X Claeys

Quasi-local multi-trace boundary integral formulations

In the context of time harmonic wave scattering by piecewise homogeneous penetrable objects, we present a new variant of the multi-trace boundary integral formulation, that differs from the local multi-trace approach of [Jerez-Hanckes & Hiptmair, 2012] by the presence of regularisation terms involving boundary integral operators and localised at junctions i.e. points where at least three subdomains abut. We prove well-posedness and quasi-optimal convergence of conforming Galerkin discretisations for this new formulation, and present numerical results.

Introduction

Due to its potentially large scope of application, the analysis of boundary integral formulation for wave scattering by piecewise homogeneous objects has received a lot of attention in past decades. Several approaches, such as PMCHWT [START_REF] Miller | Computer Techniques for Electromagnetics, chapter Integral equation solution of three-dimensional scattering problems[END_REF][START_REF] Chang | A surface formulation or characteristic modes of material bodies[END_REF][START_REF] Harrington | Boundary integral formulations for homogeneous material bodies[END_REF][START_REF] Petersdorff | Boundary Integral Equations for Mixed Dirichlet, Neumann and Transmission Problems[END_REF][START_REF] Buffa | Remarks on the discretization of some noncoercive operator with applications to the heterogeneous Maxwell equations[END_REF] or the BETI method [START_REF] Langer | Boundary Element Tearing and Interconnecting Methods[END_REF][START_REF] Of | The all-floating boundary element tearing and interconnecting method[END_REF][START_REF] Of | Boundary element tearing and interconnecting domain decomposition methods[END_REF], have been derived rather early for treating scattering by geometrically complex structures by means of boundary integral operators. However, the existing approaches did not lend themselves (at least as local solvers) to some of the most popular preconditioning techniques such as Calderón's preconditioner [START_REF] Christiansen | Des préconditionneurs pour la résolution numérique des équations intégrales de frontière de l'acoustique[END_REF][START_REF] Christiansen | A preconditioner for the electric field integral equation based on Calderon formulas[END_REF][START_REF] Steinbach | The construction of some efficient preconditioners in the boundary element method[END_REF].

This was the main motivation for the recent development of the so-called multi-trace formulations that were prone to Calderón preconditioning and, at the same time, adapted to boundary integral formulation of wave propagation in piecewise homogeneous media admitting points where three or more subdomains could be adjacent to each other. The so-called local multi-trace formulation, first proposed in [START_REF] Hiptmair | Multiple traces boundary integral formulation for Helmholtz transmission problems[END_REF], consists in writing wave equations separately in each subdomain of the propagation medium by means integral operators, and to couple subdomain together by means of a local transmission operator coming into play via a relaxation term of the equation. This formulation has been dubbed local because no integral operator is required to enforce transmission conditions.

As regards effective numerical performance, this formulation looks appealing. But few theory could be established. This is mostly due to the transmission operator that acts on Sobolev trace spaces, and involves at the same time restrictions and extensions by zero which are not continuous operations in natural trace norms. So far this formulation has been proved to admit a unique solution for general multi-dimensional cases, but no Garding's inequality has been established, and discrete inf-sup conditions for conformal discretisations of this formulation has been studied only in a 2-D case.

In the present article we introduce a regularised version of the local multi-trace formulation. The regularisation terms are localised at the junction points, and this new variant of the multi-trace formulation of [START_REF] Hiptmair | Multiple traces boundary integral formulation for Helmholtz transmission problems[END_REF] lends itself to much more standard analysis: all operators are continuous in standard Sobolev trace spaces, a Garding inequality can be proved, which implies quasi-optimal approximation property for conformal Galerkin discretisations. As regards numerical performances, this new formulation also appears slightly more accurate compared to the original local multi-trace formulation. On the other hand, iterative solvers may be a little bit slower.

The outline of this article is as follows. In section 1 and 2, we set some notations and describe precisely the problem under consideration: a standard scalar wave scattering by an arrangement of homogeneous objects. In section 3 and 4 we recall well established results of potential theory and provide a brief review of the derivation of the local multi-trace formulation considered in [START_REF] Hiptmair | Multiple traces boundary integral formulation for Helmholtz transmission problems[END_REF]. In Section 5 we introduce a new way of enforcing transmission conditions and, in Section 6, we deduce from this a regularised local multi-trace formulation.

In the last section we present numerical results. We conclude in Section 7 with numerical experiments.

Setting of the problem

We first present in detail the problem that we will consider in this document. For d = 1, 2, 3, . . . , consider a partition of the free space as R d := ∪ n j=0 Ω j where the Ω j 's are Lipschitz domains. We assume that each Ω j is bounded except Ω 0 . In the sequel we shall refer to the boundary of each subdomain by Γ j := ∂Ω j , and also set Γ j,k := Γ j ∩ Γ k = ∂Ω j ∩ ∂Ω k for reference to interfaces. The union of all interfaces will be denoted

Σ := n ∪ j=0 Γ j = ∪ 0≤j<k≤n Γ j,k .
Following the standard terminology of domain decomposition, this set will be called the skeleton of the partition. As regards function spaces we follow the usual notations; given some open subset ω ⊂ R d , we define H 1 (ω

) := {v ∈ L 2 (ω) | ∇v ∈ L 2 (ω)} with v 2 L 2 (ω) := v 2 L 2 (ω) + ∇v 2 L 2 (ω) , and H 1 (∆, ω) := {v ∈ H 1 (ω) | ∆v ∈ L 2 (ω)}. If H(ω) is any one of these spaces, H loc (ω) := {v | ϕ v ∈ H(ω) ∀ϕ ∈ C ∞ K (R d )}, where C ∞ K (R d ) refers to the space of C ∞ function with compact support.
We consider a very standard wave scattering problem (so-called transmission problem), imposing Hemlholtz equation in each subdomain, as well as transmission conditions across interfaces:

find u ∈ H 1 loc (R d ) such that                -∆u -κ 2 j u = 0 in Ω j ∀j = 0 . . . n u -u inc is κ 0 -outgoing in Ω 0 u| Γ j -u| Γ k = 0 ∂ n j u| Γ j + ∂ n k u| Γ k = 0 on Γ j,k = Γ j ∩ Γ k , ∀j, k = 0 . . . n (1)
where ∂ n j refers to the outgoing normal derivative on ∂Ω j . In the equation above u inc ∈ H 1 loc (R d ) is a known source term of the problem satisfying -∆u incκ 2 0 u inc = 0 in R d . In addition, we assume that κ j > 0 for all j = 0 . . . n. The outgoing radiation condition refers to Sommerfeld's condition i.e. we say that v is κ-outgoing if

lim r→∞ ˆ∂Br |∂ r v -ıκv| 2 dσ r = 0,
where B r is the ball of centre 0 and radius r, and ∂ r is the radial derivative. Problem ( 1) is known to admit a unique solution, see [START_REF] Petersdorff | Boundary Integral Equations for Mixed Dirichlet, Neumann and Transmission Problems[END_REF] for example.

As this problem involves transmission conditions, and since we are interested in boundary integral formulations of it, we need to introduce corresponding trace spaces and trace operators.

Given any Lipschitz open subset ω ⊂ R d with bounded boundary, recall (see e.g. [25, Thm. 2.6.8]) that the map ϕ → ϕ| ∂ω can be extended as a continuous map from H 1 loc (ω) to L 2 (∂ω). The set of Dirichlet traces on ∂ω, denoted H 1/2 (∂ω), is the range of this map. It is a Banach space when equipped with the norm v H 1/2 (∂ω) := inf{ ϕ H 1 (ω) , ϕ ∈ H 1 (ω) and ϕ| ∂ω = v}. The set of Neumann traces on ∂ω, denoted H -1/2 (∂ω), is the topological dual to this space, equipped with the canonical dual norm

q H -1/2 (∂ω) := sup v∈H 1/2 (∂ω) | q, v | v H 1/2 (∂ω)
.

Recall also (see e.g. [START_REF] Sauter | Boundary element methods[END_REF]Thm 2.7.7]) that the normal derivative ϕ → n • ∇ϕ| ∂ω can be extended by continuity as an operator mapping H 1 loc (∆, ω) onto H -1/2 (∂ω).

As an application of the preceding remarks, every subdomain Ω j gives rise to continuous boundary trace operators γ j d : H 1 loc (Ω j ) → H 1/2 (∂Ω j ) and γ j n : H 1 loc (∆, Ω j ) → H -1/2 (∂Ω j ) (so-called Dirichlet and Neumann traces) uniquely defined by

γ j d (ϕ) := ϕ| ∂Ω j and γ j n (ϕ) := n j • ∇ϕ| ∂Ω j ∀ϕ ∈ C ∞ (Ω j ).
In the definition above n j refers to the vector field normal to ∂Ω j pointing toward the exterior of Ω j . We will also need a notation to refer to an operator gathering both traces in a single array

γ j (u) := (γ j d (v), γ j n (v)). Define γ j d,c , γ j n,c
in the same manner as γ j d , γ j n with traces taken from the exterior of Ω j , and set in addition

γ j c (v) := (γ j d,c (v), γ j n,c (v)
). We will also need to refer to mean values and jumps to these trace operators, so we set

{γ j (u)} := 1 2 γ j (u) + γ j c (u)
and [γ j (u)] := γ j (u)γ j c (u).

