \rangle - f(v-u)\} \, dv \ge 0.
  \end{array}
\right.
\end{aligned}
\end{equation}
$$
The solution of $\rho_1$ is also the solution of a mixed problem : a Dirichlet condition is fulfilled on the subset of $\partial\Omega$ where the constraint $\{u \ge 0 \text{ on } \partial\Omega\}$ is satu-rated, and a Neumann condition elsewhere : this partition of the boundary is an a priori unknown and constitutes, jointly with the distributed function $u$, the solution of the problem ; because of this geometrical unknown, we shall say that Problem $\rho_1$ is a Free Boundary type Problem. From this point of view, it should be convenient to set the problem in such a way that it appears as

$$ (*) \quad \left\{ \begin{array}{l} \text{A well-posed classical elliptic problem except for one geometrical} \\ \text{unknown, to be determined by means of a supplementary equation.} \end{array} \right. $$

This is not possible for Problem $\rho_1$ : we introduce a second problem - Problem $\rho_2$ - strongly related with $\rho_1$ and fulfilling this condition.

Before a perturbation study of $\rho_2$, it is necessary to look for conditions of local well-posedness ; by analogy with the bifurcation theory we give a characterization of the singular points.

The perturbation results are then obtained for the so-called "Free Boundary" and derived for the distributed solution.

Our point of view is very similar to D.G. SCHAEFFER's in [1] ; it is useful for the sequel to recall some features of SCHAEFFER's work, in which is studied the Obstacle Problem

$$ (\rho_1') \quad \left\{ \begin{array}{l} K' = \{v \in H^1(\Omega) ; v=1 \text{ on } \partial\Omega ; v \ge 0 \text{ a.e. on } \Omega\} \\ \text{to find } u \in K' \text{ s.t. } \forall v \in K' \\ \qquad \int_{\Omega} \{ \langle \text{grad } u, \text{grad}(v-u) \rangle - f(v-u) \} \, dv \ge 0. \end{array} \right. $$

Under certain regularity assumptions, a solution of $\rho_1'$ is a solution of the following problem $\rho_2'$ ($\rho_2'$ fulfils condition (*) ; $\partial\Omega_+$ is the geometrical unknown i.e. the Free Boundary)

$$ (\rho_2') \quad \left\{ \begin{array}{l} \Delta u = f \text{ on } \Omega_+ \\ u=1 \text{ on } \partial\Omega \\ u = \displaystyle\frac{\partial u}{\partial n} = 0 \text{ on } \partial\Omega_+ \\ u=0 \text{ in } \Omega-\Omega_+ , \end{array} \right. $$
but also verify the following *sign condition*

(s.c.) $u > 0$ on $\Omega_+$.

Now, by an Implicit Function Theorem, SCHAEFFER shows that under some assumptions $\rho_2'$ is *locally* (with respect to f) *well-posed* and that the solution depends continuously of f ; then, in a *last step*, it is shown that the sign condition (s.c.) is conserved locally so that the perturbation result stated for $\rho_2'$ holds too for $\rho_1'$.

In our work, we follow the same process, except the last step in place of
which we give some information concerning the relation between $\rho_1$ and $\rho_2$.

This section uses some results of the second one ; it is important to note
that such problems where singularities of the unknown function occur,
lead to inextricable proofs and even results if we attempt to attain enough
generality so that we shall explain it in a very simple case.

Let us note at last that this question of dependence with respect to data
of the solution of this S.E.V.I. has been studied by F. MIGNOT [1].

3.1. *Formulation of the two problems*

The geometrical assumptions are the following : $\Omega$ is a $C^\infty$ ($^5$) bounded domain of $\mathbb{R}^2$ ; $\partial\Omega$ is subdivided into three non empty connected arcs $\Gamma_N, \Gamma_D$, $\Sigma$ and $\Sigma'$ is a straight segment AB ; a system of cartesian coordinates is introduced as in Section 2 (see Fig. 2).

