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

BIFURCATION
OF FREE PLASMA
EQUILIBRIA

Michel SERMANGE

Rapport de Recherche N° 365

Octobre 1979

30p.
BIFURCATION OF FREE PLASMA
EQUILIBRIA

Michel SERMANGE.

(Presented by Roger TEMAM)

ABSTRACT

By comparing eigenvalues of elliptic problems with discontinuous coefficients, we prove the existence of a bifurcation of free boundary plasma equilibria in a "hourglass-shaped" cavity.

RESUME

En utilisant la comparaison des valeurs propres de problèmes elliptiques à coefficients discontinus, nous prouvons l'existence d'une bifurcation des équilibres d'un plasma à frontière libre dans une cavité en "forme de sablier".
O. INTRODUCTION

In this paper, we study a non-linear Dirichlet problem introduced by R. TEMAM [19] as a special case of free boundary plasma equilibrium. Given a domain $\Omega \subset \mathbb{R}^2$ and a positive real number $\lambda$, we look for $u \in H^1(\Omega)$ (Sobolev space of order 1 on $L^2(\Omega)$) such that

$$
\begin{array}{ll}
(0.1) & \left\{
\begin{array}{ll}
-\Delta u = \lambda u_+ & \text{a.e. in } \Omega, \\
\\
u = \text{ct. (unknown)} & \text{a.e. on } \Gamma = \partial\Omega, \\
\\
-\int_{\Gamma} \frac{\partial u}{\partial n} d\sigma = 1, &
\end{array}
\right. \\
\text{where } & \Delta = \sum_{i=1}^{2} \frac{\partial^2}{\partial x_i^2} \quad \text{and} \quad u_+ = \max(u, 0).
\end{array}
$$

From a physical point of view, the sets $\Omega$ and $\Omega_p = \{x \in \Omega \mid u(x) > 0\}$ are respectively the sections of the TOKAMAK and of the plasma, supposed to be infinite straight cylinders ; the distribution $u$ is the poloidal magnetic flux. Thus, the set $\Omega_p$ occupied by the plasma is not given beforehand.

R. TEMAM [19] first proved the existence and regularity of solutions of (0.1) for every $\lambda > 0$, and uniqueness for $0 < \lambda < \lambda_2$, where $\lambda_2$ is the second eigenvalue of the Dirichlet problem on $\Omega$.

Here we prove the existence of bifurcation of solutions of (0.1) for certain symmetrical domains. The domains considered are hourglass-shaped and they were first introduced by D.G. SCHAEFFER [14] in order to give an example of non-uniqueness of solutions of (0.1). Let us also mention that numerical computations of the bifurcated solutions for such domains were presented in a previous paper [15].

Related to the present work are formulations of sufficient conditions of bifurcation for the problem (0.1), obtained by A. DERVIEUX [4], B. NICOLAENKOR. TEMAM [9], J. SIJBRAND [16] [17]; these results are different from the present result, for it has not been proved so far that there exist domains $\Omega$ where their conditions are satisfied ; while we do not know if our example enters their framework, we do show that bifurcation occurs.
Our method of proof is as follows. Comparing eigenvalues of elliptic operators
with discontinuous coefficients and making use of the disconnectedness of Ω<sub>p</sub>
for the symmetrical solution for λ sufficiently large (proved by D.G. SCHAEFFER
[14]), we obtain some estimates on the eigenvalues of the linearized problem.
Then we prove the bifurcation by a degree argument. A technical difficulty is
that, in general, the free boundary of the plasma is not globally regular.

In a last part, we show that the present method of proof of bifurcation can
also be applied in a broader framework than the "hourglass-shape" case.

The author thanks R. TEMAM for directing this work.

# 1. THE BIFURCATION THEOREM.

Let us introduce

$\sigma$ : the symmetry operator $(x_1, x_2) \longmapsto (-x_1, x_2),$

$S_\varepsilon: \{ (x_1, x_2) \in \mathbb{R}^2, |x_1| < 2, |x_2| < \varepsilon \},$

$B_r$: the ball of radius $r$, center $(x_1, x_2) = (3,0)$.

As in D.G. SCHAEFFER [14], we shall consider the domains Ω ⊂ ℝ² satisfying
(cf. fig. 1) the following conditions :

(1.1) Ω is a bounded domain of ℝ², ∂Ω is a C∞ curve and
Ω is locally on 'one side of ∂Ω,

(1.2) $\epsilon_0, r_0$ \quad $\Omega = \mathcal{O} \cup S_\epsilon \cup \sigma(\mathcal{O})$ where $\mathcal{O}$ is an open set of $\mathbb{R}^2$ such that
$B_1 \subset \mathcal{O} \subset B_{r_0}$ and $\epsilon$ is a real number such that $0 < \epsilon < \epsilon_0$.

<figure><img src="image_2.png" /><figcaption>Figure 1</figcaption></figure>
As in J.P. PUEL [11], we introduce the following auxiliary problem :
to find $v \in H^1(\Omega)$, $v \neq -1$, such that

$$
(1.3) \quad \left\{
\begin{array}{ll}
- \Delta v = \lambda v_+ & \text{a.e. in } \Omega, \\
v = -1 & \text{a.e. on } \Gamma.
\end{array}
\right.
$$

From a physical point of view, both problems (0.1) and (1.3) are interesting :
for a flux function $u$, the parameter $- \int_{\Gamma} \frac{\partial u}{\partial n} d\sigma$ is the total current crossing the plasma, while the parameter $u|_{\Gamma}$ is the magnetic flux between the wall $\Gamma$ and the boundary of the plasma. In the sequel, we shall prove bifurcation for (1.3) and, in consequence, bifurcation for (0.1).

The main result is

THEOREM. There exist $\epsilon_0$ and $r_0 > 1$ sufficiently small such that, for any given domain $\Omega$ satisfying (1.1), $(1.2)_{\epsilon_0, r_0}$, denoting by $\lambda_i$ the eigenvalues of the Dirichlet problem on $\Omega$,

i) the second eigenvalue $\lambda_2$ is simple,

ii) for $\lambda$ such that : $\lambda_2 \le \lambda \le \lambda_3$, there exists only one symmetrical solution of (0.1) (respectively only one symmetrical solution $v \ne -1$ of (1.3)),

iii) there exists at least one bifurcation point $(\alpha, u)$ of (0.1) (respectively $(\alpha, v)$ of (1.3) with $v \ne -1$) in $(\lambda_2, \lambda_3) \times H^2(\Omega)$ with respect to the branch of symmetrical solutions.

By bifurcation, we mean that there exist a symmetrical solution $(\alpha, u)$ of (0.1)
with $\lambda_2 < \alpha < \lambda_3$ and a sequence of non-symmetric solutions $(\alpha_n, u_n)$ of (0.1)
such that

$u_n \longrightarrow u$ in $H^2(\Omega)$ and $\alpha_n \longrightarrow \alpha$ as $n \to \infty$.