Trace spaces

We want to recast Problem (1) into variational boundary integral equations. We aim for boundary integral equations set in natural trace spaces. The most fundamental trace space we can introduce consists in the multi-trace space [8, Sect. 2.1], the Cartesian product of local traces:

H(Σ) := H(Γ 0 ) × • • • × H(Γ n ) where H(Γ j ) := H + 1 2 (Γ j ) × H -1 2 (Γ j ) . (2)
We endow each H(Γ j ) with the norm (v, q) H(Γ j ) := ( v 2 H 1/2 (Γ j ) + q 2 H -1/2 (Γ j ) )1/2 , and equip H(Σ) with the norm naturally associated with the cartesian product

u H(Σ) := u 0 2 H(Γ 0 ) + • • • + u n 2 H(Γn) 1 2
for u = (u 0 , . . . , u n ) ∈ H(Σ) 1 . We write , Γ j for the duality pairing between H 1/2 (Γ j ) and H -1/2 (Γ j ). In the sequel we shall repeatedly refer to the continuous operator γ : Π n j=0 H 1 loc (∆, Ω j ) → H(Σ) defined by γ(u) := (γ 0 (u), . . . , γ n (u)), where Π n j=0 H 1 loc (∆, Ω j ) should be understood as the set of u ∈ L 2 loc (R d ) such that u| Ω j ∈ H 1 loc (∆, Ω j ) for all j. We also need a bilinear duality pairing for H(Γ j ) and H(Σ); we opt for the skew-symmetric version

u, v := n j=0 [u j , v j ] Γ j where u j p j , v j q j Γ j := u j , q j Γ j -v j , p j Γ j . (3)
This particular choice is well adapted to the forthcoming analysis. Note that under the pairing , , the space H(Σ) is its own topological dual, and it is easy to show, using the duality between H 1/2 (Γ j ) and H -1/2 (Γ j ), that the pairing , induces an isometric isomorphism between H(Σ) and its dual H(Σ) ′ , equivalent to the inf-sup condition inf

v∈H(Σ) sup u∈H(Σ) | u, v | u H(Σ) v H(Σ) = 1. (4)
Next, as in [8, Sect. 2.2], [10, Sect. 3.1], we introduce the so-called single-trace space that consists in collections of traces that comply with transmission conditions. This space can be defined by

X(Σ) := clos H(Σ) { γ(u) = (γ j (u)) n j=0 | u ∈ C ∞ (R d ) } (5)
where clos H(Σ) refers to the closure with respect to the norm of H(Σ). By construction, this is a closed subspace of H(Σ). Note also that a function

u ∈ H 1 loc (∆, Ω 0 ) × • • • × H 1 loc (∆, Ω n ) satisfies the transmission conditions of (1), if and only if γ(u) = (γ j (u)) n j=0 ∈ X(Σ). In particular, if u ∈ H 1 (∆, R d ) then γ(u) = (γ j (u)) n j=0 ∈ X(Γ), see [8, §2.2
] and [7, Lemma 7.4]. In the sequel, we will use this space to enforce transmission conditions. The single-trace space admits a simple weak characterisation, see [8, Prop.2.1].

Lemma 2.1.

For any u ∈ H(Σ) we have: u ∈ X(Σ) ⇐⇒ u, v = 0 ∀v ∈ X(Σ).

Summary of potential theory

In this paragraph, we shall remind the reader of well established results concerning the integral representation of solutions to homogeneous Helmholtz equation in Lipshitz domains. A detailed proof of the statements contained in the present paragraph can be found for example in [START_REF] Sauter | Boundary element methods[END_REF]Chap.3].

Let the function G κ (x) refer to the outgoing Green's kernel associated to the Helmholtz operator -∆κ 2 . For each subdomain Ω j , for any (v, q) ∈ H(Γ j ) and any x ∈ R d \ Γ j , define the potential operator

G j κ (v, q)(x) := ˆΓj q(y) G κ (x -y) + v(y) n j (y) • (∇G κ )(x -y)dσ(y) . (6)
The operator G j κ maps continuously [START_REF] Sauter | Boundary element methods[END_REF]Thm 3.1.16]. In particular G j κ can be applied to a pair of traces of the form u = γ j (v). This potential operator can be used to write a representation formula for solutions to homogeneous Helmholtz equations, see [START_REF] Sauter | Boundary element methods[END_REF]Thm 3.1.6].

H(Γ j ) into H 1 (∆, Ω j ) × H 1 (∆, R d \ Ω j ), see
Proposition 3.1. Let u ∈ H 1 loc (Ω j ) satisfy ∆u + κ 2 j u = 0 in Ω j . Assume in addition that u is κ j -outgoing if j = 0. We have the representation formula G j κ j (γ j (u))(x) = u(x) for x ∈ Ω j 0 for x ∈ R d \ Ω j . Similarly, if v ∈ H 1 loc (R d \ Ω j ) satisfies ∆v + κ 2 j v = 0 in R d \ Ω j ,
as well as a radiation condition in the case j = 0, then we have

G j κ j (γ j c (v))(x) = -v(x) for x ∈ R d \ Ω j , and G j κ j (γ j c (v))(x) = 0 for x ∈ Ω j .
The potential operator (6) also satisfies a remarkable identity, known as jump formula, describing the behaviour of G j κ j (v)(x) as x crosses Γ j = ∂Ω j . The notations that we have introduced allow to write it in a very condensed manner, namely

[γ j ] • G j κ j (v) = v ∀v ∈ H(Γ j ) . (7)
This can also be rewritten [γ j ] • G j κ j = Id. We refer the reader to [START_REF] Sauter | Boundary element methods[END_REF]Thm.3.3.1] (the jump formula are more commonly written in the form of four equations in the literature). Proposition 3.1 shows that, if u is solution to a homogeneous Helmholtz equation in Ω j (and is outgoing if

j = 0) then γ j • G j κ j (γ j (u)) = γ j (u)
. This actually turns out to be a characterisation of solution to homogeneous Helmholtz equation.

Proposition 3.2. Define C in κ (Γ j ) := {γ j (u) | u ∈ H 1 loc (Ω j ), ∆u + κ 2 u = 0 in Ω j , u κ-outgoing if j = 0 }. Then γ j • G j κ : H(Γ j ) → H(Γ j ) is a continuous projector, called Calderón projector interior to Ω j , whose range coincides with C in κ (Γ j ) i.e. for any v ∈ H(Γ j ) γ j • G j κ (v) = v ⇐⇒ v ∈ C in κ (Γ j ) .
Similarly, defining

C out κ (Γ j ) := {γ j c (u) | u ∈ H 1 loc (Ω j ), ∆u+κ 2 u = 0 in R d \Ω j , u outgoing if j = 0 }, we have γ j • G j κ (v) = 0 if and only if v ∈ C out κ (Γ j ).
For a detailed proof of this proposition, see [START_REF] Sauter | Boundary element methods[END_REF]Prop.3.6.2]. A direct consequence of this is the so-called Calderón identity ({γ j } • G j κ ) 2 = Id/4. We shall repeatedly use this characterisation as a convenient way to express wave equations in the subdomains Ω j . Here is another characterisation of the space of Cauchy data which was established in [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF]Lemma 6.2]. Lemma 3.1. Consider any j = 0, . . . n, and any κ ∈ C \ {0} such that ℜe{κ} ≥ 0, ℑm{κ} ≥ 0. Then for any u ∈ H(Γ j ) we have:

u ∈ C in κ (Γ j ) ⇐⇒ [u, v] Γ j = 0 ∀v ∈ C in κ (Γ j ). Similarly we have u ∈ C out κ (Γ j ) ⇐⇒ [u, v] Γ j = 0 ∀v ∈ C out κ (Γ j ).
The results that we have stated above hold for any j = 0 . . . n. They can be re-expressed in a matrix form, which will help reducing notations. First we introduce the continuous operator A :

H(Σ) → H(Σ) defined by 1 2 Au, v := n j=0 {γ j } • G j κ j (u j ), v j Γ j . for all u = (u j ) n j=0 , v = (v j ) n j=0 ∈ H(Σ). We also set C α (Σ) := C α κ 0 (Γ 0 ) × • • • × C α κn (Γ n ) for α = in, out. Observe that (Id ± A)/2 are projectors, according to Proposition 3.2, with Ker(A -Id) = Range(A + Id) = C in (Σ) and Ker(A + Id) = Range(A -Id) = C out (Σ).
The notations we have just introduced lead to the following compact reformulation of the well-posedness of (1), see [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF]Prop.6.1] for a detailed proof.

Lemma 3.2. X(Σ) ⊕ C in (Σ) = H(Σ).

Local multi-trace formulation

Now we would like to recall the derivation of the local multi-trace formulation, that was first introduced in [START_REF] Hiptmair | Multiple traces boundary integral formulation for Helmholtz transmission problems[END_REF]. In the present section we will only recall already established results concerning local multi-trace formulation, not giving a complete overview, and refer the reader to [START_REF] Hiptmair | Multiple traces boundary integral formulation for Helmholtz transmission problems[END_REF] for detailed proof of these results.

