ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 10, Pages 135–141 (December 10, 2004) S 1079-6762(04)00138-6

COMPACTNESS AND GLOBAL ESTIMATES FOR THE GEOMETRIC PANEITZ EQUATION IN HIGH DIMENSIONS

EMMANUEL HEBEY AND FRÉDÉRIC ROBERT

(Communicated by Tobias Colding)

Abstract. Given (M, g), a smooth compact Riemannian manifold of dimension n ≥ 5, we investigate compactness for the fourth order geometric equation P_{g}u = u^{2^{♯}−1}, where P_{g} is the Paneitz operator, and 2^{♯} = 2n/(n − 4) is critical from the Sobolev viewpoint. We prove that the equation is compact when the Paneitz operator is of strong positive type.

In 1983, Paneitz [9] introduced a conformally fourth order operator defined on 4-dimensional Riemannian manifolds. Branson [3] generalized the definition to n-dimensional Riemannian manifolds, n ≥ 5. While the conformal Laplacian is associated to the scalar curvature, the geometric Paneitz-Branson operator is associated to a notion of Q-curvature. The Q-curvature in dimension 4, and for conformally flat manifolds, is the integrand in the Gauss-Bonnet formula for the Euler characteristic. In this article we let (M, g) be a smooth compact conformally flat Riemannian n-manifold, n ≥ 5, and consider the geometric Paneitz equation

(0.1)  P_{g}u = u^{2^{♯}−1} ,

where P_{g} is the Paneitz operator in dimension n ≥ 5, u is required to be positive, and 2^{♯} = \frac{2n}{n−4} is the critical exponent for the Sobolev embedding. The Paneitz operator in dimension n ≥ 5 reads as

P_{g}u = Δ_{g}^{2}u − div_{g} (A_{g}du) + \frac{n − 4}{2}Q_{g}u ,

where Δ_{g} = −div_{g}∇ is the Laplace-Beltrami operator, Q_{g} is the Q-curvature of g, A_{g} is the smooth symmetrical (2, 0)-tensor field given by

A_{g} = \frac{(n − 2)^{2} + 4}{2(n − 1)(n − 2)}S_{g}g − \frac{4}{n − 2}Rc_{g} ,

and Rc_{g} and S_{g} are respectively the Ricci curvature and scalar curvature of g. The Paneitz operator is conformally invariant in the sense that if g̃ = u^{4/(n−4)}g is conformal to g, then Pg̃(f) = u^{s}P_{g}(uf) for all f ∈ C^{∞}(M), where s = 1−2^{♯}. From the viewpoint of conformal geometry, equation (0.1) turns out to be the natural fourth order analogue of the second order Yamabe equation. We refer to Chang [4] and Chang and Yang [5] for more details on the above definitions.

\begin{matrix}c \\ ◯ \end{matrix}2004 American Mathematical Society

Reverts to public domain 28 years from publication

Received by the editors October 7, 2004.

2000 Mathematics Subject Classification. Primary: 58E30, 58J05.

Key words and phrases. Blow-up behavior, compactness, Paneitz operator.

In what follows we let H_{2}^{2}(M) be the Sobolev space consisting of functions in L^{2}(M) with two derivatives in L^{2}. As shown in Hebey and Robert [7], up to passing to a subsequence, bounded sequences (u_{α}) in H_{2}^{2}(M) of nonnegative solutions of (0.1) split into the sum of a nonnegative solution u^{0}, namely the weak limit of the u_{α}, a finite sum of k bubbles (B_{α}^{i}), obtained by rescaling positive solutions of the Euclidean equation Δ^{2}u = u^{2^{♯}−1}, and a remainder R_{α} which converges strongly to zero in H_{2}^{2}(M) as α → +∞. This splitting provides exact asymptotics for the u_{α} in the Sobolev setting. Following standard terminology, we say that equation (0.1) is compact if for any bounded sequence (u_{α}) in H_{2}^{2}(M) of nonnegative solutions of (0.1), we necessarily have that k = 0 in such decompositions. Regularity theory (see for instance Esposito-Robert [6]) holds for (0.1). Then, thanks to Agmon-Douglis-Nirenberg-type estimates [1, 2], and estimates like the ones developed in Hebey, Robert, and Wen [8], an equivalent definition for compactness is that bounded sequences in H_{2}^{2}(M) of nonnegative solutions of (0.1) are actually bounded in C^{4,θ}(M). A major stress in studying compactness is to understand large solutions. Namely, solutions with large energies which, in studying their possible blow-up, involve multi-bubbles which may interact with each other on the pointwise level.

