Ahmad El

S A Ïd

DOMAIN DEFORMATIONS AND EIGENVALUES OF THE DIRICHLET LAPLACIAN IN A RIEMANNIAN MANIFOLD

Keywords: 2000 Mathematics Subject Classification. 49R50, 35P99, 58J50, 58J32 Eigenvalues, Laplacian, Dirichlet problem, Domain deformation, Heat trace

For any bounded regular domain Ω of a real analytic Riemannian manifold M , we denote by λ k (Ω) the k-th eigenvalue of the Dirichlet Laplacian of Ω. In this paper, we consider λ k and as a functional upon the set of domains of fixed volume in M . We introduce and investigate a natural notion of critical domain for this functional. In particular, we obtain necessary and sufficient conditions for a domain to be critical, locally minimizing or locally maximizing for λ k . These results rely on Hadamard type variational formulae that we establish in this general setting.

As an application, we obtain a characterization of critical domains of the trace of the heat kernel under Dirichlet boundary conditions.

Introduction

Isoperimetric eigenvalue problems constitute one of the main topics in spectral geometry and shape optimization. Given a Riemannian manifold M , a natural integer k and a positive constant V , the problem is to optimize the k-th eigenvalue of the Dirichlet Laplacian, considered as a functional upon the set of all bounded domains of volume V of M .

The first result in this subject is the famous Faber-Krahn Theorem [START_REF] Faber | dass unter allen homogenen membranen von gleicher fläche und gleicher spannung die kreisför mige den tiefsten grundon gibt[END_REF][START_REF] Krahn | über eine von Rayleigh formulierte minimaleigenschaft des kreises[END_REF], originally conjectured by Rayleigh, stating that Euclidean balls minimize the first eigenvalue of the Dirichlet Laplacian among all domains of given volume. Extensions of this classical result to higher order eigenvalues, combinations of eigenvalues as well as domains of other Riemannian manifolds or subjected to other types of constraints, have been obtained during the last decades and a very rich literature is devoted to this subject (see for instance [START_REF] Ashbaugh | Isoperimetric and universal inaqualities for eigenvalues[END_REF][START_REF] Ashbaugh | A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions[END_REF][START_REF] Berger | Sur les premières valeurs propres des variétés Riemanniennes[END_REF][START_REF] Berard | Sur une inégalité isopérimétrique qui généralise celle de Paul Lévy-Gromov[END_REF][START_REF] Bérard | Iné galités isopérimétriques et applications[END_REF][START_REF] Cheng | Eigenvalue comparison theorems and its geometric applications[END_REF][START_REF] Escobar | Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities, and an eigenvalue estimate[END_REF][START_REF] Soufi | Immersions minimales, premire valeur propre du Laplacien et volume conforme[END_REF][START_REF] Henrot | Minimization problems for eigenvalues of the Laplacian[END_REF][START_REF] Nadirashvili | Berger's isoperimetric problem and minimal immersion of surfaces[END_REF][START_REF] Osgood | Extremals of determinants of Laplacians[END_REF][START_REF] Reilly | Applications of the Hessian operator in a Riemannian manifold[END_REF][START_REF] Shimakura | La première valeur propre du Laplacien pour le problème de Dirichlet[END_REF][START_REF] Szegö | Inequalities for certain eigenvalues of a membrane of given area[END_REF][START_REF] Weinberger | An isoperimetric inequality for the n-dimensional free membrane problem[END_REF] and the references therein).

A fundamental tool in the proof of many results concerning the first Dirichlet eigenvalue is the following variation formula, known as Hadamard's formula [START_REF] Hadamard | Mémoire sur le problème d'analyse relatif à l'équilibre des plaques élastiques encastrées[END_REF][START_REF] Garabedian | Convexity of domain functionals[END_REF][START_REF] Schiffer | Partial differential equations of elliptic type[END_REF][START_REF] Schiffer | Hadamard's formula and variation of domain functions[END_REF]:

d dε λ 1 (Ω ε ) ε=0 = - ∂Ω0 v ∂φ ∂ν 2 dσ,
where λ 1 (Ω ε ) stands for the first Dirichlet eigenvalue of the domain Ω ε , ∂φ ∂ν denotes the normal derivative of the first normalized eigenfunction φ of the Dirichlet Laplacian on Ω 0 and v is the normal displacement of the boundary induced by the deformation. This formula shows that a necessary and sufficient condition for a domain Ω ⊂ R n to be critical for the Dirichlet first eigenvalue functional under fixed volume variations, is that its first Dirichlet eigenfunctions are solutions of the following overdetermined problem:

           ∆φ = λ 1 (Ω)φ in Ω φ = 0 on ∂Ω | ∂φ ∂ν | = c
on ∂Ω, for some constant c. Since a first Dirichlet eigenfunction does not change sign in Ω, it follows from the well known symmetry result of Serrin [START_REF] Serrin | A symmetry problem in potential theory[END_REF] that φ is radial and Ω is a round ball. Therefore, Euclidean balls are the only critical domains of the Dirichlet first eigenvalue functional under fixed volume deformations.

Notice that Hadamard's formula remains valid for any higher order eigenvalue λ k as far as λ k (Ω) is simple. Nevertheless, when λ k (Ω) is degenerate, a differentiability problem arises and our first aim in this paper (see Section 3) is to introduce, in spite of this non-differentiability problem, a natural and simple notion of critical domain.

Indeed, using perturbation theory of unbounded self-adjoint operators in Hilbert spaces, we will see that, for any deformation Ω ε , analytic in ε, of a domain Ω of a real analytic Riemannian manifold M , and any natural integer k, the function ε → λ k (Ω ε ) admits a left sided and a right sided derivatives at ε = 0. Of course, when Ω is a local extremum of λ k , these derivatives have opposite signs. This suggests us to define critical domains of λ k to be the domains Ω such that, for any analytic volume-preserving deformation Ω ε of Ω, the right sided and the left sided derivatives of λ k (Ω ε ) at ε = 0 have opposite signs. That is,

d dε λ k (Ω ε ) ε=0 + × d dε λ k (Ω ε ) ε=0 -≤ 0. which means that λ k (Ω ε ) ≤ λ k (Ω) + o(ε) or λ k (Ω ε ) ≥ λ k (Ω ε ) + o(ε) as ε → 0.
After giving, in Section 2, a general Hadamard type variation formula, we derive, in Section 3, necessary and sufficient conditions for a domain Ω of the Riemannian manifold M to be critical for the k-th Dirichlet eigenvalue functional under volume-preserving domain deformations.

For instance, we show that (Theorem 3.3) if Ω is a critical domain of the k-th Dirichlet eigenvalue under volume-preserving domain deformations, then there exists a family of eigenfunctions φ 1 , . . . , φ m satisfying the following system:

(1)

             ∆φ i = λ k (Ω) φ i in Ω, ∀i ≤ m, φ i = 0 on ∂Ω, ∀i ≤ m, m i=1 ∂φi ∂ν 2 = 1 on ∂Ω.
Moreover, this necessary condition is also sufficient when either λ k (Ω) > λ k-1 (Ω) or λ k (Ω) < λ k+1 (Ω), which means that λ k (Ω) corresponds to the first one or the last one in a cluster of equal eigenvalues. On the other hand, we prove that if λ k (Ω) > λ k-1 (Ω) (resp. λ k (Ω) < λ k+1 (Ω)) and if Ω ⊂ M is a local minimizer (resp. maximizer) of the k-th Dirichlet eigenvalue functional under volume-preserving domain deformations, then λ k (Ω) is simple and the absolute value of the normal derivative of its corresponding eigenfunction is constant along the boundary ∂Ω (see Theorem 3.1).

The last section deals with the trace of the heat kernel under Dirichlet boundary conditions defined for a domain Ω ⊂ M by

Y Ω (t) = Ω H(t, x, x)v g = k≥1 e -λ k (Ω)t ,
where H is the fundamental solution of the heat equation in Ω under Dirichlet boundary conditions. Indeed, Luttinger [START_REF] Luttinger | Generalized isoperimetric inequalities[END_REF] proved an isoperimetric Faber-Krahn like result for Y (t) considered as a functional upon the set of bounded Euclidean domains, that is, for any bounded domain Ω ⊂ R n and any t > 0, one has Y Ω (t) ≤ Y Ω * (t), where Ω * is an Euclidean ball whose volume is equal to that of Ω.