A key ingredient of the local multi-trace theory is an operator yielding a characterisation of transmission conditions of (1). Let us denote

L 2 (Σ) := Π n j=0 (L 2 (Γ j ) × L 2 (Γ j )). For u = (u j , p j ) n j=0 ∈ L 2 (Σ), and v = (v j , q j ) n j=0 ∈ L 2 (Σ)
, we define the transmission operator Π :

L 2 (Σ) → L 2 (Σ) by v = Π(u) ⇐⇒ v j = + u k q j = -p k on Γ j,k ∀j, k = 0 . . . n. (8)
Clearly, for any function

u ∈ H 2 loc (R d ) we have γ(u) := (γ j (u)) n j=0 ∈ L 2 (Σ) and γ(u) = Π(γ(u)). Conversely, considering any function u ∈ L 2 loc (R d ) such that u| Ω j ∈ H 2 loc (Ω j ) for all j = 0 . . . n, then γ(u) ∈ L 2 (Σ) is well defined, and if γ(u) = Π(γ(u)) then we have u ∈ H 2 loc (R d ).
Routine calculus shows that the transmission operator satisfies the following remarkable identities

Π 2 = Id , Π(v) = Π(v) and Π(u), v = Π(v), u ∀u, v ∈ L 2 (Σ). ( 9
)
As is readily checked, the operator Π maps continuously H(Σ) onto H(Σ) under the assumption that Γ j,k is a Lipschitz manifold without boundary for any j, k, which corresponds to situations where there is no junction point i.e. points where at least three subdomains abut. Without this assumption, Π does not map H(Σ) into itself. We actually have the continuous mapping Π : H(Σ) → H(Σ) ′ where H(Σ) ′ is the topological dual to the space

H(Σ) := { v = (v j , q j ) n j=0 ∈ H(Σ) | v j | Γ j,k ∈ H + 1 2 (Γ j,k ) , q j | Γ j,k ∈ H -1 2 (Γ j,k ) ∀j, k = 0 . . . n } equipped with v H(Σ) := n j=0 n k=0 k =j v j 2 H 1/2 (Γ j,k ) + q j 2 H -1/2 (Γ j,k ) 1/2
where, following [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]Chap.3], for σ = ±1/2 we adopt the standard definitions

H σ (Γ j,k ) = {ϕ| Γ j,k ϕ ∈ H σ (Γ j )} and H σ (Γ j,k ) := H -σ (Γ j,k ) ′ .
The space H(Σ) equipped with the norm above is a Banach space. We have

H(Σ) ⊂ H(Σ) = H(Σ) ′ ⊂ H(Σ) ′ . Besides, H(Σ) is dense in H(Σ) in the sense of H(Σ)
, and the norm H(Σ) is strictly stronger than the norm

H(Σ) , see [17, §2.2.2]
. Elementary arguments on trace spaces show that, for any u ∈ H(Σ), we have u ∈ X(Σ) ⇐⇒ u = Π(u). Since Π 2 = Id, this can be simply rewritten in the following manner. Now consider u = (γ j (u)) n j=0 the traces of the unique solution u to Problem (1), and denote u inc := (γ 0 (u inc ), 0, . . . , 0). The homogeneous wave equation satisfied by u in each subdomain can be reformulated by means of Calderón projectors (A-Id)(u-u inc ) = 0. Moreover we must have u = Π(u) since u satisfies the transmission conditions of (1). Setting f := (A -Id)u inc , this implies u ∈ H(Σ) and

(A -Π)u, v = f, v ∀v ∈ H(Σ). (10)
For this variational formulation, there is existence [START_REF] Hiptmair | Multiple traces boundary integral formulation for Helmholtz transmission problems[END_REF]Thm.11] and uniqueness [17, Thm.9] to its solution.

Lemma 4.2.

For any f ∈ H(Σ), the formulation (10) admits a unique solution u ∈ H(Σ) that depends continuously on f. In particular Ker(A -Π) = {0}.

Whether this formulation satisfies a Garding inequality remains an open question though. For this reason, so far, uniform discrete inf-sup conditions (and hence quasi-uniform approximation property) for Galerkin discretisations of (10) could only be established in the particular case of 2-D scalar problems.

Reformulation of transmission conditions

The difficulties arising in the analysis of [START_REF] Claeys | Multi-trace boundary integral equations[END_REF] are mainly due to the presence of junction points. Indeed, with such points, the operator Π is no more continuous in H(Σ) that, at the same time, appears as a natural functional setting for the boundary integral formulations of the transmission problem (1). In the present paper, we propose a specific additional treatment of junction points that leads to a regularised operator Π for which continuity in the norm of H(Σ) is restored.

Treatment of junction points

The particular treatment that we propose will rely on integral operators. In this paragraph, K 0 will refer to a Green kernel of an Helmholtz equation i.e. we will assume that it satisfies -∆K 0λ 2 K 0 = δ 0 in R d in the sense of distributions, for some λ ∈ C. The choice of λ can be decorrelated from the wave numbers κ j coming into play in [START_REF] Boffi | Finite element approximation of eigenvalue problems[END_REF]. So the Green kernels K 0 and G are independent.

We also consider a function

K ∈ C ∞ (R d \ {0}
) such that K coincides with K 0 in a neighbourhood of 0. It is important to note that, the kernel K (x) shares the same singularity at x = 0 as K 0 (x). Except in a neighbourhood of 0, we do not impose that K (x) satisfies any particular equation. It does not a priori need to be the Green kernel of any equation.

In addition, the K may be chosen so as to admit a bounded support located closely around

x = 0. For any (v, q) ∈ H(Γ j ), set K j (v, q)(x) := ˆ∂Ω j K (x -y) q(y) + n j (y) • (∇K )(x -y) v(y) dσ(y) (11)
Because the kernel K only differs from a Green's kernel up to a smooth factor, the operators K j satisfy many remarkable properties inherited from standard potential operators. For example, they satisfy a symmetry property, and jump formula.

Lemma 5.1.

The operator

K j maps continuously H(Γ j ) into H 1 (∆, Ω j ) × H 1 loc (∆, R d \ Ω j ). Moreover, for any u, v ∈ H(Γ j ), w ∈ H(Γ k ) with k = j, we have the following elementary properties: i) [γ j ] • K j (u) = u, ii) [{γ j } • K j (u), v] Γ j = [{γ j } • K j (v), u] Γ j , iii) [γ j • K k (w), u] Γ j = [γ k • K j (u), w] Γ k .

Proof:

Let K j 0 refer to the same operator as K j but constructed with the Green kernel K 0 instead of K , and set K j * := K j -K j 0 . The properties announced above already hold with K j 0 instead of K j , see for example [START_REF] Costabel | Boundary integral operators on Lipschitz domains: elementary results[END_REF]. Besides the kernel of K j * is C ∞ since K and K 0 coincide in a neighbourhood of 0. This proves the continuity properties announced for K j , and shows in addition that [γ j ] • K j * = 0. Moreover, it is a straightforward consequence of Fubini's theorem, that ii) and iii) are verified with K j replaced by K j * . Since we also know that ii) and iii) are verified with K j 0 (see lemmas 3.6 and 3.7 in [START_REF] Claeys | Integral equations for acoustic scattering by partially impenetrable composite objects[END_REF] for example), this concludes the proof.

We may rely on combinations of all operators K j for reformulating transmission conditions across interfaces. Indeed consider the sum of the K j over all subdomains i.e. for any u = (u j ) n j=0 ∈ H(Σ), set

K(u)(x) := n j=0 K j (u j )(x), for x ∈ R d \ Σ. ( 12
)
This operator is close to the multi-potential operator introduced in [START_REF] Claeys | A single trace integral formulation of the second kind for acoustic scattering[END_REF][START_REF] Claeys | A second-kind galerkin boundary element method for scattering at composite objects[END_REF]. It maps continuously

H(Σ) into Π n j=0 H 1 loc (∆, Ω j ).
It is important to note, though, that the same kernel has been considered for all terms in the sum [START_REF] Colton | Integral equation methods in scattering theory[END_REF].

Recall that we have introduced a global trace operator defined by γ(v) := (γ 0 (v), . . . , γ n (v)) ∈ H(Σ). We can combine this global trace with the operator (12) which yields a continuous map γ • K : H(Σ) → H(Σ). This operator satisfies several remarkable properties.

Lemma 5.2. γ • K(u), v = γ • K(v), u + u, v
This result is obtained straightforwardly by expanding K according to [START_REF] Colton | Integral equation methods in scattering theory[END_REF], and then applying the properties stated in Lemma 5.1, together with the simple identity

γ j = {γ j } + 1 2 [γ j ]. Lemma 5.3. K(u)(x) = 0 ∀u ∈ X(Σ), ∀x ∈ R d .

Proof:

Choose an arbitrary x ∈ R d \ Σ, and let χ ∈ C ∞ (R d ) refer to a cut-off function such that χ(y) = 1 for y in a neighbourhood of Σ, χ(y) = 0 for y in a neighbourhood of x and supp(χ) is bounded. Denote K x (y) := χ(y)K (xy) so that K x ∈ C ∞ (R d ) and K x has compact support. Take an arbitrary u ∈ C ∞ (R d ). Using the explicit definitions [START_REF] Claeys | A second-kind galerkin boundary element method for scattering at composite objects[END_REF] and [START_REF] Colton | Integral equation methods in scattering theory[END_REF] we obtain

K(γ(u))(x) = n j=0 ˆ∂Ω j γ j d (K x )γ j n (u) -γ j n (K x )γ j d (u)dσ = n j=0 ˆΩj K x ∆u -u ∆K x dy = ˆRd K x ∆u -u ∆K x dy = 0.
Since this holds for any x ∈ R d \ Σ, we conclude that K(γ(u)) for any u ∈ C ∞ (R d ). In addition, since the elements of the form γ(u) for u ∈ C ∞ (R d ) are dense in X(Σ), according to [START_REF] Christiansen | A preconditioner for the electric field integral equation based on Calderon formulas[END_REF], this concludes the proof.