<figure><img src="image_5.png" /><figcaption>Figure 2</figcaption></figure>
We consider the following problem

$$
\begin{equation}
\begin{aligned}
K &= \{ v \in H^1(\Omega) \mid v|_{\Sigma} \ge 0 \quad v|_{\Gamma_D} = 0 \} \\
(\rho_I) \quad \begin{cases}
\text{For a given } f \text{ in } L^2(\Omega), \\
\displaystyle \int_{\Omega} \langle \text{grad } u(f), \text{grad}(v-u(f)) \rangle dx dy \ge - \int_{\Omega} f(v-u(f)) dx dy.
\end{cases}
\end{aligned}
\end{equation}
$$

Let us recall some classical results concerning Problem $\rho_I$ (see J.L. LIONS [2]) :

**Proposition 5** : Problem $\rho_I$ has an unique solution $u(f)$ characterized by the
following interpretation

$$
\text{(3.1)} \quad \left\{
\begin{array}{@{}l@{\quad}l@{}}
\text{(a)} &
\left\{
\begin{array}{@{}l@{}}
\Delta u(f) = f \text{ in } \Omega \\
u(f) = 0 \text{ on } \Gamma_D
\end{array}
\right. \\
\text{(b)} &
\left\{
\begin{array}{@{}l@{}}
\displaystyle\int_{\Omega} \langle \nabla u(f), u(f) \rangle \,dx \,dy > 0 \\
\forall P \in \Sigma, P \notin A, B, \text{ there exists a neighbourhood } \mathcal{V}(P) \\
\text{of } P \text{ such that } u(f) \in H^2(\Omega \cap \mathcal{V}(P)) . \quad \blacksquare
\end{array}
\right.
\end{array}
\right.
$$

Let us denote by $\Sigma_+(f)$ the set defined by

$$
(3.2) \quad \Sigma_+(f) = \{ (x,o) \in \Sigma \; ; \; u(f)(x,o) > 0 \} ,
$$

and let us assume that (a highly <u>simplifying assumption</u> which can be satisfied)

$$
(3.3) \quad \left\{
\begin{array}{@{}l@{\quad}l@{}}
\text{For an } f_o \text{ belonging to } H^2(\Omega), & \Sigma_+(f_o) \text{ is an interval} \\
\mathbb{P}_o \text{ with } \mathbb{P}_o = (h_o, 0) . &
\end{array}
\right.
$$

Then the "Free Boundary" is in this particular case reduced to only <u>one point</u> $P_o$.
For every real h small enough, let us introduce

$$
(3.4) \quad \Sigma_h = \{ (x,o) \in \Sigma , x > h \}
$$

and $z(f,h)$ as the solution of the following mixed problem

$$
(3.5) \quad \left\{
\begin{array}{ll}
z(f,h) \in K_h = \{ \phi \in H^1(\Omega) , \phi|_{\Sigma_h \cup \Gamma_D} = 0 \} , & \\
\forall \phi \in K_h & \\
\int_{\Omega} <\mathrm{grad } z, \mathrm{grad } \phi> dx dy = - \int_{\Omega} f \phi dx dy.
\end{array}
\right.
$$

Then

$$
(3.6) \quad u(f_o) = z(f_o, h_o) ,
$$

and the function $u(f_o)$ is part of the solution for $f=f_o$ of the following Free Boundary type Problem $\mathcal{P}_2$

$$
(\mathcal{P}_2) \quad \left\{
\begin{array}{ll}
\text{for } f \text{ given in } H^2(\Omega), & \\
\text{find } H(f) \in \mathbb{R} \text{ and } W(f) \in H^1(\Omega) \text{ such that} & \\
W(f) = z(f,H(f)) \text{ solution of (3.5) , and} & \\
\text{if } P_f = (H(f),o), \text{ there exists a neighbourhood } \mathcal{V}(P) \text{ of } P_f & \\
\text{such that } W(f) \text{ belongs to } H^2(\Omega \cap \mathcal{V}(P_f)).
\end{array}
\right.
$$

Problem $\mathcal{P}_2$ can be compared with (3.1) : the difference between $\mathcal{P}_1$ and $\mathcal{P}_2$
consists of the **sign conditions** :

$$
(3.7) \quad u(f) \ge 0 , \frac{\partial u(f)}{\partial y} \ge 0 \text{ on } \Sigma .
$$