For any smooth deformation Ω ε of Ω, the corresponding heat trace function Y ε (t) is always differentiable w.r.t. ε and the domain Ω will be said critical for the trace of the heat kernel under the Dirichlet boundary condition at time t if, for any volume-preserving deformation Ω ε of Ω, we have

d dε Y ε (t) ε=0 = 0
After giving the first variation formula for this functional (Theorem 4.1), we show that a necessary and sufficient condition for a domain Ω to be critical for the trace of the heat kernel under Dirichlet boundary condition at time t is that the Laplacian of the function x → H(t, x, x) must be constant along the boundary ∂Ω (Corollary 4.1). Using Minakshisundaram-Pleijel asymptotic expansion of Y (t), one can derive necessary conditions for a domain to be critical for the trace of the heat kernel under Dirichlet boundary condition at any time t > 0. For instance, we show that the boundary of such a domain necessarily has constant mean curvature (Theorem 4.2).

Thanks to Alexandrov type results (see [START_REF] Alexandrov | Uniqueness theorems for surfaces in the large[END_REF][START_REF] Montiel | Compact hypersurfaces: The Alexandrov theorem for higher order mean curvatures[END_REF]), one deduces that when the ambient space M is Euclidean, Hyperbolic or a standard hemisphere, then geodesic balls are the only critical domains of the trace of the heat kernel under Dirichlet boundary condition at any time t > 0 (Corollary 4.3).

Hadamard type variation formulae

Let Ω be a regular bounded domain of a Riemannian oriented manifold (M, g). We will denote by ḡ the metric induced by g on the boundary ∂Ω of Ω. Let us start with the following general formula. Proposition 2.1. Let (g ε ) be a differentiable variation of the metric g. Let φ ε ⊂ C ∞ (Ω) be a differentiable family of functions and Λ ε a differentiable family of real numbers such that, ∀ε, ||φ ε || L 2 (Ω,gε) = 1 and

   ∆ gε φ ε = Λ ε φ ε in Ω φ ε = 0 on ∂Ω. Then, d dε Λ ε ε=0 = Ω φ 0 ∆ ′ φ 0 v g = - Ω dφ 0 ⊗ dφ 0 + 1 4 ∆φ 2 0 g, h v g ,
where h := d dε g ε ε=0 , ∆ ′ := d dε ∆ gε ε=0 and , is the inner product induced by g on the space of covariant tensors.

Proof. For simplicity, let us introduce the following notations: λ := Λ 0 , φ := φ 0 , φ ′ := d dε φ ε ε=0 and Λ ′ := d dε Λ ε ε=0 . Differentiating the two sides of the equality ∆ gε φ ε = Λ ε φ ε we obtain

∆ ′ φ + ∆φ ′ = Λ ′ φ + Λφ ′ .
After multiplication by φ and integration we get

Ω φ∆ ′ φv g + Ω φ∆φ ′ v g = Λ ′ + λ Ω φφ ′ v g .

Integration by parts gives

Ω φ∆φ ′ v g = λ Ω φφ ′ v g + ∂Ω ( ∂φ ∂ν φ ′ - ∂φ ′ ∂ν φ)v ḡ .
Thus,

Λ ′ = Ω φ∆ ′ φv g + ∂Ω (φ ′ ∂φ ∂ν -φ ∂φ ′ ∂ν )v ḡ .
It is clear that the boundary integral in this last equation vanishes (since φ ε = 0 on ∂Ω).

In conclusion, we have

(2) Λ ′ = Ω φ∆ ′ φv g .
Now, the expression of ∆ ′ is given by (see [START_REF] Berger | Sur les premières valeurs propres des variétés Riemanniennes[END_REF])

(3) ∆ ′ φ = D 2 φ, h -dφ, δh + 1 2 d h ,
where h is the trace of h w.r.t. g (that is h = g, h ). Integration by parts yields

Ω φ dφ, δh v g = 1 2 Ω D 2 φ 2 , h v g (4) = Ω dφ ⊗ dφ + φD 2 φ, h v g and Ω φ dφ, d h v g = 1 2 Ω h∆φ 2 v g (5)
Combining (2), (3), ( 4) and ( 5) we obtain

Λ ′ = - Ω dφ ⊗ dφ + 1 4 ∆φ 2 g, h v g
which completes the proof of the proposition.

In the particular case of domain deformations, Proposition 2.1 gives rise to the following variation formulae.

Corollary 2.1. Let Ω ε = f ε (Ω) be a deformation of Ω. Let φ ε ∈ C ∞ (Ω ε ) and Λ ε ∈ R be two differentiable curves such that, ∀ε, ||φ ε || L 2 (Ωε,g) = 1 and    ∆φ ε = Λ ε φ ε in Ω ε φ ε = 0 on ∂Ω ε . Then, d dε Λ ε ε=0 = - ∂Ω v ∂φ ∂ν 2 v ḡ,
where φ = φ 0 and v = g d dε f ε ε=0 , ν is the normal component of the variation vector field of the deformation Ω ε .

Proof. Let us apply Proposition 2.1 with g ε = f * ε g and φε = φ ε • f ε . Indeed, one can easily check that φε L 2 (Ω,gε) = 1, ∆ gε φε = Λ ε φε in Ω and φε = 0 on ∂Ω. Hence, [START_REF] Blazic | The spectral geometry of the Laplacian and the conformal Laplacian for manifolds with boundary[END_REF]

d dε Λ ε ε=0 = - Ω dφ ⊗ dφ + 1 4 ∆φ 2 g, h v g with φ := φ 0 = φ0 and h = d dε f * ε g ε=0 = L V g,
where L V g is the Lie derivative of g w.r.t. the vector field V = d dε f ε ε=0 . Using the expression of L V g in terms of the covariant derivative ∇V of V and integrating by parts, we obtain

Ω dφ ⊗ dφ, L V g v g = Ω L V g(∇φ, ∇φ)v g = 2 Ω ∇ ∇φ V, ∇φ v g = Ω div( V, ∇φ ∇φ)v g + 2 Ω V, ∇φ ∆φv g -2 Ω D 2 φ(V, ∇φ)v g = 2 ∂Ω V, ∇φ ∂φ ∂ν v ḡ + λ Ω V, ∇φ 2 v g -2 Ω D 2 φ(V, ∇φ)v g ,
with λ := Λ 0 , and

1 4 Ω ∆φ 2 g, L V g v g = 1 2 Ω ∆φ 2 divV v g = λ Ω φ 2 divV v g - Ω |∇φ| 2 divV v g = Ω (-λ V, ∇φ 2 + 2D 2 φ(V, ∇φ))v g + ∂Ω (λφ 2 -|∇φ| 2 ) V, ν v ḡ .
Replacing in (6), we get

d dε Λ ε ε=0 = ∂Ω {-2 V, ∇φ ∂φ ∂ν + V, ν |∇φ| 2 -λ V, ν φ 2 }v ḡ.
Since φ is identically zero on the boundary, we have at any point of ∂Ω,

∇φ = ∂φ ∂ν ν. In particular, |∇φ| 2 = ∂φ ∂ν 2 and V, ∇φ = V, ν ∂φ ∂ν = v ∂φ ∂ν . Thus, d dε Λ ε ε=0 = - ∂Ω v ∂φ ∂ν 2 v ḡ .

Critical domains

Throughout this section, the ambient Riemannian manifold (M, g) is assumed to be real analytic.

Preliminary results and definitions.

Let Ω be a regular bounded domain of a Riemannian manifold (M, g). An analytic deformation (Ω ε ) of Ω is given by an analytic 1-parameter family of diffeomorphisms f ε : Ω → Ω ε such that f ε (∂Ω) = ∂Ω ε and f 0 = Id. Such a deformation is called volumepreserving if the Riemannian volume of Ω ε w.r.t the metric g does not depend on ε.