Compactness for fourth order equations like (0.1) was recently studied in Hebey, Robert, and Wen [8]. It was shown in [8] that equations like (0.1) are compact as long as they are not close to the geometric equation (0.1). In this note we investigate compactness for the geometric equation (0.1) and show how the analysis developed in [8] provides an answer to the question of whether (0.1) is compact or not. The result we get is the fourth order analogue of the result proved in Schoen [10], where the second order Yamabe equation was investigated. As in Hebey, Robert, and Wen [8], and also Schoen [10], we assume in what follows that (M, g) is conformally flat (and hence, since n ≥ 5, that the Weyl tensor of g is zero). We let G_{g} be the Green’s function of P_{g}. The Green’s function is unique if P_{g} is positive. By conformal invariance of the Paneitz operator, if g̃ = u^{4/(n−4)}g is a conformal metric to g, then

(0.2)  Gg̃(x, y) = \frac{G_{g}(x, y)}{u(x)u(y)} .

It is known that if g̃ is a flat metric around some x_{0} ∈ M, then

(0.3)  Gg̃(x_{0}, x) = \frac{λ_{n}}{dg̃(x_{0}, x)^{n−4}} + μg̃(x_{0}, x) ,

where λ_{n}^{−1} = 2(n−2)(n−4)ω_{n−1}, ω_{n−1} is the volume of the unit (n−1)-sphere, and the function x → μg̃(x_{0}, x) is smooth on M. Combining the above two equations, noting that conformal changes of metrics which leave a metric flat around one point come from conformal diffeomorphisms of the Euclidean space, we easily get (as in Schoen and Yau [11] for the conformal Laplacian) that if g and g̃ = u^{4/(n−4)}g are conformal metrics, both being flat around x_{0}, then

(0.4)  μg̃(x_{0}, x_{0}) = \frac{μ_{g}(x_{0}, x_{0})}{u(x_{0})^{2}} .

In particular, by (0.4), the sign of μ_{g}(x_{0}) = μ_{g}(x_{0}, x_{0}) does not depend on the choice of the metric in [g]_{x_{0}}, where [g]_{x_{0}} stands for the set of conformal metrics to g which are flat around x_{0}.

In what follows we say that P_{g} is of strong positive type if P_{g} is positive, G_{g} is positive, and for any x ∈ M there exists g̃ ∈ [g]_{x} such that μg̃(x) > 0. For example, the Paneitz operator on quotients of the unit sphere is of strong positive type. Positivity of the Paneitz operator was studied in Xu and Yang [12]. The main result of this note is:

Theorem 0.1. The geometric equation (0.1) is compact on compact conformally flat manifolds of dimensions n ≥ 5 with Paneitz operator of strong positive type.

Let (M, g) be a smooth compact conformally flat manifold of dimension n ≥ 5 with positive Paneitz operator P_{g}, and positive Green’s function G_{g}. Given S ⊂ M, let [g]_{S} be the set of conformal metrics to g which are flat in a neighborhood of S. We prove Theorem 0.1 by proving that if (u_{α}) is a bounded sequence in H_{2}^{2}(M) of nonnegative solutions of (0.1) which blows up with geometric blow-up points S = {x_{1}, …, x_{N}}, then u_{α} ⇀ 0 in H_{2}^{2}(M) as α → +∞, and for any g̃ ∈ [g]_{S}, there exist λ_{i,j} > 0 such that for any i = 1, …, N,

(0.5)  μg̃(x_{i}) + ∑_{j≠i} λ_{i,j}Gg̃(x_{i}, x_{j}) = 0 .

Theorem 0.1 clearly follows from (0.5). We prove equation (0.5) in the rest of this note, using the material proved in Hebey, Robert, and Wen [8].