Using the results and notations given in Section 2, Problem $\mathcal{P}_2$ is seen to
be equivalent to

$$
(3.8) \quad \left\{
\begin{array}{ll}
\text{For } f \text{ given in } H^2(\Omega), & \\
\text{find } H(f) \text{ and } W(f) \text{ such that} & \\
W(f) = z(f, H(f)), & \\
\text{and the first singular coefficient of } W(f) \text{ near } P_f \text{ vanishes, i.e.} & \\
W_{\frac{1}{2}}(f) (= z_{\frac{1}{2}}(f, H(f))) = 0.
\end{array}
\right.
$$
Now, we note that, using (3.5), we can consider that there is only one (real)
unknown, the "Free Boundary" represented by H(f) : if we introduce the func-
tional (for ε small enough)

$$
\Psi : H^2(\Omega) \times ]h_o^-\epsilon, h_o^+\epsilon[ \to \mathbb{R}
$$

$$
\Psi(f, h) = z_{\frac{1}{2}}(f, h),
$$

then we get a new formulation of (P<sub>2</sub>) :

$$
(3.9) \quad \left\{
\begin{array}{@{}l@{}}
\text{For a given } f \text{ in } H^2(\Omega), \\
\text{find } H(f) \text{ such that} \\
\Psi(f, H(f)) = 0.
\end{array}
\right.
$$

Our purpose is now to give a very short sketch of bifurcation study of (3.9)
when f is perturbed near f₀ (imagine for instance that f depends of one real
parameter).

3.2. Singular points for Problem ρ₂

We shall call a singular point a pair (f,h) satisfying (3.9) for which the
Implicit Function theorem cannot be applied to Ψ ; to make this notion precise, we
need to prove the following proposition

**Proposition 6** : the mapping Ψ is continuously differentiable, so that
(f,h) is a singular point if and only if ∂Ψ/∂h (f,h) = 0.

Proof :

Let $h_1$ be a fixed real belonging to $]h_o^-, h_o^+[$. From $q = r^{\frac{1}{2}} \sin \frac{\theta}{2} \phi(r)$ where $\phi(r)$ is equal to 2 for $r<r_o$ and to 0 for $r<r_1$ with $0<r_o<r_1$, it is possible to construct a pair $(q,g)$ such that

$$
(3.10) \quad \left\{
\begin{array}{@{}l@{}}
\displaystyle g \in C^\infty(\bar{\Omega}) , \quad q \in H^1(\Omega), \quad q = z(g,h_1) \quad (\text{cf. (3.5)}) \quad \text{and} \\
\displaystyle q \notin H^2(\Omega) , \quad \text{i.e. } q_{\frac{1}{2}} = z_{\frac{1}{2}}(g,h_1) \neq 0.
\end{array}
\right.
$$

Now if we put

$$
(3.11) \quad \tilde{j}(h) = \int_{\Omega} \{ |\operatorname{grad} z(g,h)|^2 + |z(g,h)|^2 \} \, dx \, dy,
$$
then, from Proposition 4, $\tilde{j}$ is $C^\infty$, and

$$ \frac{d\tilde{j}}{dh}(h) = \frac{\pi}{4} [z_{\frac{1}{2}}(g,h)]^2 $$

thus $h \to z_{\frac{1}{2}}(g,h)$ is a $C^\infty$, non vanishing function in a neighbourhood of $h_1$.
If we consider

$$ (3.12) \qquad \bar{j}(f,h) = \int_{\Omega} g z(f,h) \, dx \, dy, $$

it is not difficult to show (extending the proof of Proposition 4) that $\bar{j}$ is a $C^2$ mapping from $H^2(\Omega) \times ]h_o-\epsilon, h_o+\epsilon[$ into $\mathbb{R}$, and that

$$ (3.13) \qquad \left\{ \begin{aligned} \frac{\partial \bar{j}}{\partial h}(f,h) &= \frac{\pi}{4} z_{\frac{1}{2}}(f,h) z_{\frac{1}{2}}(g,h) \\ &= \frac{\pi}{4} \Psi(f,h) z_{\frac{1}{2}}(g,h) \end{aligned} \right. $$

is a $C^1$ function, from which, jointly with the properties of $h \to z_{\frac{1}{2}}(g,h)$, the regularity of $\Psi$ follows. ■