The spectrum of the Laplace operator ∆ g under the Dirichlet boundary condition will be denoted

Sp D (∆ g , Ω ε ) = { λ 1,ε < λ 2,ε ≤ • • • ≤ λ k,ε ↑ + ∞ }
The functions ε → λ k,ε is continuous but not differentiable in general, excepting λ 1,ε which is always differentiable since it is simple. However, as we will see hereafter, the general perturbation theory of unbounded self-adjoint operators enables us to show that the function λ k,ε admits a right sided and a left sided derivatives at ε = 0. In all the sequel, a family of functions

φ ε ∈ C ∞ (Ω ε ) will be said differentiable (resp. analytic) w.r.t ε, if that is the case for φ ε • f ε ∈ C ∞ (Ω). Lemma 3.1. Let λ ∈ Sp D (∆ g , Ω)
be an eigenvalue of multiplicity p of the Dirichlet Laplacian in Ω. For any analytic deformation Ω ε of Ω, there exist p families (Λ i,ε ) i≤p of real numbers and p families

(φ i,ε ) i≤p ⊂ C ∞ (Ω ε ) of functions, depending analytically on ε and satisfying, ∀ε ∈ (-ε 0 , ε 0 ) and ∀i ∈ {1, . . . , p}, (a) Λ i,0 = λ (b) the family {φ 1,ε , • • • , φ p,ε } is orthonormal in L 2 (Ω ε , g) (c)    ∆ φ i,ε = Λ i,ε φ i,ε in Ω ε φ i,ε = 0 on ∂Ω ε
The proof is based on perturbation theory of unbounded self-adjoint operators in Hilbert spaces. Results concerning the differentiability of eigenvalues and eigenvectors have been first obtained by Rellich [START_REF] Rellich | Perturbation theory of eigenvalue problems[END_REF] and then by Kato [START_REF] Kato | Perturbation theory for linear operators[END_REF] in the analytic case. Many results were also obtained under weaker differentiability conditions (see for instance [START_REF] Kriegl | The convenient setting of global analysis[END_REF][START_REF] Kriegl | Differentiable perturbation of unbounded operators[END_REF] for recent contributions in this subject). Nevertheless, even a smooth curve ε → P ε of self-adjoint operators may lead to noncontinuous eigenvectors w.r.t. ε (see Rellich's example [19, chap II, Example 5.3]). Since we need to differentiate eigenvectors w.r.t. ε, we imposed analyticity assumptions in order to get analytic curves of operators.

Proof of Lemma 3.1. In order to get into the framework of perturbation theory, we first need to modify our operators so that they all have the same domain. Indeed, for any ε we set g ε = f * ε g and denote ∆ ε the Laplace operator of (Ω, g ε ). Clearly, we have

Sp D (∆ g , Ω ε ) = Sp D (∆ ε , Ω).
Notice that since f ε depends analytically on ε and that g is real analytic, the curves ε → g ε and, hence, ε → ∆ ε are analytic w.r.t. ε.

The operator ∆ ε is symmetric w.r.t. the inner product in L 2 (Ω, g ε ), but not necessarily w.r.t. the inner product in L 2 (Ω, g). Therefore, we need to introduce a conjugation as follows. Let U ε : L 2 (Ω, g) → L 2 (Ω, g ε ) be the unitary isomorphism given by

U ε : v → |g| |g ε | 1 4 v,
where |g| = det(g ij ) is the determinant of the matrix (g ij ) of the components of g in a local coordinate system. We define the operator P ε to be

P ε = U -1 ε • ∆ ε • U ε . Therefore, we have Sp D (P ε , Ω) = Sp D (∆ ε , Ω) and, if v ε ∈ C ∞ (Ω) is an eigen- function of P ε , then φ ε = U ε (v ε ) • f -1 ε ∈ C ∞ (Ω ε
) is an eigenfunction of ∆ g with the same eigenvalue. Again, since ∀ε, (M, g ε ) is real analytic, the curves ε → U ε and ε → P ε are analytic. The result of the lemma then follows from the Rellich-Kato theory applied to ε → P ε . Now, let us fix a positive integer k and let Λ 1,ε , . . . , Λ p,ε be the family of eigenvalues associated with λ k by Lemma 3.1. Using the continuity of λ k,ε and the analyticity of Λ i,ε w.r.t. ε, we can easily see that there exist two integers i ≤ p and j ≤ p such that

λ k,ε =    Λ i,ε if ε ≤ 0 Λ j,ε if ε ≥ 0.
Hence, λ k,ε admits a left sided and a right sided derivatives with

d dε λ k,ε ε=0 + = d dε Λ j,ε ε=0 and d dε λ k,ε ε=0 -= d dε Λ i,ε ε=0 . Definition 3.1.
The domain Ω is said to be "critical" for the k-th eigenvalue of Dirichlet problem if, for any analytic volume-preserving deformation Ω ε of Ω, the right sided and the left sided derivatives of λ k,ε at ε = 0 have opposite signs. That is,

d dε λ k,ε ε=0 + × d dε λ k,ε ε=0 -≤ 0.
It is easy to see that

d dε λ k,ε ε=0 + ≤ 0 ≤ d dε λ k,ε ε=0 -⇐⇒ λ k,ε ≤ λ k,0 + o(ε) and d dε λ k,ε ε=0 -≤ 0 ≤ d dε λ k,ε ε=0 + ⇐⇒ λ k,ε ≥ λ k,0 + o(ε).
Therefore, the domain Ω is critical for the k-th eigenvalue of Dirichlet problem if and only if one of the following inequalities holds:

λ k,ε ≤ λ k,0 + o(ε) λ k,ε ≥ λ k,0 + o(ε).
Remark 3.1. Suppose that for an integer k we have λ k < λ k+1 , then, for sufficiently small ε, we will have

λ k,ε = max i≤p Λ i,ε
, where Λ 1,ε , . . . , Λ p,ε are the eigenvalues associated to λ k by Lemma 3.1 (indeed, Λ i,0 = λ k < λ k+1 for any

1 ≤ i ≤ p). Hence, d dε λ k,ε | ε=0 -≤ d dε λ k,ε | ε=0 + . In particular, Ω is critical for the functional Ω → λ k (Ω) if and only if d dε λ k,ε | ε=0 -≤ 0 ≤ d dε λ k,ε | ε=0 + (or, equivalently, λ k,ε ≤ λ k,0 + o(ε)). Similarly, if λ k-1 < λ k , then, for sufficiently small ε, λ k,ε = min i≤p Λ i,ε and d dε λ k,ε | ε=0 + ≤ d dε λ k,ε | ε=0 -. Lemma 3.2. Let λ ∈ Sp D (∆ g , Ω)
be an eigenvalue of multiplicity p of the Dirichlet Laplacian in Ω and let us denote by E λ the corresponding eigenspace.

Let Ω ε = f ε (Ω) be an analytic deformation of Ω and let (Λ i,ε ) i≤p and

(φ i,ε ) i≤p ⊂ C ∞ (Ω ε ) be as in Lemma 3.1. Then Λ ′ 1 := d dε Λ 1,ε ε=0 , • • • , Λ ′ p := d dε Λ p,ε ε=0
are the eigenvalues of the quadratic form q v defined on the space E λ ⊂ L 2 (Ω, g) by

q v (φ) = - ∂Ω v ∂φ ∂ν 2 v ḡ , where v = g d dε f ε ε=0 , ν . Moreover, the L 2 -orthonormal basis φ 1,0 , • • • , φ p,0 diagonalizes q v on E λ .
Proof. For simplicity, we set

g ε := f * ε g, ∆ ′ := d dε ∆ gε ε=0 , Λ i := Λ i,0 , φ i := φ i,0 and Λ ′ i := d dε Λ i,ε ε=0 . From ∆ gε (φ i,ε ) = Λ i,ε (φ i,ε ), we deduce ∆ ′ φ i + ∆φ ′ i = Λ ′ i φ i + Λ i φ ′ i
We multiply by φ j and integrate to get

Ω φ j ∆ ′ φ i v g + Ω φ j ∆φ ′ i v g = Λ ′ i Ω φ i φ j v g + λ Ω φ i φ ′ j v g .
Integration by parts gives ( since

φ j = φ ′ i = 0 on ∂Ω)) Ω φ j ∆φ ′ i v g = λ Ω φ i φ ′ j v g .
Hence,

Ω φ j ∆ ′ φ i v g = Λ ′ i Ω φ i φ j v g .
Therefore

Ω φ j ∆ ′ φ i v g = Λ ′ i Ω φ i φ j v g . It follows that the L 2 -orthonormal basis φ 1 , • • • , φ p diagonalizes the quadratic form φ → Ω φ∆ ′ φv g on E λ , the corresponding eigenvalues being Λ ′ 1 , • • • , Λ ′ p .
As we have seen in the proof of the corollary 2.1, this last quadratic form coincides with q v on E λ .