Lemma 5.4. u -γ • K(u) ∈ X(Σ) ∀u ∈ H(Σ).
Proof: Take u ∈ H(Σ). According to Lemma 5.2 and Lemma 5.3, for any v ∈ X(Σ) we have

u -γ • K(u), v = u, v -γ • K(u), v = -γ • K(v)
, u = 0. Now there only remains to apply Lemma 2.1 which characterises the elements of X(Σ).

Corollary 5.1. The operator γ • K : H(Σ) → H(Σ) is a continuous projector with Ker(γ • K) = X(Σ).

Proof:

Lemma 5.3 clearly shows that X(Σ) ⊂ Ker(γ • K). On the other hand, take any u ∈ Ker(γ • K). Applying Lemma 5.4, we obtain directly u = uγ • K(u) ∈ X(Σ). We conclude that Ker(γ • K) = X(Σ). This, combined with Lemma 5.4, proves that γ

• K • (Id -γ • K) = 0. We deduce that (γ • K) 2 = γ • K • (γ • K -Id) + γ • K = γ • K.
This last result shows that the operator γ • K can be used to characterise the elements of X(Σ), and to propose alternatives to operator Π. The following proposition yields definition to possible alternative operators. The proof is a direct consequence of Lemma 5.2 to 5.4 and Corollary 5.1.

Proposition 5.1. Consider the operator K given by [START_REF] Colton | Integral equation methods in scattering theory[END_REF], and define the continuous operator

Q := Id -2γ • K. We have Q 2 = Id and Q(u), v = Q(v), u for all u, v ∈ H(Σ). Moreover for any u ∈ H(Σ), we have u ∈ X(Σ) ⇐⇒ u = Q(u).
Clearly the operator Q introduced above is an isomorphism since Q 2 = Id. It actually satisfies in addition a generalised Garding inequality. Proposition 5.2. Define Θ : H(Σ) → H(Σ) by Θ(u) = (-u j , p j ) n j=0 for any u = (u j , p j ) ∈ H(Σ). Then there exists a compact operator R : H(Σ) → H(Σ) and a constant C > 0 such that

ℜe (-Q + R)u, Θ(u) ≥ C u 2 H(Σ) ∀u ∈ H(Σ). (13)
Proof:

Let K • refer to the unique distributional solution of -∆K • + K • = δ 0 in R d and lim |x|→∞ K • (x) = 0. Denote K • and Q • the operator defined by ( 11)-( 12) and Proposition 5.1 constructed with this kernel. The operator K -K • is compact, as it involves smoother kernels of the form K (x) -K • (x), see [25, Lemma 3.9.8] for example. This implies that Q -Q • is compact. Finally, since the operator Q • satisfies a Garding inequality such as [START_REF] Costabel | Boundary integral operators on Lipschitz domains: elementary results[END_REF] according to [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF]Prop.10.3], this concludes the proof.

Quasi-local transmission operator

In the case where the skeleton Σ does not involve any junction point (i.e. points where three or more subdomains abut), the local multi-trace operator A -Π maps H(Σ) onto itself, and satisfies a Garding inequality. Hence, the main motivation for considering Q, is the treatment of junction points. A fortiori, for writing transmission conditions, it would seem desirable to use Q only at junction points, and use the operator Π anywhere else on Σ. In this section, we present the construction of an operator Π similar to Π, but involving the operator Q in fixed arbitrarily small neighbourhoods of junction points. Define

J = 0≤i<j<k≤n ∂Ω i ∩ ∂Ω j ∩ ∂Ω k .
By definition, these are the points adjacent to at least three subdomains. For R d = R 2 it consists in a finite set of points, and for R d = R 3 it consists in a finite union of Lipschitz curves. For ǫ > 0 define V ǫ := {x ∈ R d | dist(x, J ) < ǫ}. We suppose that ǫ is chosen small enough to guarantee that Γ j,k \ V 2ǫ = ∅ ∀j, k = 0 . . . n.

We consider a C ∞ cut-off function ψ : R d → R such that ψ(x) = 1 for x ∈ V ǫ and ψ(x) = 0 for x ∈ R d \ V 2ǫ , and define χ := 1ψ 2 . In particular χ vanishes over a neighbourhood of junctions points. In a two dimensional geometrical context, the situation looks as depicted in Figure 1 below.

In the sequel, for any u = (u j , p j ) n j=0 ∈ H(Σ), we will denote χu := (χu j , χp j ) n j=0 . And we adopt a similar notation with ψ. This implies that χu, v = u, χv , and also u, v = χu, v + ψu, ψv . Next we define the continuous operator Π :

H(Σ) → H(Σ) by Π(u), v := Π(χu), v + Q(ψu), ψv ∀u, v ∈ H(Σ). (14)
. Clearly Π : H(Σ) → H(Σ) is continuous since χ vanishes at junction points. It coincides with Π except in V 2ǫ , where it admits a non-local expression given by the operator Q. This is the reason why Π may be considered "quasi-local". The operator Π is constructed as an interpolation between Π and Q. As such, it shares several properties with these operators.

V 2ǫ V ǫ Ω 1 Ω 2 Ω 3 Ω 0
Proposition 5.3.

We have u + Π(u) ∈ X(Σ) for all u ∈ H(Σ), and Ker(Id -Π) = X(Σ).

Proof:

To prove the first property pick any u ∈ H(Σ), and take an arbitrary v ∈ X(Σ). From Lemma 2.1 and Lemma 4.1 we conclude that χu + Π(χu), v = 0. Note in addition that ψv ∈ X(Σ). As a consequence, since (-Id + Q)w = 0 ∀w ∈ X(Σ) according to Proposition 5.1, we obtain

u + Π(u), v = χu + Π(χu), v + ψu + Q(ψu), ψv = -ψv + Q(ψv), ψu = 0
Since v ∈ X(Σ) was chosen arbitrarily, we conclude from Lemma 2.1 that u + Π(u) ∈ X(Σ).

As this holds for any u ∈ H(Σ), this proves the first statement. Now let us prove the second property of the proposition. Pick any u ∈ X(Σ), and take an arbitrary v ∈ H(Σ). Note that χu ∈ X(Σ) and χu vanishes in the neighbourhood of junction points, so that Π(χu) = χu which implies χu -Π(χu), v = 0. Similarly we have (-Id + Q)(ψu) = 0 since ψu ∈ X(Σ), so ψu -Q(ψu), ψv = 0. The definition (14

) of Π then yields u -Π(u), v = χu -Π(χu), v + ψu -Q(ψu), ψv = 0 ∀v ∈ H(Σ).
This shows that X(Σ) ⊂ Ker(Id -Π). Now let us take u ∈ Ker(Id -Π) which can be written 2u = u + Π(u). The first part of the proof then shows that u ∈ X(Σ).

Proposition 5.4.

We have ( Π) 2 = Id, and

Π(u), v = Π(v), u ∀u, v ∈ H(Σ). Moreover for any u ∈ H(Σ), we have u ∈ X(Σ) ⇐⇒ u = Π(u).

Proof:

It is a direct consequence of the previous proposition that (Id-Π)(Id+ Π) = 0 = Id-( Π) 2 . To prove the second property, observe directly from the definition (8) of Π that χΠ = Πχ. Moreover we know from ( 9) and Proposition 5.

1 that Π(u), v = Π(v), u and Q(u), v = Q(v), u for any u, v ∈ H(Σ).
Thanks to these remarks, the second property is obtained by direct calculus from [START_REF] Geuzaine | Gmsh: A 3-D finite element mesh generator with builtin pre-and post-processing facilities[END_REF]. Finally Ker(Id -Π) = X(Σ) according to Proposition 5.3, which can be rewritten u ∈ X(Σ) ⇐⇒ u = Π(u) .

Quasi-local multi-trace formulation

The operator Π satisfies the same elementary properties as Π. It involves non-local contributions but maps continuously H(Σ) onto itself. As a consequence we can derive a new (so called "quasi-local") multi-trace formulation from it, mimicking the derivation of the local multi-trace formulation presented in [START_REF] Hiptmair | Multiple traces boundary integral formulation for Helmholtz transmission problems[END_REF].

Assume that u = (γ j (u)) n j=0 are the traces of the unique solution u to Problem [START_REF] Boffi | Finite element approximation of eigenvalue problems[END_REF]. Then the wave equations in each subdomain can be rewritten (A -Id)(uu inc ) = 0, and the transmission conditions can be written u = Π(u). So it satisfies u ∈ H(Σ) and

(A -Π)u, v = f, v ∀v ∈ H(Σ). (15)
where f := (A -Id)u inc . A striking point of the preceding construction is that it leads to a (generalised) Garding inequality for the formulation above.

Theorem 6.1. Define Θ : H(Σ) → H(Σ) by Θ(u) = (-u j , p j ) n j=0 for any u = (u j , p j ) ∈ H(Σ). Then there exists a constant C > 0 and a compact operator R :

H(Σ) → H(Σ) such that ℜe (A -Π + R)u, Θ(u) ≥ C u 2 H(Σ)
∀u ∈ H(Σ).