This proof gives also an expression of $\frac{\partial \Psi}{\partial h}$ when its arguments are solutions of (3.9) :

**Proposition 7** : If $\Psi(f,h) = 0$, then

$$ (3.14) \qquad \frac{\partial \Psi}{\partial h}(f,h) = \frac{4}{\pi} \frac{\partial^2 \bar{j}}{\partial h^2}(f,h) / z_{\frac{1}{2}}(g,h); $$

a solution of (3.9) is a stationary point for $\bar{j}$; it is singular if the second derivative of $\bar{j}$ also vanishes. ■

**Proof** : Differentiating (3.13) we get

$$ \frac{\partial^2 \bar{j}}{\partial h^2}(f,h) = \frac{\pi}{4} \frac{\partial \Psi}{\partial h}(f,h) z_{\frac{1}{2}}(g,h) + \frac{\pi}{4} \Psi(f,h) \frac{\partial}{\partial h} [z_{\frac{1}{2}}(g,h)]; $$

the second term of the right hand sum vanishes if $\Psi(f,h) = 0$ and we get (3.14).
In the following section, we shall be concerned with *regular points*, i.e. pairs $(f,h)$ so that $\Psi(f,h) = 0$ but $\frac{\partial\Psi}{\partial h}(f,h) \neq 0$. This restriction can be intuitively justified by the following "extreme case" : assuming (3.3), if $\Psi(h_o,f)$ and $\frac{\partial\Psi}{\partial h}(h_o,f)$ vanish identically for every $h$ belonging to an interval $]h_o-\epsilon,h_o+\epsilon[$, then the corresponding solution of $\wp_1$ is a regular solution of the mixed problem for every $h$ in this interval ; thus the set where constraints are saturated is *not well defined* because its boundary can be any point with abscissa in $]h_o-\epsilon,h_o+\epsilon[$.

### 3.3. Perturbation results for regular points of $\wp_2$

**Theorem 3** : Let $(h_o, h_o)$ be a pair such that

$$ (h_o, o) \in ]A, B[ \\ f_o \in H^2(\Omega_o) \\ \Psi(h_o, h_o) = 0, \quad \frac{\partial \Psi}{\partial h}(h_o, h_o) \neq 0; $$

then there exists a neighbourhood $\mathcal{V}$ of $h_o$ in $H^2(\Omega)$ and exactly one continuous mapping

$$ H : \mathcal{V} \to \mathbb{R} $$

such that

$$ H(h_o) = h_o, \quad \Psi(f, H(f)) = 0 \quad \forall f \in \mathcal{V}. $$

Moreover, $H$ is $C^1$ and the mapping

$$ W : \mathcal{V} \to H^1(\Omega) ; W(f) = z(f, H(f)) \quad (\text{cf. (3.5)}) $$

is $C^1$ from $\mathcal{V}$ into $L^2(\Omega)$ ; the derivatives of these mappings can be expressed as follows :

$$ (3.15) \qquad \left\{ \begin{array}{l} \displaystyle \forall \delta f \in H^2(\Omega) \\[1em] \displaystyle \frac{\partial H}{\partial f}(f) \cdot \delta f = - \left[ \frac{\partial \Psi}{\partial h}(h, f) \right]^{-1} \left[ \frac{\partial \Psi}{\partial f}(h, f) \cdot \delta f \right] \\[1em] \displaystyle \frac{\partial W}{\partial f}(f) \cdot \delta f \text{ is the (weak) solution of the mixed system} \end{array} \right. $$

$$ (3.16) \qquad \frac{\partial W}{\partial f}(f) \cdot \delta f = z(\delta f, H(f)) $$
**Proof** : This is a simple application of the Implicit Function Theorem to the function $\Psi$; statement (3.15) is thus classical. Let us show the differentiability of $W$ and (3.16) : considering the construction of $W$, we need differentiability in $L^2(\Omega)$ for the mapping