Any volume-preserving deformation Ω

ε = f ε (Ω) induces a function v := g( d dε f ε ε=0 , ν) on ∂Ω satisfying ∂Ω v v ḡ = 0 (indeed, this last integral is equal up to a constant to d dε vol(Ω ε ) ε=0
). In all the sequel, we will denote by A 0 (∂Ω) the set of regular functions on ∂Ω such that ∂Ω v v ḡ = 0. The following elementary lemma will be useful in the proof of our main results. Lemma 3.3. Let v ∈ A 0 (∂Ω). Then there exists an analytic volume-preserving deformation

Ω ε = f ε (Ω) so that v = g( d dε f ε ε=0 , ν). Proof. Let U ⊂ M be
an open neighborhood of Ω and let ṽ and ν be smooth extensions to U of v and ν respectively. For ε sufficiently small, the map ϕ ε (x) = exp x ε ṽ(x) ν(x) is a diffeomorphism from Ω to ϕ ε (Ω). Moreover, since (M, g) is real analytic, the curve ε → ϕ ǫ is analytic w.r.t. ε. The deformation ϕ ε (Ω) is not necessarily volume-preserving. However, let X be any analytic vectorfield on U such that Ω divXv g = 0 and denote by (γ t ) t the associated 1-parameter local group of diffeomorphisms. The function (t, ε)

→ F (t, ε) = vol(γ t • ϕ ε (Ω)) satisfies ∂ ∂t F (0, 0) = Ω divXv g = 0.
Applying the implicit function theorem in the analytic setting, we get the existence of a function t(ε) depending analytically on ε ∈ (-η, η), for some η > 0 sufficiently small, such that F (t(ε), ε) = F (0, 0), ∀ε ∈ (-η, η). The deformation f ε = γ t(ε) • ϕ ε is clearly analytic and volume-preserving. Moreover, one has

t ′ (0) = - d dε vol(ϕ ε (Ω)) ε=0 d dt vol(γ t (Ω)) t=0 = -Ω divṽν v g Ω divXv g = - ∂Ω v v ḡ ∂Ω X, ν v ḡ = 0. Therefore, ∀x ∈ ∂Ω, d dε f ε (x) ε=0 = t ′ (0)X(x) + dϕ ǫ (x) dε ε=0 = v(x)ν(x).

3.2.

Critical domains for the k-th eigenvalue of the Dirichlet Laplacian. In all the sequel, we will denote by λ k the k-th eigenvalue of the Dirichlet problem in Ω and by E k the corresponding eigenspace.

In the following results, a special role is played by the eigenvalues λ k satisfying λ k > λ k-1 or λ k < λ k+1 . This means that the index k is the lowest or the highest one among all the indices corresponding to the same eigenvalue. Let us start with the following necessary condition to be satisfied by a locally minimizing or locally maximizing domain. Here, a local minimizer (resp. maximizer) for the k-th eigenvalue of the Dirichlet Laplacian is a domain Ω such that, for any volume-preserving deformation Ω ε , the function ε → λ k,ε admits a local minimum (resp. maximum) at ε = 0. Theorem 3.1. Let k be a natural integer such that λ k > λ k-1 (resp. λ k < λ k+1 ) and assume that Ω is a local minimizer (resp. local maximizer) for the k-th eigenvalue of the Dirichlet Laplacian. Then λ k is simple and the absolute value of the normal derivative of its corresponding eigenfunction is constant on ∂Ω. That is, there exists a unique (up to sign) function φ satisfying

           ∆φ = λ k φ in Ω φ = 0 on ∂Ω | ∂φ ∂ν | = 1 on ∂Ω.
Proof. Suppose that λ k > λ k-1 and let Ω ε = f ε (Ω) be a volume preserving analytic deformation of Ω. Let (Λ i,ε ) i≤p and (φ i,ε ) i≤p be families of eigenvalues and eigenfunctions associated to λ k according to Lemma 3.1. Since Λ i,0 = λ k > λ k-1 , we have, for sufficiently small ε, for continuity reasons,

Λ i,ε > λ k-1,ε . Hence, Λ i,ε ≥ λ k,ε .
As the function ε → λ k,ε admits a local minimum at ε = 0 with Λ i,0 = λ k,0 = λ k , it follows that the differentiable function ε → Λ i,ε achieves a local minimum at ε = 0 and that d dε Λ i,ε ε=0 = 0. Applying Lemma 3.2, we deduce that the quadratic form q v is identically zero on the eigenspace E k , where v = g( d dε f ε ε=0 , ν). The volume-preserving deformation being arbitrary, it follows that the form q v vanishes on E k for any v ∈ A 0 (∂Ω) (Lemma 3.3). Therefore, ∀φ ∈ E k and ∀v ∈ A 0 (∂Ω), we have ∂Ω v ∂φ ∂ν 2 v ḡ = 0, which implies that ∂φ ∂ν is locally constant on ∂Ω for any φ ∈ E k . Now, if φ 1 and φ 2 are two eigenfunctions in E k , one can find a linear combination φ = αφ 1 +βφ 2 so that ∂φ ∂ν vanishes on at least one connected component of ∂Ω. We apply Holmgren uniqueness theorem (see for instance [27, Theorem 2, p. 42], and recall that (M, g) is assumed to be real analytic) to deduce that φ is identically zero in Ω and that λ k is simple.

To finish the proof, we must show that, ∀φ ∈ E k , | ∂φ ∂ν | takes the same constant value on all the components of ∂Ω. Indeed, let Σ 1 and Σ 2 be two distinct connected components of ∂Ω and let v ∈ A 0 (∂Ω) be the function given by v = vol(Σ 2 ) on Σ 1 , v = -vol(Σ 1 ) on Σ The criticality of the domain Ω for the k-th eigenvalue of Dirichlet Laplacian is closely related to the definiteness of the quadratic forms q v introduced in Lemma 3.2 above, on the eigenspace E k . Indeed, we have the following Theorem 3.2. Let k be any natural integer.

(

) 1
If Ω is a critical domain for the k-th eigenvalue of the Dirichlet Laplacian, then, ∀v ∈ A 0 (∂Ω), the quadratic form

q v (φ) = -∂Ω v ∂φ ∂ν 2 v ḡ is not definite on E k . (2) Assume that λ k > λ k-1 or λ k < λ k+1 and that ∀v ∈ A 0 (∂Ω), the quadratic form q v (φ) = -∂Ω v ∂φ ∂ν 2 v ḡ is not definite on E k ,
then Ω is a critical domain for the k-th eigenvalue of the Dirichlet Laplacian.

Proof. (1) Consider a function v ∈ A 0 (∂Ω) and let Ω ε = f ε (Ω) be an analytic volume-preserving deformation of Ω so that v := g( d dε f ε ε=0 , ν) (Lemma 3.3). Let (Λ i,ε ) i≤p and (φ i,ε ) i≤p be families of eigenvalues and eigenfunctions associated to λ k according to Lemma 3.1. As we have seen above, there exists two integers i ≤ p and j ≤ p so that d dε λ k,ε ε=0 -= d dε Λ i,ε ε=0 and d dε λ k,ε ε=0 + = d dε Λ j,ε ε=0 . The criticality of Ω then implies that d dε Λ i,ε ε=0 × d dε Λ j,ε ε=0 ≤ 0. Applying Lemma 3.2, we deduce that the quadratic form q v admits both nonnegative and nonpositive eigenvalues on E k which proves Assertion 1.

(2) Assume that λ k > λ k-1 and let Ω ε = f ε (Ω) be a volume-preserving deformation of Ω. Let (Λ i,ε ) i≤p and (φ i,ε ) i≤p be families of eigenvalues and eigenfunctions associated to λ k according to Lemma 3.1. As we have seen in Remark 3.1, we have, for sufficiently small ε, λ k,ε = min i≤p Λ i,ε . Hence,

d dε λ k,ε ε=0 + = min i≤p d dε Λ i,ε ε=0 and d dε λ k,ε ε=0 -= max i≤p d dε Λ i,ε ε=0 .
Now, the non definiteness of q v on E k means that its smallest eigenvalue is nonpositive and its largest one is nonnegative. According to Lemma 3.2, this implies that d dε λ k,ε ε=0 + = min i≤p d dε Λ i,ε ε=0 ≤ 0 and d dε λ k,ε ε=0 -= max i≤p d dε Λ i,ε ε=0 ≥ 0 which implies the criticality of the domain Ω. The case λ k < λ k+1 can be handled similarly.