Proof:

We already know that such an inequality is satisfied by A, see [23, §4.1] for example. Thus it suffices to study u → ℜe{ Π(u), Θ(u) }. Note that ψΘ(v) = Θ(ψv) and χΘ(v) = Θ(χv) for any v ∈ H(Σ). Applying the definition of Π yields

Π(u), Θ(u) = χΠ(u), Θ(u) + Q(v), Θ(v) . (16)
with v = ψu. Elementary calculus on the definition (8) of Π shows that ΘΠ = ΠΘ. Moreover we have Θ(w), w =w, Θ(w) . So we obtain χΠ(u), Θ(u) = χΠ(Θu), u = Θ(χΠ(u)), u = -χΠ(u), Θ(u) . As a consequence ℜe{ χΠ(u), Θ(u) } = 0. Using this to evaluate the real part of ( 16), and according to Proposition 5.2, we see that there exists a compact operator R ′ :

H(Σ) → H(Σ) such that ℜe (-Π + R ′ )u, Θ(u) ≥ 0 which concludes that proof.
As a first consequence of the previous result, the operator associated to the quasi-local multitrace formulation ( 15) is Fredholm of index 0. In particular, it is an isomorphism if and only if it is one-to-one. We now prove injectivity.

Theorem 6.2.

Assume that the kernel K coming into play in ( 11) is real valued. Then, for κ 0 , . . . , κ n taking any real values, the operator A -Π : H(Σ) → H(Σ) is a continuous isomorphism.

Proof:

As mentioned before, we only need to prove injectivity. So take any u = (u j ) n j=0 ∈ H(Σ) satisfying (A -Π)u = 0. Thus we have w := ( Π + Id)u = (A + Id)u ∈ range(A + Id) ∩ X(Σ). Since C in (Σ) = range(A + Id) and C in (Σ) ∩ X(Σ) = {0} according to Lemma 3.2, we conclude that w = 0. Set ψ j (x) := G j κ j (u j )(x). We have (Id + A)u = 0 which implies ψ j (x) = 0 for x ∈ Ω j . So if we can prove that ψ j (x) = 0 for x ∈ R d \ Ω j , this will show that [γ j (ψ j )] = [γ j ] • G j κ j (u j ) = u j = 0.

Since the kernel K coming into play in [START_REF] Claeys | A second-kind galerkin boundary element method for scattering at composite objects[END_REF] is real valued, we have Π(u) = Π(u). In addition we have Π 2 = Id and Π(u) = -u. As a consequence 2u = u -Π(u), and u + Π(u) = 0. From this and (9), we obtain 2 u, u = u-Π(u), u = -u+ Π(u), u = 0. Since

u j = [γ j ]•G j κ j (u j ) = -γ j c • G j κ j (u j ), this can be rewritten n j=0 ℑm ˆΓj γ j d,c (ψ j )γ j n,c (ψ j )dσ = 0. (17)
Take r > 0 sufficiently large to guarantee that R d \ Ω 0 ⊂ B r where B r ⊂ R d refers to the ball centred at 0 with radius r. Since -∆ψ jκ 2

j ψ j = 0 in R d \ Ω j , applying Green's formula in each B r \ Ω j yields ˆ∂Br ψ j ∂ r ψ j dσ = ˆBr\Ωj |∇ψ j | 2 -κ 2 j |ψ j | 2 dx + ˆ∂Ω j γ j d,c (ψ j )γ j n,c (ψ j )dσ 0 = ˆBr\Ω0 |∇ψ 0 | 2 -κ 2 0 |ψ 0 | 2 dx + ˆ∂Ω 0 γ 0 d,c (ψ 0 )γ 0 n,c (ψ 0 )dσ
In the equations above "∂ r " refer to the radial derivative. Take the imaginary part of the identities above, and sum over j = 0 . . . n, taking account of [START_REF] Hiptmair | Multiple traces boundary integral formulation for Helmholtz transmission problems[END_REF]. This leads to

n j=0 ℑm ˆ∂Br ψ j ∂ r ψ j dσ = ℑm n j=0 ˆ∂Ω j γ j d,c (ψ j )γ j n,c (ψ j )dσ = 1 2ı u, u = 0
where ı refers to the imaginary unit. By construction, the functions ψ j are outgoing radiating, so that 0 = lim r→∞ ´∂Br |∂ r ψ jıκ j ψ j | 2 dσ = 0. As a consequence, we finally obtain

n j=0 1 κ j ˆ∂Br |∂ r ψ j | 2 + κ 2 j |ψ j | 2 dσ = n j=1 1 κ j ˆ∂Br |∂ r ψ j -ıκ j ψ j | 2 dσ -2 n j=1 ℑm ˆ∂Br ψ j ∂ r ψ j dσ = n j=1 1 κ j ˆ∂Br |∂ r ψ j -ıκ j ψ j | 2 dσ -→ r→∞ 0
This shows in particular that lim r→∞ ´∂Br |ψ j | 2 dσ = 0 for all j = 1 . . . n. As a consequence we can apply Rellich's lemma, see [START_REF] Colton | Integral equation methods in scattering theory[END_REF]Lemma 3.11], which implies that ψ j = 0 in the unbounded connected component of each R d \ Ω j . Let us prove that ψ j also vanishes in bounded connected components of R d \ Ω j . First of all we have u = -Π(u). Besides, according to the definition ( 14) of Π, we have Π(v) = Π(v) for all v ∈ H(Σ) supported on Σ \ V 2ǫ . Since u j = -γ j c (ψ j ), this implies in particular

γ j d,c (ψ j ) = -γ k d,c (ψ k ) γ j n,c (ψ j ) = γ k n,c (ψ k ) on Γ j,k \ V 2ǫ = ∅ (18)
Take an arbitrary j, and let O be a bounded connected component of R d \ Ω j . We have [START_REF] Langer | Boundary Element Tearing and Interconnecting Methods[END_REF]. Finally we have -∆ψ jκ 2 j ψ j = 0 in O with γ j c (ψ j ) = 0 on Γ j,k \ V 2ǫ . We conclude by unique continuation principle (see [START_REF] Colton | Integral equation methods in scattering theory[END_REF]Thm.3.5]) that ψ j = 0 in O. We have just proved that

∂Ω j ∩ ∂Ω k = Γ j,k ⊂ ∂O for some k = 0 . . . n, k = j. Let O ′ be the unbounded connected component of R d \ Ω k . Then we have Ω j ⊂ O ′ , and ∂O ′ ∩ ∂O = ∂Ω j ∩ ∂Ω k = Γ j,k . Since ψ k = 0 in O ′ , we have γ j d,c (ψ j )| Γ j,k = -γ k d,c (ψ k ) = 0 and γ j n,c (ψ j )| Γ j,k = γ k n,c (ψ k ) = 0 on Γ j,k \ V 2ǫ = ∅ according to
ψ j = 0 in R d \ Ω j ∀j = 0 . . . n.
Another important consequence of Theorem 6.1 concerns the solvability of Formulation [START_REF] Harrington | Boundary integral formulations for homogeneous material bodies[END_REF] by means of a Galerkin approach: it guarantees a quasi-optimal convergence of the numerical solution toward the exact solution. Hence the following proposition is a direct application to Formulation (15) of Theorem 4.2.9 in [START_REF] Sauter | Boundary element methods[END_REF]. Proposition 6.1. Let (H h (Σ)) 0<h<1 be any dense sequence of finite dimensional subspaces in H(Σ) satisfying Θ(H h (Σ)) ⊂ H h (Σ). For any f ∈ H(Σ) there exists h 0 > 0 such that the following formulation admits a unique solution for any h ∈ (0, h 0 ),

Find u h ∈ H h (Σ) such that (A -Π)u h , v h = f, v h ∀v h ∈ H h (Σ) (19)
In addition there exists a constant C > 0 independent of h such that, if u ∈ H(Σ) refers to the unique solution to [START_REF] Harrington | Boundary integral formulations for homogeneous material bodies[END_REF], we have

u -u h H(Σ) ≤ C inf v h ∈H h (Σ) u -v h H(Σ) ∀h ∈ (0, h 0 ).
To conclude this section, since the operator A -Π maps H(Σ) onto itself, it perfectly fits the operator preconditioning methodology described in [START_REF] Hiptmair | Operator preconditioning[END_REF], provided that the Galerkin discretisation under consideration induces stable discrete inf-sup conditions for the duality pairing •, • i.e. satisfies assumption (2.3) in [START_REF] Hiptmair | Operator preconditioning[END_REF].

Numerical experiments

This section is dedicated to describing numerical results related to the quasi-local multi-trace formulation [START_REF] Harrington | Boundary integral formulations for homogeneous material bodies[END_REF]. We consider a 2-D problem with a scatterer corresponding to the geometry depicted in the figure below. This situation is similar to the one considered in [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF][START_REF] Claeys | A second-kind galerkin boundary element method for scattering at composite objects[END_REF][START_REF] Hiptmair | Multiple traces boundary integral formulation for Helmholtz transmission problems[END_REF], which makes comparisons with the results of these references easier. With this geometry, we consider Problem (1) with the incident field u inc (x) = exp(ıκ 0 e 1 • x), and the wave numbers κ 0 = 1, κ 1 = 5, κ 2 = 0.5. As regards discretisation, we consider a uniform panelling Σ h ≃ Σ which induces a mesh for each of the subdomains Γ h j ≃ Γ j , Γ h j ⊂ Σ h . The discrete spaces H h (Σ) are constructed on these meshes by means of P 1 -Lagrange shape functions for both Dirichlet and Neumann traces

H h (Σ) = { (u h j , p h j ) j=0,1,2 such that ∀j = 0, 1, 2, for all panel e ⊂ Γ h j , u h j | e , p h j | e ∈ P 1 (e) }. (20)
To obtain a reference solution in this experiment, we rely on the Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation. This is a widespread formulation (see [START_REF] Chang | A surface formulation or characteristic modes of material bodies[END_REF][START_REF] Harrington | Boundary integral formulations for homogeneous material bodies[END_REF][START_REF] Miller | Computer Techniques for Electromagnetics, chapter Integral equation solution of three-dimensional scattering problems[END_REF][START_REF] Wu | Scattering from arbitrarily-shaped lossy dielectric bodies of revolution[END_REF]) that is known to yield quasi-optimal convergence rate under conforming Galerkin discretisations, which justifies considering it for obtaining reference discrete solution. In the present situation (continuous) PMCHWT formulation writes