Our proof of bifurcation does not use any classical theorem :

- since the derivative of the functional along the branch of symmetrical solutions is not linear in $\lambda$, we cannot use the bifurcation theorem of M.A. KRASNOSELSKI

- since the local behaviour of the branch of symmetrical solutions is unknown,
we cannot use a local argument ; for instance, the bifurcation theorem of
C. CRANDALL-P.H. RABINOWITZ cannot be used, even after a regularization of the
functional.
Our proof is based on a *global study* of the branch of symmetrical solutions, by means of the topological degree. The plan of the proof is as follows:

a) the set of solutions of (0.1) (resp. (1.3)) when $\lambda$ varies over a compact set of $(\lambda_1, +\infty)$ is compact in $H^2(\Omega)$, which is important to separate the curves of solutions;

b) there exists only one symmetrical solution of (0.1) (resp. (1.3)) for $\lambda_2 \le \lambda \le \lambda_3$, which shows, by compactness, that the branch of symmetrical solutions is continuous in $H^2(\Omega)$;

c) we introduce a functional form of equation (1.3) and we show that the eigenvalues $\mu_i$ of the derivative of the functional satisfy

$$ \lambda_i < \mu_i < \gamma_i \qquad \forall i \ge 1 $$

where $\gamma_i$ are the eigenvalues of the Dirichlet problem on $\Omega_p$;

d) this shows that the topological degree for $\lambda = \lambda_2$ or for $\lambda = \lambda_3$ (where $\Omega_p$ is disconnected, as proved by D.G. SCHAEFFER [14]) are different, which is absurd if there is no bifurcation of (1.3).

## 2. A LEMMA.

We shall use the following lemma, which seems to be a new result<fnref n="1" />.

**Lemma 2.1.** Let $\Omega$ be a bounded open set of $\mathbb{R}^q$; if $u \in C^1(\bar{\Omega})$ and $u(x) = 0$ $\forall x \in \partial\Omega$, then $u \in H_0^1(\Omega)^{(2)}$.

**Proof.** Let $j_\epsilon(x) = \epsilon^{-q} j(\frac{x}{\epsilon})$ be a regularizing family in $\mathbb{R}^q$. For $\epsilon > 0$ given, we introduce

$$ \Omega_{\epsilon} = \{ x \in \Omega \mid \text{d}(x, \partial\Omega) > \epsilon \}, $$

$$ \varphi_{\epsilon} = j_{\epsilon} * 1_{\Omega_{2\epsilon}} $$

$$ u_{\epsilon} = \varphi_{\epsilon} \cdot u. $$

The function $u_\epsilon$ is $C^1(\bar{\Omega})$ with compact support in $\Omega$ and

<footnote n="1">¹ A variant of this result is stated without proof in G. STAMPACCHIA [18], Lemma 1.2.</footnote>

<footnote n="2">² $C^k(\bar{\Omega}) = \{\text{restriction to } \bar{\Omega} \text{ of functions of } C^k(\mathbb{R}^q)\}$, $H_0^1(\Omega) = \text{D}(\bar{\Omega})H^1(\Omega)$</footnote>
$$
(2.1) \quad \varphi_{\varepsilon}(x) = 1 \qquad \forall x \in \Omega_{3\varepsilon},
$$

$$
(2.2) \quad 0 \le \varphi_{\epsilon}(x) \le 1 \quad \forall x \in \Omega,
$$

$$
(2.3) \quad \left| \frac{\partial \varphi}{\partial x_i} (x) \right| \le \frac{M}{\epsilon} \quad \forall x \in \Omega,
$$

$$
(2.4) \quad |u(x)| \le M' \epsilon \quad \forall x \in \Omega \setminus \Omega_{3\epsilon},
$$

where M and M' are constants independent of ε.

By (2.1) (2.3) (2.4),

$$
\left| \left( \frac{\partial \varphi}{\partial x_i} \cdot u_{\varepsilon} \right) (x) \right| \leq M \cdot M' \quad \forall x \in \Omega.
$$

Now the convergence in $L^2(\Omega)$ of each term of

$$
\frac{\partial}{\partial x_i} u_{\epsilon} = \frac{\partial \varphi_{\epsilon}}{\partial x_i} \cdot u + \varphi_{\epsilon} \frac{\partial u}{\partial x_i}
$$

follows from the Lebesgue dominated convergence theorem, and we obtain

$$
u_{\epsilon} \rightarrow u \quad \text{in } H^1(\Omega) \quad \text{for} \quad \epsilon \rightarrow 0,
$$

which proves the lemma.

3. PROOF OF THE THEOREM.

3.1. Compactness of solutions.

Lemma 3.1. If Ω is any bounded smooth domain of R<sup>q</sup>, q ≤ 3, then

i) if $\lambda$ varies over a compact set of $\mathbb{R}_+$, the solutions $u$ of (0.1) are bounded in $H^3(\Omega)$,
ii) if $\lambda$ varies over a compact set of $(\lambda_1, +\infty)$, the solutions $v$ of (1.3) are bounded in $H^3(\Omega)$.

..

**Proof i) For a given solution $(\lambda, u)$ of (0,1), we introduce as in [15]**

$$ \Psi = u_+ \quad \text{and} \quad \tilde{u} = u - u(\Gamma) $$

where $u(\Gamma)$ denotes the real number $u|_{\Gamma}$.

Obviously, $\tilde{u} \in H_0^1(\Omega)$ and

$$ (3.1) \qquad - \Delta \tilde{u} = \lambda \Psi \quad \text{a.e. in } \Omega. $$

From the regularity of solutions for elliptic equations, there exists a constant $c_1$, depending only upon $\Omega$ and $q$, such that

$$ \|\tilde{u}\|_{H^2(\Omega)} \le c_1 \lambda |\Psi|_{L^2(\Omega)}. $$

Since $q \le 3$, the injection of Sobolev

$$ H^2(\Omega) \subset C^\circ(\bar{\Omega}) $$

implies that $\tilde{u}$ and $\Psi \in L^\infty(\Omega)$, and that

$$ (3.2) \qquad |\tilde{u}|_{L^\infty(\Omega)} \le c_2 \lambda |\Psi|_{L^2(\Omega)}. $$

On the other hand, all $\tilde{u} \in L^\infty(\Omega)$ and $a \in \mathbb{R}$ such that

$$ \int_{\Omega} (\tilde{u}(\tau) + a)_+ d\tau = \frac{1}{\lambda}, $$

satisfy

$$ |a| \le |\tilde{u}|_{L^\infty(\Omega)} + \frac{1}{\lambda \text{meas}(\Omega)}. $$

Therefore, by integration of the first equation of (0.1) on $\Omega$,

$$ |u(\Gamma)| \le |\tilde{u}|_{L^\infty(\Omega)} + \frac{1}{\lambda \text{meas}(\Omega)}. $$
and, since $\Psi = (\tilde{u} + u(\Gamma))_+$, we have

$$ (3.3) \qquad |\Psi|_{L^\infty(\Omega)} \le 2 |\tilde{u}|_{L^\infty(\Omega)} + \frac{1}{\lambda \operatorname{meas}(\Omega)}. $$