The indefiniteness of q v for any v ∈ A 0 (∂Ω) can be interpreted intrinsically in the following manner: Lemma 3.4. Let k be a natural integer. The two following conditions are equivalent:

(i) ∀v ∈ A 0 (∂Ω), the quadratic form q v is not definite on E k (ii) there exists a finite family of eigenfunctions

(φ i ) i≤m ⊂ E k satisfying m i=1 ∂φ i ∂ν 2 = 1 on ∂Ω.
Proof. To see that (ii) implies (i), it suffices to notice that, for any v ∈ A 0 (∂Ω)

i≤m q v (φ i ) = - i≤m ∂Ω v ∂φ i ∂ν 2 v ḡ = - ∂Ω vv ḡ = 0.
Therefore, q v is not definite on E k . The proof of "(i) implies (ii)" uses arguments similar to those used in the case of closed manifolds by Nadirashvili [START_REF] Nadirashvili | Berger's isoperimetric problem and minimal immersion of surfaces[END_REF] and the authors [START_REF] Soufi | Riemannian manifolds admitting isometric immersions by their first eigenfunctions[END_REF]. Let K be the convex hull of

∂φ ∂ν 2 , φ ∈ E k in C ∞ (∂Ω)
. Then, we need to show that the constant function 1 belongs to K.

Let us suppose for a contradiction that 1 ∈ K, then, from the Hahn-Banach theorem (applied in the finite dimensional vector space spanned by K and 1 and endowed with the L 2 (∂Ω, ḡ) inner product), there exists a function

v ∈ C ∞ (∂Ω) such that ∂Ω v v ḡ > 0 and, ∀φ ∈ E k , ∂Ω v ∂φ ∂ν 2 v ḡ ≤ 0.
Hence, the zero mean value function

v o = v - 1 vol(∂Ω) ∂Ω v v ḡ satisfies, ∀φ ∈ E k , q v0 (φ) = - ∂Ω v o ∂φ ∂ν 2 v ḡ = - ∂Ω v ∂φ ∂ν 2 v ḡ + 1 vol(∂Ω) ∂Ω v v ḡ ∂Ω ∂φ ∂ν 2 v ḡ ≥ 1 vol(∂Ω) ∂Ω v v ḡ ∂Ω ∂φ ∂ν 2 v ḡ ,
with ∂Ω ∂φ ∂ν 2 v ḡ > 0 for any non trivial Dirichlet eigenfunction φ (due to Holmgren uniqueness theorem). In conclusion, the function v o ∈ A 0 (∂Ω) is such that the quadratic form q v0 is positive definite on E k , which contradicts Condition (i).

A consequence of this lemma and Theorem 3.2 is the following: Theorem 3.3. Let k be any natural integer.

(1) If Ω is a critical domain for the k-th eigenvalue of Dirichlet Laplacian, then there exists a finite family of eigenfunctions

(φ i ) i≤m ⊂ E k satisfying m i=1 ∂φi ∂ν 2 = 1 on ∂Ω, that is, (φ i ) i≤m are solutions of the following system              ∆φ i = λ k φ i in Ω, ∀i ≤ m, φ i = 0 on ∂Ω, ∀i ≤ m, m i=1 ∂φi ∂ν 2 = 1 on ∂Ω.
(2) Assume that λ k > λ k-1 or λ k < λ k+1 and that there exists a finite family of eigenfunctions (φ i ) i≤m ⊂ E k such that m i=1 ∂φi ∂ν 2 is constant on ∂Ω, then the domain Ω is critical for the k-th eigenvalue of the Dirichlet Laplacian.

Corollary 3.1. Assume that λ k is simple. The domain Ω is critical for the k-th eigenvalue of the Dirichlet Laplacian if and only if the following overdetermined Pompeiu type system admits a solution

           ∆φ = λ k φ in Ω φ = 0 on ∂Ω | ∂φ ∂ν | = 1 on ∂Ω. 3.3.
Nonexistence of critical domains under metric variations. In this paragraph, we point out the non consistency of the notion of critical domains w.r.t. metric variations under Dirichlet boundary condition. Indeed, if g ε is an analytic variation of the metric g, then we can associate to each eigenvalue λ k of the Dirichlet problem in Ω, analytic families (Λ i,ε ) i≤p ⊂ R and (φ i,ε ) i≤p ⊂ C ∞ (Ω) (where p is the multiplicity of λ k ) satisfying, for sufficiently small ε,

(1) (φ i,ε ) i≤p is L 2 (Ω, g ε ) orthonormal. (2) ∀ i ∈ {1, . . . , p}, Λ i,o = λ k . (3) ∀ i ≤ p,    ∆ gε φ i,ε = Λ i,ε φ i,ε in Ω φ i,ε = 0 on ∂Ω
Therefore λ k,ε admits a left sided and a right sided derivatives at ε = 0, and we can mimic Definition 3.1 to introduce the notion of critical domain for the k-th eigenvalue of Dirichlet problem w.r.t. volume-preserving variations of the metric. Thanks to Proposition 2.1 and using arguments similar to those used above (see also [START_REF] Soufi | Riemannian manifolds admitting isometric immersions by their first eigenfunctions[END_REF][START_REF] Nadirashvili | Berger's isoperimetric problem and minimal immersion of surfaces[END_REF]), we can show that, if the domain (Ω, g) is critical for the k-th eigenvalue of Dirichlet problem, then there exists a family of eigenfunctions φ 1 , . . . , φ m ∈ E k satisfying [START_REF] Branson | The asymptotics of the Laplacian on a manifold with boundary[END_REF] m i=1 dφ i ⊗ dφ i = g. Now, if we consider only volume-preserving conformal variations g ε of g (that is g ε = α ε g with Ω α n 2 ε v g = V ol(Ω, g)), then the necessary condition [START_REF] Branson | The asymptotics of the Laplacian on a manifold with boundary[END_REF] for (Ω, g) to be critical w.r.t such variations becomes m i=1

φ 2 i = 1 in Ω.
As the eigenfunctions of the Dirichlet Laplacian vanish on the boundary ∂Ω, this last condition can never be fulfilled by functions of E k . Thus, we have the following:

Proposition 3.1. There is no critical domain (Ω, g) for the k-th eigenvalue of the Dirichlet Laplacian under conformal volume-preserving variations of the metric g.

Applications to the trace of the heat kernel

This section deals with critical domains of the trace of the heat kernel under Dirichlet boundary condition.

Recall that the heat kernel H of (Ω, g) under the Dirichlet boundary condition is defined to be the solution of the following parabolic problem:

           ( ∂ ∂t -∆ y )H(t, x, y) = 0 H(0, x, y) = δ x
∀y ∈ ∂Ω, H(t, x, y) = 0 Its trace is the function

Y (t) = Ω H(t, x, x)v g
The relationship between this kernel and the spectrum of the Dirichlet Laplacian is given by

H(t, x, y) = k≥1 e -λ k t φ k (x)φ k (y)
where (φ) k≥1 is an L 2 (Ω, g)-orthonormal family of eigenfunctions satisfying