$$ (f,h) \to z(f,h); $$

but this is easily derived by arguments similar to the proof of Theorem 2 (i) (it is enough to set $f$ as a supplementary variable) ; then we can write a composite derivative :

$$ \frac{\partial W}{\partial f}(f) = \frac{\partial z}{\partial f}(f, H(f)) + \frac{\partial z}{\partial h}(f, H(f)) \frac{\partial H}{\partial f}(f); $$

the first term of the right hand sum can be easily recognized as the right hand side of (3.16) ; to see that the second term vanishes, we substitute $z$ by zero in (2.5). ■

### 3.4. Relations between $\mathcal{P}_1$ and $\mathcal{P}_2$ : some remarks and hints.

Suppose that the pair $[H(f_o), W(f_o, H(f_o))]$ is a solution of both Problems $\mathcal{P}_1$ and $\mathcal{P}_2$ ; it is interesting to wonder in which cases the pair $[H(f), W(f, H(f))]$ will not be a solution of $\mathcal{P}_1$ in any neighbourhood of $f_o$ : this happens for example when the "Free Boundary" of $\mathcal{P}_1$ becomes several points. To be sure that such a case does not happen - under assumptions concerning $f_o$, $H(f_o)$, $W(f_o, H(f_o))$ - it would be interesting to do as D.G. SCHAEFFER [1], i.e. to show that the sign condition (3.7) is still fulfilled for $f$ near of $f_o$ : such a study should need (at least) much more precise results than those of Section 2.

We shall give here a weaker result since we shall assume that :

$$ (3.17) \quad \left\{ \begin{array}{l} f = f_{\theta} \text{ depends smoothly in } H^2(\Omega) \text{ of a real } (6) \text{ parameter } \theta \text{ with} \\ f_o = f_{\theta} \text{ for } \theta=0 \text{ (see 3.3)} \\ \text{there exists a segment } [-a,+a] \text{ (a} < \alpha) \text{ such that for every } \theta \text{ in a} \\ \text{neighbourhood of 0, the Free Boundary of } \mathcal{P}_1 \text{ can be described as} \\ \text{only one point with an abscissa in } [-a,+a]^{(7)}. \end{array} \right. $$
Examples for which this assumption holds can easily be constructed(8); on the
other hand it is reasonable in physical applications.

**Proposition 8** : Under the assumptions of Theorem 3, assuming moreover that (3.17) holds, then, in a neighbourhood of θ=0 the pair [H(f_θ), W(f_θ, H(f_θ))] is a solution of both problems P_1 and P_2. ■

For the proof, we need the following Lemma :

**Lemma 6** : Under the assumption (3.17), and assuming also that for $f=f_0$,
the solution $u_0$ of problem $\mathcal{P}_1$ is such that there is no segment $[h_1,h_2]$ in [-a,+a]
containing $h_0$ on which both Neumann and Dirichlet boundary conditions are verified,
then the corresponding "Free Boundary" depends continuously on $\theta$ in a neighbourhood
of zero.

**Proof** : We shall prove it by contradiction : let us assume that there exists a sequence $f_{\theta_v}$, a real $\eta > 0$ such that

$$
(3.18) \quad \left| f_{\theta_v} - f_o \right|_{L^2(\Omega)} < \frac{1}{v}, \quad \forall v \in \mathbb{N},
$$

and such that the corresponding Free Boundaries have abscissae h verifying
$v$

$$
(3.19) \quad |h_v - h_o| > \eta .
$$

Since [-a,+a] is compact, there exists a subsequence (still denoted v) such
that

$$
h_v \to h^* \in [-a, +a] \text{ as } v \to \infty
$$

and since the continuity of the solution of $\mathcal{P}_1$ with respect to $f$ is classical
(and immediate), we have

$$
u_v \to u^* = u(f_0) \quad \text{strongly in } H^1(\Omega);
$$
on the other hand, from (3.19) we have

$$h^* \neq h_o.$$

**First case** : $h^* > h_o$ ; there exist $h_1$ and $h_2$ such that $h_o < h_1 < h_2 < h^*$ ;
for $\nu$ large enough, $h_\nu$ is outside of $[h_1, h_2]$, so that