By virtue of the relation

$$ |\Psi|_{L^1} = \frac{1}{\lambda}, $$

Hölder's inequality yields

$$ (3.4) \qquad |\Psi|_{L^2}^2 \le \frac{1}{\lambda} |\Psi|_{L^\infty}. $$

From (3.2) (3.3) (3.4),

$$ |\Psi|_{L^2}^2 \le 2 c_2 |\Psi|_{L^2} + \frac{1}{\lambda^2 \mathrm{meas}(\Omega)}, $$

which implies

$$ |\Psi|_{L^2(\Omega)} \le R(\lambda) $$

where

$$ R(\lambda) = c_2 + \sqrt{c_2^2 + \frac{1}{\lambda^2 \mathrm{meas}(\Omega)}} $$

Therefore, if $\lambda$ varies over a compact subset of $\mathbb{R}_+$, then

$$ ||u||_{H^1(\Omega)} \le \text{ct.} $$

Thus, since $\Psi = u_+$,

$$ ||\Psi||_{H^1(\Omega)} \le \text{ct.}; $$

once again, it follows from (3.1) that
$$||\Psi||_{H^3(\Omega)} \le \text{ct.}.$$

ii) Let $\lambda > \lambda_1$ be given. If $u$ is a solution of (0.1), then (cf. H. BERESTCZEKI and H. BREZIS [2])

$$ (3.5) \qquad u(\Gamma) < 0. $$

Hence, the mapping

$$ (3.6) \qquad u \longmapsto v = - \frac{u}{u(\Gamma)} $$

is a one-to-one mapping from the set of solutions of (0.1) onto the set of solutions $v \ne -1$ of (1.3), whose inverse is given by

$$ (3.7) \qquad v \longmapsto u = \left( \lambda \int_{\Omega} v_+ dx \right)^{-1} \cdot v. $$

If $\lambda$ varies over a compact subset of $(\lambda_1, +\infty)$, from (3.5) and from the compactness proved in i), it follows that

$$ (3.8) \qquad |u(\Gamma)| \ge \text{ct.} > 0. $$

Then ii) follows from i), (3.6) and (3.8).

### 3.2. Uniqueness of symmetrical solutions.

The parts i) and ii) of the theorem are derived from the following lemmata.

**Lemma 3.2.** Let us suppose that $\tau$ is an isometry of $\mathbb{R}^q$ and that $\Omega$ is a smooth bounded domain of $\mathbb{R}^q$ invariant by $\tau$. We denote by $\lambda_i^*$ the eigenvalues of the Dirichlet problem on $\Omega$ corresponding to at least one eigenfunction invariant by $\tau$.

If $\lambda_1 < \lambda \le \lambda_2^*$, then there exists exactly one solution of (0.1) invariant by $\tau$ (resp. exactly one solution $v \ne -1$ of (1.3) invariant by $\tau$).

We follow the method of proof of uniqueness of J.P. PUEL and A. DAMLAMIAN [11].
**Proof.** Let $\lambda_1 < \lambda \le \lambda_2^*$ be given and let $v_1, v_2 \ne 0$ be two solutions of (1.3) invariant by $\tau$. We recall (cf. R. TEMAM [19]) that $v_i \in C^{2+\eta}(\bar{\Omega}) \forall \eta > 0 \le \eta < 1$. We denote by $\omega_i = \{x \in \Omega, v_i(x) > 0\}, w = v_1 - v_2$, and we introduce

$$h(x) = \begin{cases} 0 & \text{if } w(x) = 0 \\ \frac{v_{1+}(x) - v_{2+}(x)}{w(x)} & \text{if } w(x) \ne 0. \end{cases}$$

From this definition, $w \in H_0^1(\Omega)$ is invariant by $\tau$ and satisfies

$$(3.9) \qquad - \Delta w = \lambda h \cdot w \quad \text{a.e. } x \in \Omega.$$

The function $h$ is measurable, invariant by $\tau$ and

$$(3.10) \qquad 0 \le h(x) \le 1 \quad \text{a.e. } x \in \Omega.$$

Since $\bar{\omega}_1, \bar{\omega}_2 \subset \Omega$, we have

$$(3.11) \qquad h(x) < 1 \quad \text{a.e. on a neighbourhood of } \partial\Omega.$$

We denote by $\mu_i^*$ the eigenvalues corresponding to at least one eigenfunction $\varphi \in H_0^1(\Omega)$ invariant by $\tau$ of

$$- \Delta \varphi = \mu_i^* h \varphi.$$

From (3.10) (3.11), we can compare the eigenvalues $\lambda_i^*$ and $\mu_i^*$ (cf. Lemma 5.2)

$$\lambda_i^* < \mu_i^* \quad \forall i \ge 1.$$

By (3.9), $\lambda$ is an eigenvalue of $\mu_i^*$; since $\lambda \le \lambda_2^*$, we have

$$\lambda = \mu_i^*$$

and $w$ is a first eigenfunction.

Hence (cf. G. STAMPACCHIA [18] cor. 7.1)

$$w(x) > 0 \quad (\text{or } < 0) \quad \forall x \in \Omega,$$
which yields

$$ \bar{\omega}_2 \subset \omega_1 $$

The open set $\omega_1$ is not necessarily connected but, since $\omega_2 \neq \emptyset$, there must exist a connected component $\omega'_1$ of $\omega_1$ such that $\omega'_2 = \omega_2 \cap \omega'_1 \neq \emptyset$.

Hence

$$ (3.12) \quad \bar{\omega}'_2 \subset \omega'_1 \quad \text{and} \quad \omega'_1 \text{ is connected.} $$

Since $v_i \in C^1(\bar{\Omega})$ and $v_i = 0$ on $\partial\omega'_i$, it follows from lemma 2.1 that $v_i \in H_0^1(\omega'_i)$. From (1.3),

$$ - \Delta v_i' = \lambda v_i \quad \text{in } \mathcal{D}'(\omega_i'). $$

Thus $\lambda$ is an eigenvalue of the Dirichlet problem in both open sets $\omega'_1$ and $\omega'_2$. It is the first one because $v_i > 0$ in $\omega'_i$. Since the first eigenvalue of the Dirichlet problem in a connected open set is simple, (3.12) leads to a contradiction ($v = v_2$ in $\omega'_2$, $0$ in $\omega'_1 \setminus \omega'_2$ is also eigenfunction).

We have proved uniqueness for (1.3) and, by (3.6) (3.7), for (0.1).

The existence of solution for (0.1) or (1.3) invariant by $\tau$ can be proved by a classical fixed point method (cf. [15], §3).

**Lemma 3.3.** There exist constants $\varepsilon_0 > 0$, $r_0 > 1$, $0 < \underline{\lambda} < \lambda$, such that, for any domain $\Omega$ satisfying (1.1) (1.2) $\varepsilon_0, r_0$, we have (denoting by $\lambda_i^*$ the eigenvalues corresponding to at least one eigenfunction of the Dirichlet problem on $\Omega$ invariant by $\sigma$)

$$ i) \lambda_2 \le \underline{\lambda} < \bar{\lambda} \le \lambda_3, $$

$$ ii) \lambda_3 \le \lambda_2^*. $$

**Proof.** The proof is based on lemma 5.1 and 5.2 in the appendix. Let $\varepsilon_n \to 0$, $r_n \to 1$ be two decreasing sequences.
We consider the sequence of open sets $\Omega_n$

$$ \Omega_n = B_{r_n} \cup S_{\varepsilon_n} \cup \sigma(B_{r_n}) $$

and

$$ \Omega' = B_1 \cup \sigma(B_1), $$

which satisfy the hypotheses of lemma 5.1 in the appendix.