1. Proof of the result

In what follows we prove (0.5), and thus Theorem 0.1. For that purpose we let (u_{α}) be a bounded sequence in H_{2}^{2}(M) of nonnegative nontrivial solutions of (0.1). Since the Green’s function G_{g} of P_{g} is positive, the u_{α} are positive. In the sequel, everything is up to a subsequence. We know from Hebey and Robert [7] that there exist k ∈ ℕ, a nonnegative solution u^{0} ≥ 0 of (0.1), and k bubbles (B_{α}^{i}), i = 1, …, k, such that

(1.1)  u_{α} = u^{0} + ∑_{i=1}^{k} B_{α}^{i} + R_{α} ,

where R_{α} → 0 in H_{2}^{2}(M) as α → +∞. By contradiction, we assume that k ≥ 1, and let S = {x_{1}, …, x_{N}} be the geometric blow-up point set consisting of the limits of the centers of the bubbles (B_{α}^{i}). Since bubbles may accumulate one on another (there are such examples for equations like (0.1); we refer to Hebey, Robert, and Wen [8]), N might be less than k. By conformal invariance we may assume that g = g̃ is flat around the points in S. Then the geometric Paneitz equation (0.1) reduces to Δ_{g}^{2}u_{α} = u_{α}^{2^{♯}−1} around the points in S.

As a preliminary step in the proof of (0.5), we come back to the estimates proved in Hebey, Robert, and Wen [8] and explain why they are still valid in the present context. A rough argument would be that blow-up phenomena are local in nature, while the Paneitz operator on conformally flat manifolds is locally, up to conformal changes of the metric, like the Paneitz operator on the sphere (and hence with positive constant coefficients as in [8]). More details follow. First we note that the standard procedure to get rescaling invariant pointwise estimates, as developed in [8] for fourth order operators, together with Agmon-Douglis-Nirenberg-type estimates for fourth order operators, give that there exists C > 0 such that,

(1.2)  ( min_{1≤i≤k} d_{g}(x_{α}^{i}, x))^{\frac{n−4}{2}} |u_{α}(x) − u^{0}(x)| ≤ C

for all α and all x, where u^{0}, the weak limit of the u_{α}, is as in (1.1), and where the x_{α}^{i}, i = 1, …, k, are the centers of the bubbles (B_{α}^{i}) in (1.1). In particular, we get with (1.2) that u_{α} → u^{0} in C_{loc}^{4}(M\S) as α → +∞. In addition to (1.2), if Φ_{α}(x) stands for the left hand side in (1.2), we also have that

(1.3)  lim_{R→+∞} lim_{α→+∞} sup_{x∈M\Ω_{α}(R)} Φ_{α}(x)^{\frac{n−4}{2}} = 0 ,

where, for R > 0, Ω_{α}(R) is given by Ω_{α}(R) = ⋃_{i=1}^{k} B_{x_{α}^{i}}(Rμ_{α}^{i}), and the μ_{α}^{i} are the weights of the bubbles (B_{α}^{i}) in (1.2). Going on with the estimates in Hebey, Robert, and Wen [8] we may assume, up to renumbering, and up to passing to a subsequence, that (B_{α}^{1}) is the bubble in (1.1) with the largest weight. Then we let the x_{α} and μ_{α} be such that x_{α} = x_{α}^{1} and μ_{α} = μ_{α}^{1} for all α. A preliminary remark is that the global splitting estimate ∥u_{α}∥_{p_{1},p_{2},μ_{α}^{−1}} ≤ C in [8] easily follows from the positivity of the Green’s function with only slight modifications of the arguments in [8]. The arguments in [8] used the decomposition of the fourth order operator into the product of two second order operators. We may use instead Agmon-Douglis-Nirenberg-type estimates and note that the positivity of the Green’s function G_{g} implies that P_{g} satisfies the comparison principle. As an independent easy remark, since Δ_{g}(Δ_{g}u_{α}) ≥ 0 around the points in S, there exists C > 0 such that Δ_{g}u_{α} ≥ −C in M, for all α. Then, with such an estimate, and the analysis developed in [8], we easily get that the integral and asymptotic estimates in Hebey, Robert, and Wen [8] follow from (1.1)–(1.3). In several places the computations in [8] simplify because of the simple nature of the geometric equation around the points in S. By the asymptotic estimates, if we let ũ_{α} be the rescaled function obtained from u_{α} by ũ_{α}(x) = u_{α} (exp_{x_{α}}(\sqrt{μ_{α}}x)), we get that there exist δ > 0, A > 0, and a biharmonic function φ ∈ C^{4} (B_{0}(2δ)) such that, up to a subsequence,