$$\nu > N_o \Rightarrow \frac{\partial u_\nu}{\partial n} = 0 \text{ for } y=0, x \in ]h_1, h_2[ ;$$

now, from (3.18), we have

$$\frac{\partial u_\nu}{\partial n} \rightarrow \frac{\partial u(f_o)}{\partial n} \text{ strongly in } H^{-1/2}(\partial\Omega),$$

thus

$$\frac{\partial u(f_o)}{\partial n} = 0 \text{ for } y=0, x \in ]h_1, h_2[ ;$$

but since $h_1 > h_o$, we also have

$$u(f_o) = 0 \text{ for } y=0, x \in ]h_1, h_2[$$

and this contradicts the assumption of the lemma.

**Second case** : $h_o > h^*$ we can similarly pass to the limit with the restriction of $u_\nu$ on $\{y=0, x \in ]h_1, h_2[\}$. ■

**Proof of Proposition 8** :

Let $\tilde{H}(f_\theta)$ be the abscissa of the Free Boundary solution of Problem $\mathcal{P}_1$ ; it is also a solution of Problem $\mathcal{P}_2$. From Theorem 3, $H(f_\theta)$ is the unique continuous branch of solutions of Problem $\mathcal{P}_2$ containing $H(f_o) = \tilde{H}(f_o)$; thus it is enough to show that $\theta \rightarrow \tilde{H}(f_\theta)$ is continuous with respect to $\theta$ : from the assumption $\frac{\partial \psi}{\partial h}(f_o, h_o) \neq 0$ of Theorem 3, as noticed at the end of Section 3.2, the Neumann and Dirichlet homogeneous conditions may not be both verified on a segment containing $h_o$ ; then Lemma 6 applies. ■
**Remark 6** : Let us note the accordance (mutatis mutandis) between the derivative of the mapping W (see Theorem 3) for problem $\rho_2$ and the derivative for Problem $\rho_1$ given by F. MIGNOT ([1] ; Theorem 6.1) ; particularly, we see that the linearized problem of both $\rho_1$ and $\rho_2$ is the corresponding mixed problem. ■

## CONCLUSION

Variational Hadamard's formula given in the first section requires only few assumptions (especially concerning the subset $\Gamma$ of $\partial\Omega$). If we compare with P.R. GARABEDIAN and M. SCHIFFER's *interior variation method* (see P.R. GARABEDIAN [2]), we can say that our formula is, from a certain point of view, a generalized Green formula.

It is also interesting to note that domain perturbation methods may lead to regularity results (for instance existence of a trace).

The method of the second paragraph can be adapted to a lot of problems in which singular geometrical data are perturbed (such as a plane without a point, without a segment, etc...).

The application in the third section gives information concerning the "boundary condition shifting point" which are, as far as we know, new. It would be interesting to extend the results of the last two sections to higher dimension. For this purpose it might be necessary to use *generalized Implicit Functions theorems* (as it is done in D.G. SCHAEFFER [1] for an other variational inequality) instead of the *classical one*.

## ACKNOWLEDGEMENTS

We are thankful to F. MURAT for reading this work and for many helpful suggestions.
## FOOTNOTES :

(1) $\mathcal{B}(\Omega)$ : space of infinitely differentiable functions with compact support in $\Omega$ ; $e_N = (0,0,...,0,1) \in \mathbb{R}^N$.

(2) This extension will be used in Lemma 2 and Theorem 2 in the sequel.

(3) See for instance F. MURAT, J. SIMON [2], paragraph 4.1.5.

(4) For the simplicity of notations ; the general case is trivially derived with a translation.

(5) In the sense of Section 1.1.

(6) Only for the sake of simplicity.

(7) This mainly means that the Free Boundary does not "burst" from one point into several when $\theta$ varies.

(8) For example, the support of the solution of the S.E.V.I. for $f = f_o^0$ is in a sufficiently small neighbourhood of 0, and $f_o^\theta$ is a translated $^o$ of $f_o$ in the direction of the x-axis.

-o-

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-o-

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