We denote by $\gamma_i^n$, $\gamma_i^{*n}$ the eigenvalues of the Dirichlet problem for $\Omega_n$, $\gamma_i$, $\gamma_i^{*}$, for $\Omega'$, We verify that

$$ \gamma_1 = \gamma_2 = \gamma_1^* \neq \gamma_3 = \gamma_4 = \gamma_2^*. $$

We choose $\underline{\lambda}$, $\bar{\lambda}$ such that

$$ \underline{\lambda} = \gamma_2 < \bar{\lambda} < \gamma_3. $$

By lemma 5.1,

$$ \gamma_3^n \rightarrow \gamma_3 \quad \text{for } n \rightarrow \infty, $$

$$ \gamma_2^{*n} \rightarrow \gamma_2^* \quad \text{for } n \rightarrow \infty, $$

Hence there exists $n_o$ such that, for $n \ge n_o$,

$$ \bar{\lambda} \le \gamma_3^n, \quad \underline{\lambda} \le \gamma_2^{*n}. $$

If $\Omega$ satisfies (2.3) (2.4)$_{\varepsilon_n r_n}$, then we have

$$ \Omega' \subset \Omega \subset \Omega_n, $$

hence

$$ \lambda_2 \le \underline{\lambda} < \bar{\lambda} \le \lambda_3^*. $$

and

$$ \bar{\lambda} \le \lambda_2^*. $$

Since $\lambda_2^*$ is an eigenvalue $\lambda_i$, ii) follows.

<figure><img src="image_3.png" /></figure>
### 3.3. A functional form of the equation.

Let us suppose that $\Omega$ is a bounded smooth domain of $\mathbb{R}^2$. For a given real number $\lambda > \lambda_1$, the solutions of (1.3) are the fixed points of the operator $T(\lambda,.)$ defined from $H^2(\Omega)$ into itself by $T(\lambda,v) = w$, where $w$ is the unique $w \in H^1(\Omega)$ such that

$$ (3.13) \quad \begin{cases} -\Delta w = \lambda v_+ & \text{a.e. in } \Omega, \\ w = -1 & \text{a.e. on } \partial\Omega. \end{cases} $$

From classical results of regularity, $w \in H^3(\Omega)$ and

$$ (3.14) \quad T : (\lambda_1, +\infty) \times H^2(\Omega) \longrightarrow H^2(\Omega) \text{ is compact.} $$

Let us suppose that $v^*$ is a solution of (1.3).

D. KINDERLEHRER and J. SPRUCK [7] have proved that the points $x \in \partial\Omega_p$ where $\nabla v^*(x) = 0$, are isolated and that elsewhere $\partial\Omega$ is locally a $C^\infty$ manifold. Hence

$$ (3.15) \quad \text{meas} \left\{ x \in \Omega \mid v^*(x) = 0 \right\} = 0. $$

From (3.15) (cf. R. TEMAM.[19]), the application

$$ \begin{aligned} & H^2(\Omega) \longrightarrow L^2(\Omega) \\ & v \longmapsto v_+ \end{aligned} $$

is Frechet differentiable in $v^*$, with differential

$$ \varphi \longmapsto H(v^*) \varphi, $$

where $H$ is the Heaviside function (we remark that $H(v^*)$ is defined almost everywhere by virtue of (3.15)).

By the chain rule, the operator $T(\lambda,.)$ from $H^2(\Omega)$ into itself is Frechet differentiable in $v^*$, and its differential is the operator $A$ defined from $H^2(\Omega)$ into itself by $A\varphi = \Psi$, where $\Psi$ is the unique function in $H_0^1(\Omega)$ such that

$$ (3.16) \quad -\Delta\Psi = \lambda H(v^*)\varphi \quad \text{a.e. in } \Omega. $$
The operator $\tilde{A}$, the restriction of $A$ to $H_0^1(\Omega)$, is self-adjoint in $H_0^1(\Omega)$ for the inner product $(\cdot, \cdot)$. Thus, except zero, all the eigenvalues of $\tilde{A}$ are algebraically simple. Consequently, except zero, all the eigenvalues of $A$ are algebraically simple.

Let us introduce

$\gamma_i$, the eigenvalues of the Dirichlet problem in $\Omega_p = \{x \in \Omega | v^*(x) > 0\}$,
$\mu_i$, the eigenvalues of the following problem : to find $\varphi \in H_0^1(\Omega)$ such that,
$\varphi \neq 0$ and

$$ (3.17) \qquad - \Delta \varphi = \mu H(v^*) \varphi \quad \text{a.e. in } \Omega. $$

Obviously, the characteristic values of the operator $A$ are the reals $\lambda/\mu_i$, $i \ge 1$. It follows (cf. lemma 5.2) from $0 \le H(v^*) \le 1$ and from $H(v^*) < 1$ a.e. in a neighbourhood of $\partial\Omega$ that $\lambda_i < \mu_i$. $\forall i \ge 1$. Since the restriction of $H$ to $\Omega_p$ is $1$, it follows (cf. lemma 5.2) from $\bar{\Omega}_p \subset \Omega$ that $\mu_i < \gamma_i$. $\forall i \ge 1$.

We have proved the important inequality :

$$ (3.18) \qquad \lambda_i < \mu_i < \gamma_i \quad \forall i \ge 1, $$

where the eigenvalues are repeated with their geometrical multiplicity.

### 3.4. Computation of a topological degree.

At first, let us suppose that $\Omega$ satisfies (1.1) (1.2) where $\varepsilon_1, r_1$ are given by lemma 3.3. Under this condition, parts i) ii) of our bifurcation theorem are proved and

$$ \lambda_1 \le \lambda < \bar{\lambda}. $$

From results of D.G. SCHAEFFER [14], there exist $0 < \varepsilon_0 \le \varepsilon_1$, $1 < r_0 \le r_1$ such that, for any $\Omega$ satisfying (1.1) (1.2) where $\varepsilon_0, r_0$, if $(\lambda, v)$ is a symmetrical solution of (1.3) with $\lambda \ge \bar{\lambda}$ and $v \ne -1$, then $\Omega_p$ is disconnected.
Let us suppose finally that Ω is a given domain satisfying (1.1) (1.2)$_{ε_o,r_o}$.

We denote (cf. Figure 2)

$$
X = [\lambda_2, \lambda_3] \times H^2(\Omega),
$$

$$
\mathcal{S} = \{(\lambda,\nu) \in X, (\lambda,\nu) \text{ solution of (1.3)}\}
$$

$$
S_1 = \{ (\lambda, \nu) \in S, \nu \neq -1, \nu \text{ symmetrical function} \}
$$

$$
S_2 = S \setminus S_1.
$$

<figure><img src="image_4.png" /><figcaption>Figure 2</figcaption></figure>

In order to prove the theorem by contradiction, let us suppose that there
is no bifurcation of non-symmetric solutions on the branch $S_1$.