Find u ∈ X(Σ), Au, v = f, v ∀v ∈ X(Σ). (21)
We refer the reader to [START_REF] Claeys | Multi-trace boundary integral formulation for acoustic scattering by composite structures[END_REF][START_REF] Petersdorff | Boundary Integral Equations for Mixed Dirichlet, Neumann and Transmission Problems[END_REF] for further details on the analysis of PMCHWT. The results presented in this section will then involve the discrete solutions to three different boundary integral formulations namely [START_REF] Of | The all-floating boundary element tearing and interconnecting method[END_REF] All computations have been achieved on a laptop equipped with a 2-core Intel i7-3520M processor at 2.9GHz with 4 GB of RAM. Meshes have been generated using Gmsh [START_REF] Geuzaine | Gmsh: A 3-D finite element mesh generator with builtin pre-and post-processing facilities[END_REF] (see also the website http://geuz.org/gmsh/) and, for linear algebra including GMRes routines, we relied on the Gmm++ library (see http://download.gna.org/getfem/html/homepage/gmm/ index.html). When assembling the matrix associated to the transmission operator Π we considered the following choice of cut-off function, which guarantees that Theorem 6.2 holds. The table below presents the relative error between u QL h and u STF h on the one hand, and the relative error between u L h and u STF h on the other hand. These results are summarised graphically in Figure 4. with a comparable level of consistency, although the approximation provided by the quasi-local formulation seems slightly more precise (about twice more precise for the same mesh width). Denote M h , B L h , B QL h the matrices obtain by Galerkin discretisation of the bilinear forms •, • , (A -Π)•, • , (A -Π)•, • . Figure 5 below represents the spectrum of the matrices M -1 h B L h and M -1 h B QL h that may be considered as approximations of the spectrum of the continuous operators A -Π and A -Π (see e.g. [START_REF] Boffi | Finite element approximation of eigenvalue problems[END_REF] for further details on the numerical analysis of eigenvalue problems in the context Galerkin discretisation). The eigenvalues have been computed by means of the Arpack++ library (see http://www.caam.rice.edu/software/ARPACK/ arpack++.html), using a mesh with 167 nodes corresponding to 672 unknowns for both local and quasi-local multi-trace discrete formulations. In both cases, eigenvalues clearly cluster in two packets symmetric to each other with respect to the origin. In Figure 6 we have represented the convergence history of GMRes iterations when applied to the matrices M -1 h B QL h and M -1 h B L h corresponding respectively to quasi-local and local multitrace. We used a restart of 20 iterations and a threshold of 10 -8 . For a complete description of the GMRes iterative solver see e.g. [START_REF] Saad | Iterative methods for sparse linear systems[END_REF]. Convergence of GMRes is stable with respect to the mesh width in both cases, and both formulations converge at a comparable speed. It is natural to ask about the influence of the choice of the cut-off function ψ on the performances of the quasi-local multi-trace formulation. Let us first examine the impact of the parameter α coming into play in Formula ( 22) on the consistency. In the table below, we compare the errors u QL hu ST F h H associated to different values of the mesh width and for α = 10, 40, 100, 1000. The value of this parameter seems to have very little impact on the accuracy. Finally, in Figure 7 below we examine the impact of a variation of the parameter α on the convergence of GMRes solver (with a restart of 20 iterations) applied to quasi-local multitrace formulation at a fixed mesh width h = 0.05. We can see that convergence deteriorates as α → 0.

ψ(x) := ψ α (x -x + ) + ψ α (x -x -) with ψ α (y) = exp(-α|y| 2 ) exp 1 1 -1/(2|y|) 2 1 2|y|<1 (
h u QL h -u ST F h H u L h -u ST F h H 0.

Error

  Lemma 4.1. range(Π + Id) = X(Σ).

Figure 1 :

 1 Figure 1: Support of cut-off functions localized at junction points in a typical 2-D multi-subdomain geometry.

1 Figure 2 :

 12 Figure 2: Geometry considered for the numerical experiments and corresponding meshes

10 Figure 3 :

 103 Figure 3: Shape of ψ α for different values of α

Figure 5 :

 5 Figure 5: Spectrum of the operator M -1 h B QL h (on the left) and M -1 h B L h (on the right).

Figure 6 :

 6 Figure 6: Convergence history of GMRes iterations (norm of the residual vs. number of iterations) for quasi-local multi-trace (left) and local multi-trace (right) for three values of the mesh width.

Figure 7 :

 7 Figure 7: Convergence history of GMRes iterations (norm of the residual vs. number of iterations) for quasi-local multi-trace with h = 0.05 and α = 10, 40, 100, 1000.

Functions in Dirichlet trace spaces like H 1/2 (∂Ωj ) will be denoted by u, v, w, whereas we use p, q, r for Neumann traces. Small fraktur font symbols u, v, w are reserved for Cauchy traces, with an integer subscript indicating restriction to a subdomain boundary.


Arnaud Marsiglietti
email: arnaud.marsiglietti@u-pem.fr

Concavity properties of extensions of the parallel volume

Keywords: parallel volume, Brunn-Minkowski inequality, s-concave measure, Hamilton-Jacobi equation

In this paper we establish concavity properties of two extensions of the classical notion of the outer parallel volume. On the one hand, we replace the Lebesgue measure by more general measures. On the other hand, we consider a functional version of the outer parallel sets.

Introduction

As an analogue of the famous concavity of entropy power in Information theory (see e.g. [START_REF] Costa | A new entropy power inequality[END_REF], [START_REF] Villani | A short proof of the 'concavity of entropy power[END_REF]), Costa and Cover [START_REF] Costa | On the similarity of the entropy power inequality and the Brunn-Minkowski inequality[END_REF] conjectured that the 1 npower of the parallel volume |A + tB n 2 | is a concave function, where | • | denotes the Lebesgue measure and where B n 2 denotes the Euclidean closed unit ball.

Conjecture 1.1 (Costa-Cover [START_REF] Costa | On the similarity of the entropy power inequality and the Brunn-Minkowski inequality[END_REF]). Let A be a bounded measurable set in R n then the function t → |A + tB n 2 |

1 n is concave on R + .

This conjecture has been studied by the author and Fradelizi in [START_REF] Fradelizi | On the analogue of the concavity of entropy power in the Brunn-Minkowski theory[END_REF], where it was shown to be true for any sets in dimension 1 and for any connected sets in dimension 2. However they showed that the conjecture fails for arbitrary sets in dimension 2 and for arbitrary connected sets in dimension greater than or equal to 3.

The notion of parallel volume can be extended by considering measures that are more general than the Lebesgue measure. An extension, provided by Borell in [START_REF] Borell | Convex measures on locally convex spaces[END_REF], [START_REF] Borell | Convex set functions in d-space[END_REF], follows from the Brunn-Minkowski inequality, which states that for every λ ∈ [0, 1] and for every compact subsets A, B of R n , |(1λ)A + λB|

1 n ≥ (1 -λ)|A| 1 n + λ|B| 1 n . (1)
Borell defined for s ∈ [-∞, +∞] the s-concave measures, satisfying by definition a similar inequality to (1):

µ((1 -λ)A + λB) ≥ M λ s (µ(A), µ(B))
for every λ ∈ [0, 1] and for every compact subsets A, B of R n such that µ(A)µ(B) > 0, where M λ s (a, b) denotes the s-mean of the non-negative real numbers a, b with weights 1λ and λ, defined as

M λ s (a, b) = ((1 -λ)a s + λb s ) 1 s if s / ∈ {-∞, 0, +∞}, M λ -∞ (a, b) = min(a, b), M λ 0 (a, b) = a 1-λ b λ , M λ +∞ (a, b) = max(a, b
). Basic properties of the s-concave measures are given in the next section. The case s = 0 corresponds to log-concave measures. The most famous example of a log-concave measure is the standard multivariate Gaussian measure

dγ n (x) = 1 (2π) n 2 e -|x| 2 2 dx,
where | • | stands for the Euclidean norm. These measures are of interest. For example, isoperimetric inequalities have been established for the Gaussian measure γ n by Borell [START_REF] Borell | The Brunn-Minkowski inequality in Gauss space[END_REF] and independently by Sudakov and Cirel'son [START_REF] Sudakov | Extremal properties of half-spaces for spherically invariant measures (Russian) Problems in the theory of probability distributions[END_REF], which state that among sets of given Gauss measure, half-spaces minimize the Gauss surface area (see also [START_REF] Kannan | Isoperimetric problems for convex bodies and a localization lemma[END_REF], [START_REF] Bobkov | Isoperimetric and analytic inequalities for log-concave probability measures[END_REF], [START_REF] Cianchi | On the isoperimetric deficit in Gauss space[END_REF], [START_REF] Castro | Quantitative isoperimetric inequalities on the real line[END_REF]).

In this paper, we pursue the study of these measures by considering the following problem:

Problem 1. Let s ∈ [-∞, +∞].
Let µ be an s-concave measure on R n and let A be a compact subset of R n . Is the function t → µ(A + tB n

2 ) s-concave on R + ?

Notice that from the Brunn-Minkowski inequality [START_REF] Artstein | The Santaló point of a function and a functional form of Santaló inequality[END_REF], the n-dimensional Lebesgue measure is 1 n -concave. Thus Problem 1 generalizes the Costa-Cover conjecture.