(1.4)  ũ_{α}(x) → \frac{A}{|x|^{n−4}} + φ(x)

in C_{loc}^{3} (B_{0}(2δ)\{0}) as α → +∞. Moreover, since we assumed that G_{g} > 0, so that either u^{0} ≡ 0 or u^{0} > 0 everywhere, we also have the important information that φ is positive in B_{0}(2δ) if u^{0} ≢ 0.

As an important step in the proof of (0.5), we claim that thanks to the asymptotics (1.4), and thanks to the property that φ is positive if u^{0} is nonzero, we necessarily have that u^{0} ≡ 0 when the u_{α} blow up. In order to do this, we use the Pohozaev [Pokhozhaev] identity as in [8], and conformal invariance. The Pohozaev identity for fourth order equations reads as

∫_{Ω} (x^{k}∂_{k}u) Δ^{2}udx + \frac{n − 4}{2} ∫_{Ω} uΔ^{2}udx

= \frac{n − 4}{2} ∫_{∂Ω} (−u\frac{∂Δu}{∂ν} + \frac{∂u}{∂ν}Δu) dσ

+ ∫_{∂Ω} (\frac{1}{2}(x, ν)(Δu)^{2} − (x, ∇u)\frac{∂Δu}{∂ν} + \frac{∂(x, ∇u)}{∂ν}Δu) dσ

(1.5) for all smooth bounded domains Ω in ℝ^{n} and all u ∈ C^{4}(\overline{Ω}), where Δ is the Euclidean Laplacian, ν is the outward unit normal of ∂Ω, and dσ is the Euclidean volume element on ∂Ω. We apply the Pohozaev identity (1.5) to the u_{α} in the ball Ω = B_{0}(δ\sqrt{μ_{α}}). In the process we assimilate x_{α} and 0 (thanks to the exponential map at x_{α}), and regard u_{α} as a function in the Euclidean space. Noting that

∫_{B_{α}} (x^{k}∂_{k}u_{α}) Δ^{2}u_{α}dx + \frac{n − 4}{2} ∫_{B_{α}} u_{α}Δ^{2}u_{α}dx = O (∫_{∂B_{α}} u_{α}^{2^{♯}}dσ) ,

where B_{α} = B_{0}(δ\sqrt{μ_{α}}), and that ∫_{∂B_{α}} u_{α}^{2^{♯}}dσ = o(μ_{α}^{(n−4)/2}) by (1.4), we get with (1.4), the Pohozaev identity, and the computations developed in [8], that

(n − 2)(n − 4)^{2}ω_{n−1}Aφ(0)μα^{\frac{n−4}{2}} + o (μα^{\frac{n−4}{2}} ) = 0 ,

where ω_{n−1} is the volume of the unit (n−1)-sphere, and A and φ are as in (1.4). In particular, φ(0) = 0, and since φ > 0 if u^{0} ≢ 0, this proves the above claim that we necessarily have that u^{0} ≡ 0 when the u_{α} blow up. With respect to the terminology in Hebey, Robert, and Wen [8], this amounts to saying that compactness reduces to pseudo-compactness for the geometric equation.

Going on with the proof of (0.5), and now that we know that u^{0} ≡ 0, we need to add one important estimate to the estimates listed above, which we proved in [8]. We claim here that