Under this hypothesis, we now show that $S_1$ and $S_2$ are compact subsets of X.

If, for $n \to \infty$, we have

$$
(\alpha_n, v_n) \in S_1 \rightarrow (\alpha, v) \text{ in } X,
$$
then, by the Sobolev imbedding theorem,

$$v_n \longrightarrow v \text{ in } C^\circ(\bar{\Omega}).$$

Thus $v \neq -1$ follows from $v_{n+} \neq 0$. Hence, $S_1$ is a closed subset of $S$ which is a compact subset of $X$ by lemma 3.1. Therefore, $S_1$ is a compact subset of $X$.

Obviously, no sequence $(\alpha_n, v_n)$ with $v_n \equiv -1$ can converge to some $(\alpha, v) \in S_1$. Thus, there is no bifurcation on the branch $S_1$, i.e. for every $\lambda_2 \le \lambda \le \lambda_3$, there exists an open neighbourhood $V_\lambda$ of $(\lambda, v) \in S_1$ in $X$ such that

$$V_{\lambda} \cap S_{2} = \emptyset.$$

We set $O = \bigcup_{\lambda_2 \le \lambda \le \lambda_3} V_\lambda$. Since $O \cap S = S_1$, $S_1$ is an open subset of $S$ and $S_2$ is a closed subset of $S$; hence $S_2$ is also a compact subset of $X$. ..

Thus the distance between $S_1$ and $S_2$ is positive and there exists an open subset $O'$ of $X$ such that

$$(3.19) \quad O' \cap S = S_1, \quad \partial O' \cap S = \emptyset.$$

We can now introduce, by virtue of (3.14) (3.19),

$$O'_{\lambda} = \{v \in H^2(O) \text{ such that } (\lambda, v) \in O'\}$$

and the topological degree of LERAY-SCHLAUDER (cf. [12])

$$d(\lambda) = d(I-T(\lambda, \cdot), O'_{\lambda}, 0).$$

For $\lambda$ given in $[\lambda_2, \lambda_3]$, the equation

$$v \in O'_{\lambda} \text{ such that } T(\lambda, v) = v$$

has exactly one solution, denoted $v^*$, and the operator $T(\lambda, \cdot)$ is differentiable in $v^*$, of differential A (cf. § 3.3). If I-A is injective, then the degree $d(\lambda)$ is equal to the index of the operator A(cf. [12]), which is given by
$$d(\lambda) = (-1)^{\beta}$$

where $\beta$ is the number of distinct characteristic values of A located in (0.1) (by making use of the simplicity of the characteristic values).

### a) Computation of $d(\lambda_2)$.

From (3.18) and from $\lambda = \gamma_1$ (cf. proof of lemma 3.2),

$$\lambda_1 < \mu_1 < \lambda = \gamma_1,$$

$$\lambda = \lambda_2 < \mu_2 < \gamma_2.$$

Hence I-A is injective and $\beta = 1$. Therefore $d(\lambda_2) = -1$.

### b) Computation of $d(\lambda_3)$.

For $\lambda = \lambda_3$, there exist two disjoint open sets $\omega_i$ such that $\Omega_p = \omega_1 \cup \omega_2$. Then, we have $\gamma_1 = \gamma_2$. Indeed, since $v^* \in C^1(\bar{\Omega})$ and $v^* = 0$ on $\partial\omega_i$, it follows that $v^*|_{\omega_i} \in H_0^1(\omega_i)$ (by lemma 2.1); hence both functions $v^*|_{\omega_i}$, $i = 1,2$, are first eigenfunctions of the Dirichlet problem in $\Omega_p$.

From (3.18)

$$\lambda_1 < \mu_1 < \lambda = \gamma_1,$$

$$\lambda_2 < \mu_2 < \lambda = \gamma_2,$$

$$\lambda_3 = \lambda < \mu_3 < \gamma_3.$$

From these inequalities, it follows that I-A is injective ($\mu_2 < \lambda < \mu_3$) and it follows that $0 < \mu_1 \le \mu_2 < \lambda < \mu_3$. But, the first eigenvalue $\mu_1$ is necessarily geometrically simple, i.e. $\mu_1 \ne \mu_2$. Hence $\beta = 2$ and $d(\lambda_3) = 1$.

### c) Conclusion.

The result $d(\lambda_2) \ne d(\lambda_3)$ is in contradiction with the invariance of the degree by generalized homotopy (cf. [12]). Therefore, there exists a symmetrical solution $(\alpha, \nu)$ of (1.3) with $\lambda_2 \le \alpha \le \lambda_3$, $\nu \ne -1$, and a sequence of non symmetrical solutions $(\alpha_n, \nu_n)$ of (1.3) with $\nu_n \ne -1$ such that
$$ \lim_{n \to \infty} \alpha_n = \alpha, \quad \lim_{n \to \infty} v_n = v \quad \text{in } H^2(\Omega) $$

It can be proved<fnref n="1" /> that a necessary condition of bifurcation is that 1 is eigenvalue of (3.17). We proved in a) and b) that it is not the case for $\lambda = \lambda_2$ or $\lambda = \lambda_3$. therefore, the bifurcation occurs with $\lambda_2 < \alpha < \lambda_3$.

Considering the sequence of non-symmetric solutions of (0.1)

$$ u_n = \left( \alpha_n \int_{\Omega} v_{n+1} \, dx \right)^{-1} v_n, $$

it is easy to show that

$$ (\alpha_n, u_n) \underset{n \to \infty}{\longrightarrow} (\alpha, u) \quad \text{in } \mathbb{R} \times H^2(\Omega) \text{ for } n \to \infty, $$

where $(\alpha, u)$ is the symmetrical solution of (0.1)

$$ u = \left( \alpha \int_{\Omega} v_+ \, dx \right)^{-1} v. $$

The proof of the bifurcation theorem is complete.

**Remark 3.1**. Let us suppose that $\Omega$ is a bounded smooth domain of $\mathbb{R}^q$ and that $(\lambda, u)$ is a solution of (0.1). Then $\Omega_p$ has less than $k$ connected components where $k$ is the smallest integer such that $\lambda \le \lambda_k$. Indeed, if $\Omega_p$ has $k$ connected components, then $\lambda = \gamma_1 = \dots = \gamma_k$ and, by $\lambda_i < \gamma_i$, $\lambda > \lambda_k$.

**Remark 3.2.** Our result of bifurcation can also be stated as follows : if $\Omega$ is a bounded smooth symmetrical domain of $\mathbb{R}^2$ such that

$$ \text{a)} \quad \lambda_3 \le \lambda_2^*, $$

b) the set $\Omega_p$, corresponding to the symmetrical solution for $\lambda = \lambda_3$, is disconnected,

then

i) the eigenvalue $\lambda_2$ is simple

ii) there exists at least one bifurcation point in $[\lambda_2, \lambda_3] \times H^2(\Omega)$ of (0.1) on the branch of symmetrical solutions.

<footnote n="1">¹ unpublished result of R. TEMAM.</footnote>
4. EXTENSION TO GENERAL GROUPS OF ISOMETRIES.