   ∆φ k = λ k φ k in Ω φ k = 0 on ∂Ω and then, (8)
Y (t) = k≥1 e -λ k t .
Let Ω ε be a smooth deformation of Ω and let Y ε (t) = k≥1 e -λ k,ε t be the corresponding heat trace function. Unlike the eigenvalues, the function Y ε (t) is always differentiable in ε and the domain Ω will be said critical for the trace of the heat kernel under the Dirichlet boundary condition at time t if, for any volume-preserving deformation Ω ε of Ω, we have

d dε Y ε (t) ε=0 = 0
From the results of section 3 above, one can deduce the variation formula for the heat trace. For this, we need to introduce the mixed second derivative d S H(t)| x of H at the point x defined as the smooth 2-tensor given by

d S H(t)| x (X, X) = ∂ 2 ∂α∂β H(t, c(α), c(β)) α=β=0 ,
where c is a curve in Ω such that c(0) = x and ċ(0) = X. It is easy to check that d S H(t) = Let Ω ε = f ε (Ω) be a volume-preserving deformation of Ω. We have, ∀t > 0,

d dε Y ε (t) ε=0 = -t ∂Ω v d S H(t)(ν, ν)v ḡ = t 2 ∂Ω v ∆H(t, x, x)v ḡ
where v = g( d dε f ε ε=0 , ν). Proof. The proof can be derived from the first variation formula of the heat kernel that can be found in the paper of Ray and Singer [START_REF] Ray | R-torsion and the Laplacian on Riemannian manifolds[END_REF]Proposition 6.1]. Nevertheless, at least in the case where the ambient manifold is real analytic, the formula of Theorem 4.1 can be obtained as an immediate consequence of Hadamard's type formula of Section 2, thanks to the relation (8) above. Indeed, in this manner we obtain, ∀t > 0,

d dε Y ε (t) ε=0 = -t k≥1 e -λ k t ∂Ω v ∂φ k ∂ν 2 v ḡ
where (λ k , φ k ) are as above. To get the desired formula for Y ε (t) it suffices to notice that

d S H(t)(ν, ν) = k≥1 e -λ k t dφ k ⊗ dφ k (ν, ν) = k≥1 e -λ k t ∂φ k ∂ν 2 .
An immediate consequence is the following

2 (Ω, g)-orthonormal basis φ 1 , • • • φ p of the eigenspace E k of λ k , i≤p ∂φi ∂ν 2 is constant on ∂Ω.
Recall that if ρ is an isometry of (Ω, g), then, ∀x ∈ Ω and ∀t > 0, H(t, ρ(x), ρ(x)) = H(t, x, x). In particular, if Ω is a ball of R n endowed with a rotationally symmetric Riemannian metric g given in polar coordinates by g = a 2 (r)dr 2 + b 2 (r)dσ 2 , where dσ 2 is the standard metric of the unit sphere S n-1 , then H(t, x, x) should be radial (that is depend only on the parameter r). Therefore, the function ∆H(t, x, x) is also radial and then it is constant on the boundary of the ball. Corollary 4.2. Let g be a a rotationally symmetric Riemannian metric on R n . The geodesic balls centered at the origin are critical domains for the trace of the Dirichlet heat kernel under volume-preserving domain deformations.

In particular, geodesic balls of Riemannian space forms are critical for the trace of the Dirichlet heat kernel under volume-preserving domain deformations.

The Minakshisundaram-Pleijel asymptotic expansion of the trace of the heat kernel can also informs us about the geometric properties of extremal or critical domains. Indeed, it is well known that there exists a sequence (a i ) i∈N of real numbers such that for sufficiently small t > 0, we have:

Y (t) = (4πt) -n 2 k≥0 a k t k 2
with (see for instance [START_REF] Blazic | The spectral geometry of the Laplacian and the conformal Laplacian for manifolds with boundary[END_REF][START_REF] Branson | The asymptotics of the Laplacian on a manifold with boundary[END_REF]):

a 0 = vol(Ω, g), a 1 = - √ π 2 vol(∂Ω, ḡ), a 2 = 1 6 Ω scal g v g + 2 ∂Ω trA v ḡ , a 3 = √ π 192 ∂Ω -16scal g -7(trA) 2 + 10|A| 2 + 8ρ g (ν, ν) v ḡ ,
where scal g and ρ g are respectively the scalar and the Ricci curvatures of (Ω, g), A is the shape operator of the boundary ∂Ω (i.e ∀X ∈ T ∂Ω, A(X) = D X ν) and trA is the trace of A (i.e (n -1)-times the mean curvature of ∂Ω).

An immediate consequence of these formulae is the following: Suppose that for any domain Ω ′ having the same volume as Ω, we have Y Ω ′ (t) ≤ Y Ω (t),∀t > 0, then vol ∂Ω ′ ≥ vol ∂Ω. Consequently, we have Proposition 4.1. If the domain Ω maximizes Y at any time t > 0 among all the domains of the same volume, then Ω is a solution of the isoperimetric problem in (M, g), that is, ∀Ω ′ ⊂ M such that volΩ = volΩ ′ , we have vol∂Ω ′ ≥ vol∂Ω.

Another consequence of the Minakshisundaram-Pleijel asymptotic expansion is the following Theorem 4.2. If the domain Ω is a critical domain of the trace of the Dirichlet heat kernel at any time t > 0, then ∂Ω has constant mean curvature. If in addition the Ricci curvature (resp. the sectional curvature) of the ambient space (M, g) is constant in a neighborhood of Ω, then tr(A 2 ) (resp. tr(A 3 )) is constant on ∂Ω.

Proof. Let Ω ε = f ε (Ω) be a volume-preserving variation of Ω and let us denote for any ε by (a i,ε ) i≥0 the coefficients of the asymptotic expansions of Y ε (t). Since d dε Y ε (t) ε=0 = 0, we have for any i ≥ 0, d dε a i,ε ε=0 = 0 (see for instance [START_REF] Gilkey | The eta invariant for a class of elliptic boundary value problems[END_REF] for an analytic justification for this last assertion). In particular, d dε vol(∂Ω ε ) ε=0 = 0 for any volume-preserving variation of Ω. This property is known to be equivalent to the fact that the mean curvature of ∂Ω is constant (see for instance [START_REF] Reilly | Variational properties of the functions of the mean curvatures for hypersurfaces in space forms[END_REF]). Now, let us suppose that the Ricci curvature of (M, g) is constant in a neighborhood of Ω, then for any small ε, we have:

a 2,ε = 1 6 scal g vol(Ω ε ) + 2 ∂Ωε (trA ε ) v ḡ = 1 6 scal g vol(Ω) + 2 ∂Ωε (trA ε ) v ḡ .
Hence, we have (see for instance [START_REF] Reilly | Variational properties of the functions of the mean curvatures for hypersurfaces in space forms[END_REF]):

d dε ∂Ωε (trA ε ) v ḡ ε=0 = ∂Ω ∆ ḡv -ρ(ν, ν)v -(trA 2 )v v ḡ + 1 2 ∂Ω trA div ḡ V T + v trA v ḡ ,
where V = dfε dε ε=0 = v ν + V T on the boundary ∂Ω. Since ∂Ω v v ḡ = 0 and trA and ρ(ν, ν) are constant on ∂Ω, we have:

d dε a 2,ε ε=0 = 1 3 ∂Ω (trA 2 )v v ḡ = 0.
It follows that trA 2 is constant on ∂Ω.

As before, we have After some straightforward but long computations we obtain, using the fact that the sectional curvature is constant in a neighborhood of Ω, and that trA and trA 2 are constant,

d dε a 3,ε ε=0 = c ∂Ω trA 3 v v ḡ = 0,
where c is a constant. This proves that trA 3 is constant.

Alexandrov's Theorem [START_REF] Alexandrov | Uniqueness theorems for surfaces in the large[END_REF] shows that in the Euclidean space, the geodesic spheres are the only embedded compact hypersurfaces of constant mean curvature. This theorem was extended to hypersurfaces of the hyperbolic space and the standard hemisphere(see [START_REF] Montiel | Compact hypersurfaces: The Alexandrov theorem for higher order mean curvatures[END_REF]). Since the boundary of a critical domain of the trace of the heat kernel is an embedded hypersurface of constant mean curvature, we have the Corollary 4.3. Let (M, g) be one of the following spaces:

• The Euclidean space.

• The Hyperbolic space.

• The standard Hemisphere.

Then a domain Ω of (M, g) is critical for the trace of the Dirichlet heat kernel if and only if Ω is a geodesic ball.

2 2 v

 22 and v = 0 on the other components. Then the condition ∂Ω v ∂φ ∂ν course, the same arguments work in the case λ k < λ k+1 .

k≥1e

  -λ k t dφ k ⊗ dφ k Theorem 4.1.

Corollary 4 . 1 .