(1.6)  λ_{α}u_{α}(x) → ∑_{i=1}^{N} λ_{i}G_{g}(x_{i}, x)

in C_{loc}^{4}(M\S) as α → +∞, where λ_{α} → +∞ as α → +∞, S = {x_{1}, …, x_{N}} is the geometric blow-up point set of the u_{α}, G_{g} is the Green’s function of P_{g}, and the λ_{i} ≥ 0, i = 1, …, N, are such that ∑_{i=1}^{N} λ_{i} = 1. In order to prove (1.6), we use the positivity of P_{g} and G_{g} as follows. By the positivity of P_{g}, the lowest eigenvalue λ of P_{g} is positive. If ψ is an eigenfunction for λ, letting P_{g}u = |P_{g}ψ|, and writing that P_{g}u ≥ P_{g}ψ and P_{g}u ≥ −P_{g}ψ, we get with the positivity of G_{g} that u ≥ |ψ|. Noting that u > 0 since P_{g}u = λ|ψ| and G_{g} > 0, plugging u into the Rayleigh characterization of λ, it follows that either ψ < 0 or ψ > 0. Without loss of generality we may assume that ψ > 0. Then the conformal metric g̃ = ψ^{4/(n−4)}g is such that Qg̃ > 0, where Qg̃ is the Q-curvature of g̃. By conformal invariance, u = ψ^{−1}u_{α} solves the equation Pg̃u = u^{2^{♯}−1}. Integrating over M, since Qg̃ > 0, we get that there exists C > 0 such that ∥u_{α}∥_{L^{1}(M)} ≤ C∥u_{α}∥_{L^{s}(M)}^{s} where s = 2^{♯} − 1. By Agmon-Douglis-Nirenberg-type estimates, noting that (1.2) gives that the u_{α} are bounded in C_{loc}^{0}(M\S) as α → +∞, we get that for any p > 1, and any δ > 0, the L^{∞}-norm of the u_{α} in sets like M\B_{δ} is controled by the L^{p}- norm of the u_{α} in M, where B_{δ} is the union over the x ∈ S of the geodesic balls B_{x}(δ). By the above estimate, using Hölder’s inequality with 1 ≤ p ≤ 2^{♯}, choosing p > 1 close to 1, and since u_{α} → 0 in L^{q}(M) for q < 2^{♯}, it easily follows that ∥u_{α}∥_{L^{s}(M\B_{δ})} = o (∥u_{α}∥_{L^{s}(M)}) for all δ > 0, where s is as above. In particular, the λ_{i} given by

λ_{i} = lim_{α→+∞} \frac{∫_{B_{x_{i}} (δ)} u_{α}^{2^{♯}−1}dv_{g}}{∫_{M} u_{α}^{2^{♯}−1}dv_{g}}

are nonnegative, independent of δ > 0 small, and such that ∑ λ_{i} = 1. In what follows we let λ_{α} = ∥u_{α}∥_{2^{♯}−1}^{1−2^{♯}}. Then λ_{α} → +∞ as α → +∞, while we can write with the Green’s representation formula that for x ∈ M\B_{δ}, and 0 < δ′ ≪ δ,

u_{α}(x) = ∫_{B_{δ′}} G_{g}(x, y)u_{α}^{2^{♯}−1}(y)dv_{g}(y) + ∫_{M\B_{δ′}} G_{g}(x, y)u_{α}^{2^{♯}−1}(y)dv_{g}(y)

= (∑_{i=1}^{N} λ_{i}G_{g}(x_{i}, x) + o_{δ′}(1)) λ_{α}^{−1} ,

where lim_{δ′→0} lim_{α→+∞} o_{δ′}(1) = 0. In particular, λ_{α}u_{α}(x) → ∑_{i=1}^{N} λ_{i}G_{g}(x_{i}, x) in C_{loc}^{0}(M\S) as α → +∞, an equation from which we easily get that (1.6) is true.

With (1.6) we can now end the proof of (0.5). If f stands for the function on the right hand side of (1.6), then Δ^{2}f = 0 in a set like Ω = ⋃_{i=1}^{N} B_{x_{i}}(δ_{0})\S, where δ_{0} > 0. We apply the Pohozaev identity (1.5) to the u_{α} in B_{x_{i}}(δ) for δ > 0 small and i in {1, …, N}. In the process we assimilate x_{i} and 0 (thanks to the exponential map at x_{i}), and regard u_{α} as a function in the Euclidean space. Noting that