<figure><img src="image_5.png" /><figcaption>Fig.3. Number of balls n = 4.</figcaption></figure>

We now rapidly discuss the case of domains Ω_ε(n) as in Figure 3. For such domains, the eigenvalues of the Dirichlet problem satisfy, for ε → 0,

$$ \lambda_1 \sim \lambda_2 \dots \sim \lambda_n \ll \lambda_{n+1} \sim \dots \sim \lambda_{2n} $$

and the disconnectedness of the plasma invariant by rotation will occur only for $\lambda \ge \lambda_{n+1}$. The previous methods are not directly applicable, since the computation of $d(\lambda_{n+1})$ requires us to know if $\mu_1, \dots, \mu_n$ are distinct or not.

In order to prove bifurcation in such domains, we shall use the following method. We suppose that the domain considered, Ω, is invariant under some group of isometries G and that there exists a subgroup H of G having the properties i), ii) below.

We denote by $\lambda_i^H$ (resp $\lambda_i^G$) the eigenvalues of the Dirichlet problem on Ω corresponding to at least one eigenfunction invariant under H(resp. under G). We suppose that
i) $\lambda_3^H \le \lambda_2^G$

ii) for $\lambda = \lambda_3^H$, the solution of (0.1) invariant under G is such that

$$ \Omega_p = \omega_1 \cup \omega_2 $$

where $\omega_i$ are disjoint open subsets of $\Omega$, invariant under H.

Under these hypotheses, by using the method of the § 3 to the problem :

to find $u$ invariant under H solution of (0.1),
it is easy to prove that $\lambda_2^H \ne \lambda_3^H$ and that there exists at least one point of
bifurcation of (0.1) on the branch of solutions invariant under G.

Let us now come back to the situation of the Figure 3. Let $P_n$ be a regular
polyhedron of $n$ sides (lenght a), whose vertices are denoted $S_n = \{M_1, \dots, M_n\}$.
We consider a domain $\Omega_\varepsilon(n)$ consisting of the $n$ balls $B(M_i, r)$, with $2r < a$,
connected by $n$ strips of thickness $\varepsilon$.

We choose for H a group (if there exists) of isometries conserving $S_n$ and
such that, under H, $S_n$ is divided into two orbits. The existence of such a group
depends on $n$.

On one hand, for every even number ($n = 2m$), we can take

$$ G = \{ \text{rotations of angle } \frac{2k\pi}{n}, k = 1, \dots, n \} $$

$$ H = \{ \text{rotations of angle } \frac{2k\pi}{m}, k = 1, \dots, m \}. $$

On the other hand, for some odd numbers $n$ (for instance $n = 5$), we can prove
that such a group does not exist.

In conclusion, if $n$ is any even number or an odd number for which there
exists H as above, then, for $\varepsilon$ small enough, there exists at least one
bifurcation point of (0.1) on the branch of solutions invariant under a rota-
tion of angle $\frac{2\pi}{n}$.
5. APPENDIX.

In this appendix are proved two properties of eigenvalues of elliptic problems with discontinuous coefficients.

Let $\Omega$ be a bounded domain in $\mathbb{R}^q$, $q \ge 1$, $a_{ij}$ and $\rho$ be functions in $L^\infty(\Omega)$ such that

$$
(5.1) \quad a_{ij}(x) = a_{ji}(x) \quad \text{a.e. } x \in \Omega,
$$

$$
(5.2) \quad \left\{
\begin{array}{@{}l@{}}
\text{there exists } \alpha > 0 \text{ such that} \\
\displaystyle \sum a_{ij}(x) \xi_i \xi_j \ge \alpha |\xi|^2 \quad \text{a.e. } x \in \Omega, \quad \forall \xi \in \mathbb{R}^q,
\end{array}
\right.
$$

$$
(5.3) \quad \rho(x) \ge 0 \quad \text{a.e. } x \in \Omega,
$$

$$
(5.4) \quad \begin{array}{l} \text{there exists a non empty subset } \omega \text{ of } \Omega \text{ such that} \\ \rho(x) > 0 \quad \text{a.e. } x \in \omega. \end{array}
$$

We call eigenvalues the real numbers $\lambda$ corresponding to a function $u \neq 0 \in H_0^1(\Omega)$ such that

$$
(5.5) \quad \sum_{i,j=1}^{n} \frac{\partial}{\partial x_j} \left( a_{ij} \frac{\partial u}{\partial x_i} \right) = \lambda \rho u \quad \text{in } \mathcal{D}'(\Omega).
$$

The following result seems new : this is the convergence of eigenvalues
with respect to a decreasing sequence of domains, the Hausdorff convergence of
the domains not being required. For the proof, we follow a method of compact-
ness of O. PIRONNEAU-C. SAGUEZ [10].

**Lemma 5.1.** If $(\Omega_n)$ is a sequence of bounded open subsets of $\mathbb{R}^q$ satisfying

i) $\Omega_{n+1} \subset \Omega_n \quad \forall n \ge 1,$

ii) meas $\{ \partial (\cap_{n \ge 1} \Omega_n) \} = 0,$

iii) $\Omega = \text{int}(\cap_{n \ge 1} \Omega_n) \neq \emptyset,$

iv) $\partial\Omega$ is a $C^1$ manifold of $\mathbb{R}^q$ of dimension $q-1$, $\Omega$ is locally on only one side of $\partial\Omega,$
if $a_{ij}$, $\rho$ are functions of $L^\infty(\Omega_1)$ satisfying (5.1)-(5.3) on $\Omega_1$ and (5.4) on $\Omega$,

then the increasing sequences $(\lambda_k^n)_{k \ge 1}$ of eigenvalues of (5.5) in $\Omega_n$ satisfy

$$ \lim_{n \to \infty} \lambda_k^n = \lambda_k \qquad \forall k \ge 1, $$

where $(\lambda_k)$ is the increasing sequence of eigenvalues of (5.5) in $\Omega$.

**Proof.** The inner products in $H_0^1(\Omega_n)$, $H_0^1(\Omega)$ are

$$ ((u,v))_n = \int_{\Omega_n} \sum_{i,j=1}^{q} a_{ij}(x) \frac{\partial u}{\partial x_i} \frac{\partial v}{\partial x_j} dx, $$

$$ ((u,v)) = \int_{\Omega} \sum_{i,j=1}^{q} a_{ij}(x) \frac{\partial u}{\partial x_i} \frac{\partial v}{\partial x_j} dx. $$

For $u \in H_0^1(\Omega)$, we denote by $\tilde{u}$ the function equal to $u$ in $\Omega$, $0$ in $\Omega_1 \setminus \Omega$. It is known (cf. J.L. LIONS [8]) that $\tilde{u} \in H_0^1(\Omega_1)$ and that

$$ \nabla \tilde{u} = \nabla u \quad \text{in } \Omega, \ 0 \text{ in } \Omega_1 \setminus \Omega. $$