 41 The following conditions are equivalent: (i) The domain Ω is critical for the trace of the Dirichlet heat kernel at the time t under volume-preserving domain deformations, (ii) ∆H(t, x, x) is constant on the boundary ∂Ω, (iii) for any positive integer k and any L

(trA ε ) 2 v ḡ ε=0 = 2 ∂Ω

 2 trA ∆ ḡ vρ(ν, ν)v -(trA 2 )v v ḡ + 1 2 ∂Ω (trA) 2 div ḡ V T + v trA v ḡ = 0since trA, trA 2 and ρ(ν, ν) are constants. Thus,

Acknowledgment

The authors would like to Thank Professors Bernard Helffer and Peter Gilkey for valuable discussions. They also thank the referee for pointing out some mistakes in the first version of the paper and for valuable comments


Ahmed Loukili

David Chopin

Abdelhafid Khelidj

Jean-Yves Le Touzo

A new approach to determine autogenous shrinkage of mortar at an early age considering temperature history

Keywords: Fresh mortar, Early age, Temperature, Autogenous shrinkage, Thermal dilation coefficient

Shrinkage, settlement, and thermal deformations of concrete start to develop as soon as the casting is finished. The majority of standard tests currently used to measure free deformations are carried out on hardened concrete. Therefore, the prediction of thermal and other stresses at an early age cannot be determined. This article describes the design and the operation of an experimental device used for measuring the volume change of mortar in relation to the real temperature history. Measurements are carried out on a mortar having a water/cement (W/C) ratio of 0.35 and started approximately 20 min after the addition of water in the mixture up to 24 h of hydration. The measured deformation includes autogenous shrinkage and thermal expansion or contraction caused by the temperature changes induced by the heat of cement hydration. An experimental method is proposed to uncouple these deformations. The investigations show that the autogenous shrinkage amplitude is strongly affected by the temperature history of mortar. Therefore, for a good estimate of the shrinkage amplitudes of cement based material, temperature must be taken into account in testing.

Introduction

For practical use of cement-based materials, good knowledge of their mechanical behavior is necessary. Shrinkage, among others, is a very important property for the good working of a structure since it very often generates either early age cracking (thermal shrinkage and autogenous shrinkage) or late cracking (drying shrinkage). The correct knowledge of the phenomena governing the free deformations of the concrete and the evaluation of their real amplitudes make it possible to avoid cracking and consequently, to guarantee a better durability of the structures.

The volume change involved by the hydration reaction of cement starts from the beginning of the making concrete. Usually, the measurement of shrinkage deformations is carried out on hardened concrete. With this test method, the deformations of the freshly mixed concrete are completely ignored and may result in the statement of inaccurate laws of behavior.

Among the methods available to assess measurement of volume change of concrete at early age, we have the measurement of buoyancy variations using Archimedes' principle [START_REF] Justnes | An evaluation of methods for measuring chemical shrinkage of cementitious paste Nord[END_REF][START_REF] Justnes | Total and external chemical shrinkage of low W/C-ratio cement pastes[END_REF]. This method consists in introducing fresh cementing material into a flexible latex mold (prophylactic) and in immersing it in a volume of water at constant temperature in order to carry out the hydrostatic weighing of the sample. Taking good care to comply with the precautions listed by Justnes et al. [START_REF] Justnes | An evaluation of methods for measuring chemical shrinkage of cementitious paste Nord[END_REF], this technique allows for a satisfactory evaluation of the variations in volume of material under isothermal conditions. On the other hand, this method does not take into account the evolution of the thermal conditions within the material. Indeed, shrinkage can be activated by a variation in temperature due to the exothermal effect of the reaction of hydration, and possibly compensated by the thermal dilation involved by the increase in temperature. Moreover, the thermal gradient, which is considerable between the periphery and the core of a concrete structure, involves a gradient of stresses and influences the total shrinkage. The study first consists in developing an experimental tool, which allows us to measure the shrinkage at an early age by considering the temperature history undergone by the cement-based materi-al. We then present an experimental method that makes a separation of autogenous shrinkage and thermal deformation possible.

The terms used in the present paper are defined as follows:

Total deformation measured deformation of the mortar caused by cement reactions including autogenous and thermal deformations. Autogenous shrinkage is the difference between total deformation defined above and thermal deformation, which is calculated from the measured Thermal Dilation Coefficient (TDC).

Materials and experimental methods

Materials

The mix composition of the studied mortar is shown in Table 1. A Portland cement (CPA CEM I 52.5 R) containing 66.9% C 3 S, 19.2% C 2 S, 4.35% C 3 A and 1% C 4 AF was used. Water/cement (W/C) ratio was selected at 0.35 in order to avoid bleeding [START_REF] Justnes | An evaluation of methods for measuring chemical shrinkage of cementitious paste Nord[END_REF]. The aggregate was quartzite sand with a maximum nominal size of 4 mm.

Cement and sand were mixed in a 5-l laboratory mixer for 1 min before the addition of water and a further mixing period of 3 min to ensure a homogeneous mix.

Experimental methods

Measurements of external volume change of mortar caused by cement reactions were taken by hydrostatic weighing. The principle of this method consists in measuring the buoyancy variations of a sample at a very early age (Fig. 1). The studied mortar is introduced into a prophylactic immediately after casting. The prophylactic is turned inside out so that the lubricant is now inside it. This reduces the friction between mortar and the membrane. The prophylactic is then filled up with about 300 to 400 g of material [START_REF] Justnes | An evaluation of methods for measuring chemical shrinkage of cementitious paste Nord[END_REF]. The latex mold is then closed using a thin stainless steel wire. The greatest care is exercised to evacuate air. The presence of air in the mold is indeed undesirable because the contraction pores, which form during the hydration reaction, suck in this air and cause a reduction in the volume of the latex envelope that can wrongly be interpreted as shrinkage. The excess latex is then cut and the sample container is cleaned and weighed. Then, the sample is placed in the nacelle, which is hung on a balance, and immersed immediately in the water bath at 20°C (Fig. 1). The external volume change of mortar results in the modification of buoyancy force and varies the mass read on the balance. The accuracy of the balance was about 0.01 g. It can be noticed that the choice of the prophylactic guarantees a good seal with respect to the surrounding medium. Moreover, the elasticity of latex membrane allows free deformations of mortar.

We then make the 110 Â 220 mm cylindrical specimen, which is introduced in a quasi-adiabatic enclosure immediately after casting (Fig. 2). It can be noticed that all mortar specimens come from a single batch. Two thermocouples are placed in the specimen at 1 cm (point A) and 5 cm (point B) below the surface, respectively so as to obtain two different temperature histories.

The shape and the position of this sample make a representation of the conditions existing in a 44-cm thick floor possible (Fig. 2).

To be able to measure the deformations under various thermal conditions, it was necessary to develop a system which allows to apply to the bath the temperature recorded into the mortar. This system works according to the principle of comparison of the two temperatures thanks to a regulator (Fig. 1). When the bath temperature is lower than that of the mortar specimen, the regulator starts a resistor and a water pump. The power of the resistor makes it possible to obtain a fast increase in the temperature and thus accurate control. The role of the water pump is to homogenize the temperature of the water bath. The experimental device was set up in a room at 20°C and 50% relative humidity. Temperature of the bath, temperature of the sample, and the mass measured by the balance are logged continuously on the computer every 10 min during 24 h. The temperature histories are measured at 1 and 5 cm below the surface for two reasons. First, the temperatures reached in the core of the cylindrical specimen are too high so that measurements remain reliable in the bath. Moreover, it makes it possible to show that the temperature histories of two very close points are eventually very different.

It is worth noting that the prophylactic used in this study cannot be used to measure the shrinkage of concrete (the aggregates would be likely to puncture the rubber bag). So, the tests are carried out on a mortar in order to obtain a behavior closer to that of the concrete than that of the cement paste.

Correction of the obtained results

During an isothermal test at 20°C, the value read on the balance gives the volume change of the sample directly (specific gravity of water is constant). In this study, the temperature of water varies significantly (from 20°C to 65°C) leading to a variation of the water density. Carrying out a correction of this value to determine the volume change of mortar is then necessary.