∫_{B} (x^{k}∂_{k}u_{α}) Δ^{2}u_{α}dx + \frac{n − 4}{2} ∫_{B} u_{α}Δ^{2}u_{α}dx = O (∫_{∂B} u_{α}^{2^{♯}}dσ) ,

where B = B_{0}(δ), it follows from (1.6) and the Pohozaev identity that

\begin{matrix}\frac{n − 4}{2} ∫_{∂B_{0}(δ)} (−f \frac{∂Δf}{∂ν} + \frac{∂f}{∂ν} Δf) dσ \\ + ∫_{∂B_{0}(δ)} (\frac{1}{2}(x, ν)(Δf)^{2} − (x, ∇f)\frac{∂Δf}{∂ν} + \frac{∂(x, ∇f)}{∂ν}Δf) dσ = 0 . \end{matrix} ; (1.7)

By (0.3), and (1.6), we can write that

f(x) = \frac{λ̂_{i}}{|x − x_{i}|^{n−4}} + R_{i}(x)

for x ≠ x_{i} close to x_{i}, where λ_{i} is as in (1.6), λ̂_{i} = λ_{n}λ_{i}, λ_{n} is as in (0.3), R_{i} is smooth around x_{i}, and

R_{i}(x_{i}) = λ_{i}μ_{g}(x_{i}) + ∑_{j≠i} λ_{j}G_{g}(x_{j}, x_{i}) .

Plugging these equations into (1.7) and letting δ → 0, we get that R_{i}(x_{i}) = 0. This equation holds for all i = 1, …, N, and we assumed that G_{g} > 0. It follows that λ_{i} > 0 for all i. This ends the proof of (0.5) and of Theorem 0.1.

As a remark, Theorem 0.1 still holds if we replace the critical exponent 2^{♯} in (0.1) by 2^{♯} − p_{α}, where p_{α} ≥ 0 is such that p_{α} → 0 as α → +∞. In this case the left hand side in (0.5) is not zero anymore, but is nonpositive. Needless to say, compactness of the subcritical equations provides a minimizing solution of (0.1).

References

[1] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR0125307 (23:A2610)

[2] S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35–92. MR0162050 (28:5252)

[3] T.P. Branson, Group representations arising from Lorentz conformal geometry, J. Funct. Anal. 74 (1987), 199–291. MR0904819 (90b:22016)

[4] S.Y.A. Chang, On Paneitz operator—a fourth order differential operator in conformal geometry, Harmonic Analysis and Partial Differential Equations, Essays in honor of Alberto P. Calderon, Eds: M. Christ, C. Kenig and C. Sadorsky, Chicago Lectures in Mathematics, 1999, 127–150. MR1743859 (2001g:58059)

[5] S.Y.A. Chang and P.C. Yang, On a fourth order curvature invariant, Comp. Math. 237, Spectral Problems in Geometry and Arithmetic, Ed: T. Branson, AMS, 9–28, 1999. MR1710786 (2001b:58056)

[6] P. Esposito and F. Robert, Mountain pass critical points for Paneitz-Branson operators, Calc. Var. Partial Differential Equations 15 (2002), 493–517. MR1942129 (2005a:58054)

[7] E. Hebey and F. Robert, Coercivity and Struwe’s compactness for Paneitz type operators with constant coefficients, Calc. Var. Partial Differential Equations 13 (2001), 491–517. MR1867939 (2003m:58051)

[8] E. Hebey, F. Robert, and Y. Wen, Compactness and global estimates for a fourth order equation of critical Sobolev growth arising from conformal geometry, Preprint of the University of Nice, 697, 2004.

[9] S. Paneitz, A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian manifolds, Preprint, 1983.

[10] R. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in Calculus of Variations (Montecatini Terme, 1987), Lecture Notes in Math. 1365, Springer-Verlag, Berlin, 1989, pp. 120–154. MR0994021 (90g:58023)

[11] R. Schoen and S.T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988), 47–71. MR0931204 (89c:58139)

[12] X. Xu and P. Yang, Positivity of Paneitz operators, Disc. Cont. Dynamical Systems 7 (2001), 329–342. MR1808405 (2002d:58043)

Université de Cergy-Pontoise, Département de Mathématiques, Site de Saint-Martin, 2 avenue Adolphe Chauvin, 95302 Cergy-Pontoise cedex, France

E-mail address: Emmanuel.Hebey@math.u-cergy.fr

Laboratoire J.A.Dieudonné, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice cedex 2, France

E-mail address: frobert@math.unice.fr