We denote by $(u_k^n)_{k \ge 1}$ an orthonormal system of eigenfunctions of (5.5) in $H_0^1(\Omega_n)$ corresponding to the sequence $(\lambda_k^n)$. Let us suppose, by induction that, for a fixed $\ell \ge 1$, there exists a subsequence $n' \to \infty$ and some eigenfunctions $(u_k)_{k=1}^{\ell-1}$ of (5.5) in $\Omega$ corresponding to $(\lambda_k)_{k=1}^{\ell-1}$ such that

$$ \tilde{u}_k^n \to \tilde{u}_k \quad \text{in } H_0^1(\Omega_1) \qquad \forall k < \ell. $$

Using the compactness of the imbedding of $H_0^1(\Omega_1)$ in $L^2(\Omega_1)$,

$$ \| |u_{\ell}^{n}| |_{n} = \| |\tilde{u}_{\ell}^{n}| |_{1} = 1 \qquad \forall n \ge 1 $$

implies the existence of a subsequence $n' \to \infty$ such that

$$ \tilde{u}_{\ell}^{n} \to v \quad \begin{cases} \text{in } H_0^1(\Omega_1) \text{ weak,} \\ \text{in } L^2(\Omega_1) \text{ strong,} \\ \text{a.e. in } \Omega_1. \end{cases} $$
By i),

$$v = 0 \quad \text{a.e. in } \Omega_1 \setminus \bigcap_{n \ge 1} \Omega_n,$$

and with ii),

$$v = 0 \quad \text{a.e. in } \Omega_1 \setminus \Omega.$$

From iv), the theorem of traces (cf. J.L. LIONS [8]) on $\partial\Omega$ can be used :

$$u = v|_{\Omega} \in H_0^1(\Omega).$$

It is now standard, by making use of

$$||u|| \le \lim_{n' \to \infty} ||u_n^{n'}|| \le 1$$

and

$$\lambda_{\ell}^{n} \le \lambda_{\ell}^{n+1} \le \dots \le \lambda_{\ell},$$

to show that $u$ is actually an eigenfunction of (5.5) in $\Omega$ corresponding to $\lambda_\ell$.

Hence

$$\lim_{n \to \infty} \lambda_{\ell}^{n} = \lambda_{\ell}$$

and

$$\lim_{n' \to \infty} \tilde{u}_{\ell}^{n'} = \tilde{u}_{\ell} \quad \text{in } H_0^1(\Omega_1),$$

which proves the lemma.

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

The monotonicity of eigenvalues with respect to the function $\rho$ and to the domain $\Omega$ is easy to establish, even in case of discontinuous coefficients (for methods of proof, cf. R. COURANT-D. HILBERT [3]). The strict monocity, stated in the lemma 5.2 below, is more difficult to obtain ; the main argument in the proof is the principle of unique continuation. The precise statement is perhaps known, but we were unable to find it in the literature.
**Lemma 5.2.** We suppose that $a_{ij} \in L^\infty(\Omega)$ satisfy (5.1)-(5.4) and

(5.6) $\Omega$ is connected, $\partial\Omega$ is a $C^2$ manifold of dimension $q-1$, $\Omega$ is locally on one side of $\partial\Omega$,

(5.7) $a_{ij} \in C^1(\bar{\Omega})$;

then

i) an eigenfunction of (5.5) cannot be a.e. equal to zero in a non empty open subset of $\Omega$,

ii) if $\Omega' \subset \Omega$ and $\text{int}(\Omega \setminus \Omega') \neq \emptyset$, then

$$ \lambda_n < \lambda_n' \quad \forall n \ge 1, $$

iii) if $0 \le \rho' \le \rho$ a.e. in $\Omega$ and $\rho' < \rho$ a.e. in a non empty open subset of $\Omega$, then

$$ \lambda_n < \lambda_n' \quad \forall n \ge 1. $$

*Proof.* i) Let $u \in H_0^1(\Omega)$ be an eigenfunction of (5.5). Since $\rho u \in L^2(\Omega)$, it follows from the regularity of solutions of the Dirichlet problem (cf. S. AGMON-A. DOUGLISS-L. NIRENBERG [1]) that $u \in H^2(\Omega)$. Since $a_{ij} \in C^1(\bar{\Omega})$, i) follows from the principle of unique continuation (cf. L. HÖRMANDER [6], th. 8.9.1).

ii) The part ii) follows from i) and from the following method of R. COURANT-D. HILBERT [3] : we suppose that the $k^{th}$ eigenvalue in $\Omega$ and $\Omega'$ are equal for some $k$ and we prove that no eigenvalue in $\Omega$ can be greater than $\lambda_k$, which is absurd.

At first, we introduce

$$ ((u,v)) = \int_{\Omega} \sum_{i,j=1}^{q} a_{ij}(x) \frac{\partial u}{\partial x_i} \frac{\partial v}{\partial x_j} dx, $$

$$ R(u) = \frac{\|u\|^2}{(\rho u, u)} \le + \infty \quad (\text{called the Rayleigh quotient}). $$
If $\Omega$, $a_{ij}$ and $\rho$ satisfy (5.1)-(5.4), it is easy to obtain the following maximum-minimum property of eigenvalues (cf. R. COURANT-D. HILBERT [3]): if $V$ is any set such that

$$H_0^1(\Omega) \subset V \subset H^1(\Omega),$$

then the eigenvalues $(\lambda_m)_{m \ge 1}$ of (5.5) are given by

$$(5.8) \quad \lambda_m = \sup_{v_1, \dots, v_{m-1} \in V} \left\{ \begin{array}{l} \inf_{u \in H_0^1(\Omega)} \text{R}(u) \quad \text{for } ((u, v_i)) = 0, i = 1, \dots, m-1 \\ u \ne 0 \end{array} \right\}.$$

By the hypothesis taken in ii), there exists a ball

$$\overline{B}(x_o, r_o) \subset \Omega \setminus \Omega'$$

We consider a sequence of open sets :

$$\Omega_n = \Omega \setminus B(x_o, \frac{r_o}{n}) \quad n \ge 1,$$

which satisfy (5.6) and

$$\Omega' \subset \Omega_1 \subset \dots \subset \Omega_n \subset \Omega_{n+1} \subset \dots \subset \Omega.$$

By monocity, the $k$-th eigenvalues on $\Omega_n$ are equal to $\lambda_k$. We denote by $u_n \in H_0^1(\Omega_n)$ the corresponding eigenfunctions, by $\tilde{u}_n$ the functions prolonged to $\Omega$ by 0 in $\Omega \setminus \Omega_n$.

Let $m$ be any fixed number. In order to prove that $\lambda_m \le \lambda_k$, it is sufficient (cf. (5.8)) to prove that for any $v_1, \dots, v_{m-1} \in H^1(\Omega)$, there exists $\varphi \in H_0^1(\Omega)$, $\varphi \ne 0$, such that

$$((\varphi, v_i)) = 0, \quad 1 \le i \le m-1, \quad \text{R}(\varphi) \le \lambda_k.$$

We choose

$$\varphi = \sum_{i=1}^{m} \gamma_i \tilde{u}_i.$$

The equations

$$((\varphi, v_i)) = 0, \quad 1 \le i \le m-1,$$

are $m-1$ homogeneous linear equations for $m$ unknowns $(\gamma_i)$, hence there exists a non trivial solution $(\tilde{\gamma}_i)$. By i), the system $(u_i)$ is free, hence $\varphi \ne 0$.
For $i \le j$, we have

$$((\tilde{u}_i, \tilde{u}_j)) = \lambda_k(\rho \tilde{u}_i, \tilde{u}_j),$$

because $\tilde{u}_i \in H_0^1(\Omega_j)$ and $u_j$ is an eigenfunction on $\Omega_j$.