Other extensions of geometric inequalities can be set up by considering functional versions. The most famous extension of this type in the Brunn-Minkowski theory is certainly the Prékopa-Leindler inequality (see [START_REF] Leindler | On a certain converse of Hölder's inequality, II[END_REF], [START_REF] Prékopa | On logarithmic concave measures and functions[END_REF], [START_REF] Borell | Convex set functions in d-space[END_REF]). Functional versions provide new proofs of geometric inequalities as well as new applications. Other examples of such extensions are a functional version of the Blaschke-Santaló inequality and a functional version of the Mahler conjecture (see e.g. [START_REF] Ball | Isometric problems in ℓ p and sections of convex sets[END_REF], [START_REF] Artstein | The Santaló point of a function and a functional form of Santaló inequality[END_REF], [START_REF] Fradelizi | Some functional forms of Blaschke-Santaló inequality[END_REF], [START_REF] Fradelizi | Functional inequalities related to Mahler's conjecture[END_REF], [START_REF] Lehec | A direct proof of the functional Santaló inequality[END_REF]).

To establish a functional version of Conjecture 1.1, we consider a functional version of parallel sets A + tB n 2 . We set up the following problem (the notion of γ-concave functions is defined in the next section):

Problem 2. Let γ ≥ -1
n . Let f : R n → R + be a measurable non-negative function and g : R n → R + be a γ-concave function. Let us denote

h (γ) t (z) = sup z=x+ty f (x)>0; g(y)>0 (f (x) γ + tg(y) γ ) 1 γ and h (0) t (z) = sup z=x+ty f (x)g(y) t . Is the function t → R n h (γ) t (z) dz γ 1+γn -concave on R + ?
The Costa-Cover conjecture is a particular case of Problem 2 by taking f = 1 A , g = 1 B n 2 and γ = 0. For γ < 0, one can connect the function h (γ) t with the Hopf-Lax solution of the Hamilton-Jacobi equation (see Section 4 for precise definitions). The Hopf-Lax solution is of interest. For example, it can be used to show that hypercontractivity of this solution is equivalent to log-Sobolev inequalities (see e.g. [START_REF] Bobkov | Hypercontractivity of Hamilton-Jacobi equations[END_REF], [START_REF] Gozlan | Hamilton Jacobi equations on metric spaces and transport-entropy inequalities[END_REF]). In Problem 2, we pursue the study of this solution by investigating concavity properties in time of the Hopf-Lax solution of the Hamilton-Jacobi equation.

We shall prove that both Problem 1 and Problem 2 have positive answers in dimension 1. However, the Costa-Cover conjecture fails in dimension n ≥ 2 in such a generality, thus we won't expect positive answers of these more general problems in dimension n ≥ 2. Using the geometric localization theorem of Kannan, Lovász and Simonovits [START_REF] Kannan | Isoperimetric problems for convex bodies and a localization lemma[END_REF] in the form established by Fradelizi and Guédon [START_REF] Fradelizi | The extreme points of subsets of s-concave probabilities and a geometric localization theorem[END_REF] (see Theorem 3.2 below), we prove:

Theorem 1. Let s ∈ [-∞, 1 2 ] ∪ [1, +∞].
Let µ be an s-concave measure on R and let A be a compact subset of R. Then the function t

→ µ(A + t[-1, 1]) is s-concave on R + . Moreover, for s ∈ ( 1 2 , 1) there exists a compact subset A of R such that t → µ(A + t[-1, 1]) is not s-concave on R + (see Proposition 3.3 below).
Using a precise analysis of the Hopf-Lax solution, we prove:

Theorem 2. Let γ ∈ [-1, 0]. Let f : R → R + be such that f γ (to be interpreted by -log(f ) when γ = 0) is a bounded Lipschitz function. Define
for every y ∈ R, V (y) = |y| p p with p ≥ 1, and

h (γ) t (z) = sup z=x+ty f (x)>0; V (y)>0 (f (x) γ + tV (y)) 1 γ , h (0)
t (z) = sup z=x+ty f (x)e -tV (y) .

If there exists t 0 > 0 such that for almost every z ∈ R, t → h

(γ)
t (z) is twice differentiable on [0, t 0 ] with derivatives that are bounded by an integrable function and if lim z→±∞ ∂h

(γ) t ∂z (z) = 0, then the function t → R h (γ) t (z) dz is concave on [0, t 0 ].
Notice that the conclusion of Theorem 2 is stronger than the expected conclusion in Problem 2 (when restricted to log-concave functions of the form |y| p /p, p ≥ 1).

In the next section, we present some notation and we explain basic properties of the s-concave measures. In section 3, we give a complete answer to Problem 1. In section 4, we give a partial answer to Problem 2. To conclude this paper, we derive a weighted Brascamp-Lieb-type inequality from our functional version.

Preliminaries

We work in the Euclidean space R n , n ≥ 1, equipped with the usual scalar product •, • and the ℓ n We denote by int(A) the interior of the set A, by A the closure of A and by ∂A = A \ int(A) the boundary of A.

For an arbitrary (non-negative) measure µ, we call outer parallel µvolume of a compact set A the function defined on R + by t → V µ A (t) = µ(A + tB n 2 ). We simply call this function outer parallel volume when µ is the Lebesgue measure.

Let us recall basic properties of the s-concave measures. For s ≤ 1 n , Borell showed in [START_REF] Borell | Convex set functions in d-space[END_REF] that any s-concave measure which is absolutely continuous with respect to the Lebesgue measure on R n admits a γ-concave density function,

with γ = s 1 -sn ∈ [- 1 n , +∞],
where a function f is said to be γ-concave,

with γ ∈ [-∞, +∞], if the inequality f ((1 -λ)x + λy) ≥ M λ γ (f (x), f (y))
holds for every λ ∈ [0, 1] and for every x, y ∈ R n such that f (x)f (y) > 0.

Notice that a 1-concave function is concave on its support, that the level sets of a -∞-concave function are convex, and that a +∞-concave function is constant on its support. For s > 1, Borell showed that every (non-null) s-concave measure is a Dirac measure. Notice that an s-concave measure is r-concave for all r ≤ s.

3 The s-concavity of the parallel µ-volume

In this section we solve Problem 1.

Let us note that since the function t → µ(A + tB n 2 ) is non-decreasing, it follows that the answer to Problem 1 is positive for s = -∞ (for every non-negative measures). Note also that Problem 1 is solved for convex sets. Indeed, if A is a compact convex subset of R n , then for every λ ∈ [0, 1] and for every t 1 , t 2 ∈ R + , one has

µ(A + ((1 -λ)t 1 + λt 2 )B n 2 ) = µ((1 -λ)(A + t 1 B n 2 ) + λ(A + t 2 B n 2 )) ≥ ((1 -λ)µ(A + t 1 B n 2 ) s + λµ(A + t 2 B n 2 ) s ) 1 s .
Before proving Theorem 1, we establish a preliminary lemma in dimension 1.

Lemma 3.1. Let µ be an s-concave measure on R which is absolutely continuous with respect to the Lebesgue measure and let A be a compact subset of R. Then the function t

→ V µ A (t) = µ(A + t[-1, 1]
) admits left and right derivatives on (0, +∞), denoted respectively by (V µ A ) ′ -and by (V µ A ) ′ + , which satisfy for every t > 0, (

V µ A ) ′ -(t) ≥ (V µ A ) ′ + (t).
Proof. Let us denote by ψ the density of the measure µ and let us denote by [α, β] the support of µ, with -∞ ≤ α < β ≤ +∞. Notice that for every t > 0, the set

A + t[-1, 1] is a finite disjoint union of intervals, thus one can assume that A = ∪ N i=1 [a i , b i ], with α ≤ a 1 < b 1 < • • • < a N < b N ≤ β, and a 1 = α when α ∈ A, b N = β when β ∈ A. Let us denote t i = a i+1 -b i 2 , i ∈ {0, • • • , N }, with the convention that b 0 = 2α -a 1 and that a N +1 = 2β -b N . Notice that V µ A is differentiable on (0, +∞) \ {t 0 , • • • , t N }
and that for every t > 0, one has

(V µ A ) ′ + (t) = a∈∂(A+t[-1,1]) ψ(a)1 (α,β) (a), (V µ A ) ′ -(t) = a∈∂(A+t(-1,1)) ψ(a)1 [α,β] (a). Notice that A + t[-1, 1] = A + t (-1, 1) thus ∂(A + t[-1, 1]) ⊂ ∂(A + t (-1, 1)). We conclude that for every t > 0, (V µ A ) ′ -(t) ≥ (V µ A ) ′ + (t).
The proof of Theorem 1 uses the following localization theorem.

Theorem 3.2 (Fradelizi, Guédon [START_REF] Fradelizi | The extreme points of subsets of s-concave probabilities and a geometric localization theorem[END_REF]). Let n be a positive integer, let K be a compact convex set in R n and denote by P(K) the set of probability measures on R n , with support contained in K. Let f : K → R be an upper semicontinuous function, let s ∈ [-∞, 1 2 ] and denote by P f the set of s-concave probability measures µ, with support contained in K, satisfying f dµ ≥ 0. Let Φ : P(K) → R be a convex w * -upper semi-continuous function. Then sup{Φ(µ); µ ∈ P f } is achieved at a probability measure ν which is either a Dirac measure at a point x such that f (x) ≥ 0, or a probability measure ν which is s-affine on a segment [α, β] ⊂ K, such that f dν = 0 and

x α f dν > 0 on (α, β) or β x f dν > 0 on (α, β).
Proof of Theorem 1. Let µ be an s-concave measure on R and let A be a compact subset of R.