The formula giving the density of water r e (T) between 17°C and 100°C [START_REF] Duriez | Nouveau Traite  de Mate Âriaux de Construction[END_REF] is expressed by Eq. ( 1):

r w T 1 À T À 3X982 2 466Y 700 T 273 T 67 350 À T 365 À T 1
It then becomes a question of writing the balance of the system of forces acting on the sample. By considering an isotropic behavior of material, the linear deformation is given by one-third of the volumetric deformation. The linear deformation is then obtained by Eq. ( 2) resulting from the calculation detailed in [START_REF] Chopin | De Âveloppement d'un dispositif de mesure du retrait des mortiers au jeune age[END_REF]:

e 1 3 r iw r w T À 1 À M r V i r w T ! 2
where T: temperature of bath (°C); r iw : initial density of water (g/cm 3 ); V i : initial volume of the sample (cm 3 ); r w (T): density of water as function of temperature (g/cm 3 ); e: linear deformation of mortar (mm/m); M r : mass read on the balance (g).

Results and discussions

Three series of tests have been carried out. The first series consists in measuring the volume change of the sample at a constant temperature (20°C). This series allows a validation of the experimental device while referring to previous studies. It also allows us to obtain a reference curve for the analysis of the results in quasi-adiabatic conditions. The two other series of tests consist in applying to the sample the temperature histories measured at 1 and 5 cm below the surface of a 110 Â 220-mm mortar specimen.

Reference test at 20°C

For this test, the sample is placed in the water bath at a temperature of 20 0.5°C. Simultaneously, Vicat test apparatus is used to determine the initial and final set of the mortar.

The results of this test (Fig. 3) show traditional kinetics of the shrinkage also observed by other authors [START_REF] Justnes | Total and external chemical shrinkage of low W/C-ratio cement pastes[END_REF]. A significant amount of shrinkage is observed during the first 4 h. Then, the shrinkage curve flattens (point A) and then the deformation evolves slowly.

It is significant to note that point A corresponds to the end of setting period (about 240 min) characterized by a quasi-constant depth of penetration of the needle into the mortar. At this time, the skeleton is formed and is stiff sufficiently to restrain deformation caused by hydration reactions.

Measurement of external volume change in quasi adiabatic conditions

Fig. 4 show the temperature evolution into the cylindrical specimen. The maximum temperature ob-served was 65°C at point B and 46°C at point A about 7 h after placing the mortar. The maximum temperature differential between the two points was about 20°C. It is important to note that the critical temperature difference to avoid cracking during cooling is 20°C [START_REF] Neville | Properties of Concrete[END_REF].

Effects of temperature history on total deformation are shown in Figs. 5 and6. It appears that the kinetics and amplitude of these curves are very different from those obtained at 20°C. Indeed, it can be observed a swelling of the sample during the period when the increase in temperature is most significant (5 to 7 h).

This swelling occurs earlier and is all the more marked since the temperature reached is high (Fig. 6). This leads to a phenomenon, which is more important for points located near the core of the structure where the temperature reaches 60°C.

The total measured deformation includes autogenous shrinkage and thermal deformation which depends on the TDC of mortar. For better analysis of these curves, it would be very instructive to separate the thermal deformation from the autogenous shrinkage. Following this work, we propose a new experimental method to determine TDC based on hydrostatic weighing method.

TDC measurements

The sample studied, placed under variable conditions of temperature, undergoes volumetric variations of thermal origin which are difficult to evaluate since the microstructure of material varies with time. Measuring this coefficient by using the hydrostatic weighing device brought under control in temperature is possible. Indeed, the thermal dilation of a sample can be caused by an increase in the temperature of the bath. If this rise in temperature is sufficiently fast, the measured deformation will be only of thermal origin.

The TDC is measured immediately after batching every hour during 6 h. Tests are carried out on six different samples. The measurement of TDC is carried out by the method of spontaneous heating of the sample. For each measurement, the temperature of the bath is initially about 20°C and increased 10°C during 10 min. The results of these experiments are reported in Table 2 and plotted in Fig. 7.

Our own measurement device is not sensitive enough to consider the low TDC values reliable. Therefore, the continuation of the TDC curve after 6 h is extrapolated from the experimental results and from the kinetics of the curves found in Ref. [START_REF] Neville | Properties of Concrete[END_REF] according to Eq. ( 3):

TDCt 77 Ã e 0X75 t 2X5 10 3
where, TDC (mm/m/°C); t: age of mortar (h). Fig. 7 shows that TDC is not constant at an early age of mortar. At 1 h, the TDC reaches 90 mm/m/°C, then it decreases very quickly and reaches 15 mm/m/°C after 6 h. These results are in agreement with the findings of Laplante [START_REF] Laplante | Proprie Âte Âs Me Âcaniques Des Be Âtons Durcissants: Analyse Compare Âe des Be Âtons Classiques et a Tres Hautes Performances[END_REF] for ordinary concrete. The amplitude, however, is different since the compositions are themselves different. It is significant to note that the rapid decrease of TDC (during the first 5 h) corresponds to the period when the material passes from a liquid state to a hardened state. Laplante [START_REF] Laplante | Proprie Âte Âs Me Âcaniques Des Be Âtons Durcissants: Analyse Compare Âe des Be Âtons Classiques et a Tres Hautes Performances[END_REF] suggests an explanation for the particular kinetics of TDC based on theory of percolation [START_REF] Acker | Comportement Me Âcanique du Be Âton: Apports de l'Approche Physico-Chimique[END_REF]. The high TDC value observed at an early age corresponds to the time when the water phase dominates. When the mortar starts to stiffen, the TDC may not be influenced by the volume changes due to the free water movements but may be essentially influenced by the characteristics of porous structure.

Proposition of a method to separate thermal and autogenous deformations

To be able to determine autogenous shrinkage in a satisfactory manner, it is necessary to calculate the deformation of thermal origin using the TDC values. For each new measurement, the thermal strain is calculated compared to the previous measurement (step-by-step) according to Eq. ( 4):

en en À 1 T n À T n À 1 TDCn TDCn À 1 2 ! 4
where T(n) is the temperature, TDC(n) the thermal dilation coefficient and e(n) the thermal deformation at the n time step. By subtracting the thermal deformation from the measured total deformation, the true self-desiccation shrinkage is obtained. The principle of this calculation is illustrated in Fig. 8 for the temperature history with 46°C maximum.

Fig. 8 shows that the material undergoes a significant thermal dilation during temperature rise period which partly compensates for the shrinkage. Then, the thermal dilation is followed by a return of dilation (thermal shrinkage) during the cooling stage, which occurs on hardened material. Therefore, thermal deformation is added to the autogenous shrinkage instead of compensating for it as what happened during the first phase.

Fig. 9 presents the calculated autogenous shrinkage for the three temperature histories using the TDC values given in Fig. 7. The main observation is that the autogenous shrinkage amplitude of sample with 65°C maximum (point A) is about three times higher than that observed in the quasi-isothermal test at 20°C. It can be concluded that the amplitude of autogenous shrinkage is strongly dependent on the temperature history the material undergoes. Therefore, for a good estimate of the shrinkage amplitudes of a cement-based material, temperature must be taken into account in testing. These results could be of great importance for practical applications; in particular, for highperformance concrete, which has a significant autogenous shrinkage at an early age.

Conclusions

The main conclusions that can be drawn from this experimental investigation are the following:

1. Design and validation of an experimental process allows us to determine total deformation of mortar at an early age under real conditions of construction. Indeed, the tests can determine the volume change vs. the temperatures recorded in mortar in real time. 2. An experimental method is proposed to determine the Thermal Dilation Coefficient of cement-based materials. This made it possible to separate thermal and autogenous deformations. 3. The amplitude of autogenous shrinkage is strongly affected by the temperature history of the mortar. 4. Not to take into account the temperature history of material in a shrinkage test at early age leads, in our opinion, to a false estimate of autogenous shrinkage amplitude.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Sample giving the temperature history.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Temperature histories in mortar specimen.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Comparison of the total deformations for different temperature histories.

Table 1

 1

	Mix composition
	Cement (kg/m 3 )	Aggregates (kg/m 3 )	Water (kg/m 3 )	W/C
	700	1216	245	0.35

Fig. 1. Schematic drawing of the experimental system.

Table 2

 2

	Experimental measurement of TDC
	Age (h)	0.75	1	2	3	4	5	6
	TDC (mm/m/°C)	87	80	61	50	25	21	15

Fig. 8. Separation of thermal and autogenous deformations for 25 46°C temperature history.