Hence $R(\varphi) = \lambda_k$, which proves ii).

iii) We follow the method of proof of ii), but there are some supplementary technical difficulties.

By hypothesis, there exists a ball $B(x_o, r_o)$ such that

$$\rho'(x) < \rho(x) \quad \text{a.e. } x \in B(x_o, r_o).$$

We consider the increasing sequence of balls $B_n$:

$$B_n = B(x_o, r_o \cdot \left(1 - \frac{1}{2n}\right))$$

and the increasing sequence of functions $\rho_n \in L^\infty(\Omega)$:

$$\rho_n = \rho \text{ on } B_n, \rho' \text{ elsewhere.}$$

If the $k^{\text{th}}$ eigenvalues are equal for $\rho$ and $\rho'$, then, by monotonicity, that is the same for all $\rho_n$. We denote by $u_n$ the corresponding eigenfunctions.

Let $m$ be any fixed number and $v_1, \dots, v_{m-1} \in H^1(\Omega)$.

We choose $\varphi = \sum_i \gamma_i u_i$. If we can prove that $(u_i)$ is a free system and that $R(\varphi) \le \lambda_k$, then iii) will be proved.

a) $(u_i)$ is a free system. Since

$$(5.9) \qquad ((u_i, v)) = \lambda_k (\rho_i u_i, v) \qquad \forall v \in H_0^1(\Omega),$$

if $\sum_{i=1}^p \alpha_i u_i = 0$ for some reals $\alpha_i$, then

$$\sum_{i=1}^{p} \alpha_i \rho_i u_i = 0$$
and

$$ \sum_{i=1}^{p} \alpha_i (\rho_i - \rho') u_i = 0. $$

Hence

$$ \alpha_p u_p = 0 \quad \text{a.e. in } B_{p-1} \setminus B_p, $$

which implies $\alpha_p = 0$ because of i) ; hence a) by induction.

$$ \text{b) } R(\varphi) \le \lambda_k. $$

By (5.9) , we have

$$ (5.10) \quad ( (\rho_i - \rho_j) u_i, u_j ) = 0 \quad \forall i,j. $$

For any sequence $(\gamma_i)$, we introduce a function $\phi$ on $U = \bigcup_{n \ge 1} B_n$ by

$$ \phi(x) = \begin{cases} 0 & \text{on } B_1 \\ j & \text{on } B_{j+1} \\ \sum_{i=1}^{n} \gamma_i u_i & \text{on } B_{n+1} \setminus B_j. \end{cases} $$

By a computation on $B_{n+1} \setminus B_n$ using (5.10), one proves by induction on $n$ that

$$ (5.11) \quad \sum_{i,j=1}^{n} \gamma_i \gamma_j \int_{B_n} (\rho - \rho_i) u_i \cdot u_j \,dx = \int_{B_n} (\rho - \rho') \phi^2(x) \,dx. $$

For any $m$ and $\varphi = \sum_{i=1}^{m} \gamma_i u_i$, we have

$$ \begin{aligned} \lambda_k(\rho \varphi, \varphi) - ||\varphi||^2 &= \lambda_k \left\{ \sum_{i,j=1}^{m} \gamma_i \gamma_j (\rho u_i, u_j) - \sum_{i,j=1}^{m} \gamma_i \gamma_j (\rho_i u_i, u_j) \right\} \\ &= \lambda_k \left\{ \int_{\Omega \setminus B_m} (\rho - \rho') \phi^2 \,dx + \sum_{i,j=1}^{m} \gamma_i \gamma_j \int_{B_m} (\rho - \rho_i) u_i u_j \,dx \right\} \end{aligned} $$

which is not less than zero by (5.11). Hence $R(\varphi) \le \lambda_k$.

<figure><img src="image_7.png" /></figure>
Remark 5.1. Part ii) and iii) of lemma 5.2 could also be proved by making use of differential properties of eigenvalues with respect to the coefficients. (cf. P.R. GARABEDIAN-M. SCHIFFER [5], B. ROUSSELET [13]).

R E F E R E N C E S.

[1] S. AGMON, A. DOUGLIS, L. NIRENBERG, "Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions", Comm. Pure Applied Math. 12, (1959) 623-727.

[2] H. BERESTYCKI and H. BREZIS, "On a free boundary problem arising in plasma physics", lecture notes 78017, University of Paris, (1978).

[3] R. COURANT and D. HILBERT, "Methods of Mathematical Physics", New-York, Interscience Publishers, (1953).

[4] A. DERVIEUX, Rapport LABORIA, to appear.

[5] P.R. GARABEDIAN and M. SCHIFFER, "Variational problems in the theory of elliptic partial differential equations, J. of Rat. Mech. and Analys. (1953), 137-171.

[6] L. HORMANDER, "Linear Partial Differential Operators", Springer Verlag, New-York (1963).

[7] D. KINDERLEHRER and J. SPRUCK, "The shape and smoothness of stable plasma configurations", Annali. Scu. Norm. Pisa, 5 (1978), 131-148.

[8] J.L. LIONS, "Problèmes aux limites dans les équations aux dérivées partielles", Presses de l'Université de Montréal, sec. ed. 1965.

[9] B. NICOLAENKO and R. TEMAM, to appear.

[10] O. PIRONNEAU and C. SAGUEZ, "Asymptotic behavior, with respect to the domain, of solutions of partial differential equations", Rapport LABORIA 218 (1977).
[11] J.P. PUEL, "Un problème de valeurs propres non linéaires et de frontière libre", lecture notes 76016, University of Paris, (1976)

[12] P.H. RABINOWITZ, "Théorie du degré topologique et application à des problèmes aux limites non linéaires", lecture notes 75010, University of Paris, 1975.

[13] B. ROUSSELET, "Identification de domaines et problèmes de valeurs propres", thèse de 3ème cycle, Université de Nice, 1977.

[14] D.G. SCHAEFFER, "Non uniqueness in the equilibrium shape of a confined plasma", Comm. in P.D.E., 2 (6), (1977), 587-600.

[15] M. SERMANGE, "Une méthode numérique en bifurcation, Application à un problème à frontière libre de la physique des plasmas", Applied Math. and Optim., 5, (1979), 127-151.

[16] J. SIJBRAND, "Computation of the bifurcation solutions of a class of problems with a free boundary", lecture notes 90, University of Utrecht, (1978).

[17] J. SIJBRAND, "Bifurcation analysis for a class of problems with a free boundary", lecture notes 91, University of Utrecht, (1978).

[18] G. STAMPACCHIA, "Équations Elliptiques du second ordre à coefficients discontinus", Presse de l'Université de Montréal, 1962.

[19] R. TEMAM, "Remarks on a free boundary value problem arising in plasma physics, Comm. in P.D.E., 2 (6), (1977), 563-585.