If s > 1 then µ is a Dirac measure δ x , x ∈ R, and one can see that the function t → δ x (A + t[-1, 1]) is constant on its support. Thus this function is +∞-concave on R + , which proves Theorem 1 for s > 1. Notice that this argument is also valid in higher dimension.

If

s = 1 then for every x ∈ R, dµ(x) = C1 [α,β] (x)dx, with C > 0 is a constant and [α, β] is an interval of R. One can see by a direct computation that t → µ(A + t[-1, 1]) is 1-concave on R + .
If s ≤ 1 2 then µ admits a density with respect to the Lebesgue measure on R which is s 1-s -concave. We assume that s = 0, the case s = 0 follows by continuity. Let t 0 > 0 such that the set

A + t 0 [-1, 1] is an interval. Notice that the function t → V µ A (t) = µ(A + t[-1, 1]) is s-concave on [t 0 , +∞). To prove that V µ
A is s-concave on (0, t 0 ), we shall reduce the problem to extremal measures ν described in Theorem 3.2. For these specific measures, the outer parallel volume is easy to compute and is twice differentiable except for a finite number of points. Then we prove by differentiation that the function

t → V ν A (t) = ν(A + t[-1, 1]
) is s-concave outside points of nondifferentiability. To conclude global s-concavity, we use Lemma 3.1.

Step 1:

Reduction to extremal measures Let t 1 , t 2 ∈ (0, t 0 ) such that µ(A + t 1 [-1, 1])µ(A + t 2 [-1, 1]) > 0.
We assume that t 1 < t 2 without loss of generality. We denote K = A + t 0 [-1, 1] and we consider the restriction of µ over K, thus it is a finite measure that we can assume to be a probability measure without loss of generality. For convenience, we still denote by µ this measure. Let 0 < ε < t 2t 1 . We apply Theorem 3.2 to f : K → R defined by

f = 1 A+t 2 [-1,1] -τ ε 1 A+(t 1 +ε)(-1,1)
and Φ ε : P(K) → R defined by

Φ ε = ρ ε 1 A+t 1 [-1,1] -1 A+ (t 1 +ε)+t 2 2 (-1,1)
where

ρ ε = 1 2 µ(A + t 2 [-1, 1]) s µ(A + (t 1 + ε)[-1, 1]) s + 1 1 s , τ ε = µ(A + t 2 [-1, 1]) µ(A + (t 1 + ε)[-1, 1])
.

The choice of introducing ε and the open interval (-1, 1) is a technical trick to get upper semi-continuous functions regarding f and Φ ε and to make the argument work for Dirac measures in Step 2 below. We shall prove that Φ ε (µ) ≤ 0, thus by letting ε go to 0 and by using that V µ A is continuous on (0, t 0 ), this will lead to the conclusion that V µ A is s-concave on [t 1 , t 2 ], for arbitrary t 1 , t 2 ∈ (0, t 0 ). To prove that Φ ε (µ) ≤ 0, we shall prove that Φ ε (ν) ≤ 0 for the extremal probability measures ν described in Theorem 3.2. First, notice that τ ε ≥ 1.

If τ ε = 1, then V µ A is constant on [t 1 + ε, t 2 ]. Thus V µ A is s-concave on [t 1 + ε, t 2 ]
. Thereafter, we assume that τ ε > 1.

Step 2: s-concavity for extremal measures -Let ν be a Dirac measure

δ x with x such that f (x) ≥ 0. The condition f (x) ≥ 0 says that 1 A+t 2 [-1,1] (x) ≥ τ ε 1 A+(t 1 +ε)(-1,1) (x). Since τ ε > 1, it fol- lows that x / ∈ A + (t 1 + ε)(-1, 1). Thus x / ∈ A + t 1 [-1, 1]. Hence Φ ε (δ x ) ≤ 0.
-Let ν be an s-affine measure with support [α, β], i.e. the density of the measure ν, denoted by ψ, satisfies for every

x ∈ R, ψ(x) = (mx + p) 1/γ 1 [α,β] (x), where γ = s/(1 -s) ∈ [-1, 1] \ {0}
and where m ∈ R \ {0}, p ∈ R are such that for every x ∈ [α, β], mx + p ≥ 0. We assume that [x,β] f dν < 0 on (α, β), which means that for every x ∈ (α, β),

ν((A + t 2 [-1, 1]) ∩ [x, β]) < τ ε ν((A + (t 1 + ε)[-1, 1]) ∩ [x, β]). (2)
If β / ∈ A + (t 1 + ε)[-1, 1], then there exists x ∈ (α, β) such that (A + (t 1 + ε)[-1, 1]) ∩ [x, β] = ∅. This contradicts (2). It follows that β ∈ A + (t 1 + ε)[-1, 1].
Notice that the function V ν A is twice differentiable on [t 1 + ε, t 2 ] outside a finite number of points s 0 , . . . , s ℓ , with s 0

:= t 1 + ε < • • • < s ℓ < s ℓ+1 := t 2 . Case 1: α ∈ A + (t 1 + ε)[-1, 1]. Let j ∈ {0, • • • , ℓ}. Notice that A + s j [-1, 1] is a finite disjoint union of intervals containing α and β, hence we can assume that A + s j [-1, 1] = ∪ N i=1 [a i , b i ], with a 1 ≤ α < b 1 < • • • < a N < β ≤ b N . We assume that N ≥ 2,
otherwise the result clearly holds. Let t ∈ [s j , s j+1 ). We get

V ν A (t) = N i=1 b i +t a i -t (mx + p) 1 γ 1 [α,β] (x)dx, (V ν A ) ′ (t) = N i=2 (m(b i-1 + t) + p) 1 γ + (m(a i -t) + p) 1 γ , (V ν A ) ′′ (t) = m γ N i=2 (m(b i-1 + t) + p) 1-γ γ -(m(a i -t) + p) 1-γ γ ≤ 0.
Hence the function

V ν A is concave on [s j , s j+1 ). We deduce that V ν A is piecewise s-concave on [t 1 + ε, t 2 ]. From Lemma 3.1, we get that for ev- ery t ∈ [t 1 + ε, t 2 ], s((V ν A ) s ) ′ -(t) = s 2 (V ν A ) ′ -(t)(V ν A ) s-1 (t) ≥ s 2 (V ν A ) ′ + (t)(V ν A ) s-1 (t) = s((V ν A ) s ) ′ + (t).
We conclude that the function

V ν A is s-concave on [t 1 + ε, t 2 ]. Case 2: α / ∈ A + (t 1 + ε)[-1, 1]. Let j ∈ {0, • • • , ℓ}. If α ∈ A + s j [-1, 1] then from the previous case we can conclude that V ν A is s-concave on [s j , s j+1 ). Thus we can assume that A + s j [-1, 1] = ∪ N i=1 [a i , b i ], with α < a 1 < b 1 < • • • < a N < β ≤ b N and that α / ∈ A + s j+1 [-1, 1]
. We assume that N ≥ 2, otherwise the result clearly holds. In the following, we denote

a i (t) = m(a i -t) + p, b i (t) = m(b i + t) + p, 1 ≤ i ≤ N -1 and a N (t) = m(a N -t) + p, b N (t) = mβ + p.
Let t ∈ [s j , s j+1 ). We get

V ν A (t) = 1 m γ 1 + γ N i=1 b i (t) 1+γ γ -a i (t) 1+γ γ , (V ν A ) ′ (t) = N -1 i=1 b i (t) 1 γ + a i (t) 1 γ + a N (t) 1 γ , (V ν A ) ′′ (t) = m γ -a 1 (t) 1-γ γ + N i=2 b i-1 (t) 1-γ γ -a i (t) 1-γ γ .
Then the function

V ν A is s-concave on [s j , s j+1 ) if and only if for every t ∈ [s j , s j+1 ), V ν A (t)(V ν A ) ′′ (t) ≤ (1 -s)(V ν A ) ′ (t) 2 if and only if N i=1 a i (t) 1+γ γ -b i (t) 1+γ γ N -1 i=1 a i (t) 1-γ γ -b i (t) 1-γ γ + a N (t) 1-γ γ ≤ N -1 i=1 b i (t) 1 γ + a i (t) 1 γ + a N (t) 1 γ 2 .
If mγ > 0, then one has

0 ≤ a 1 (t) 1-γ γ < b 1 (t) 1-γ γ < • • • < a N (t) 1-γ γ < b N (t) 1-γ γ .
Thus (V ν A ) ′′ (t) ≤ 0. Hence the function V ν A is concave on [s j , s j+1 ). Using the same argument as in Case 1, we conclude that the function

V ν A is s-concave on [t 1 + ε, t 2 ].
If mγ < 0, then one has

a 1 (t) 1-γ γ > b 1 (t) 1-γ γ > • • • > a N (t) 1-γ γ > b N (t) 1-γ γ ≥ 0.
We deduce that

N i=1 a i (t) 1+γ γ -b i (t) 1+γ γ N -1 i=1 a i (t) 1-γ γ -b i (t) 1-γ γ + a N (t) 1-γ γ ≤ a 1 (t) 1+γ γ a 1 (t) 1-γ γ ≤ N -1 i=1 b i (t) 1 γ + a i (t) 1 γ + a N (t) 1 γ 2 .
Hence the function

V ν A is s-concave on [s j , s j+1 ). We conclude that V ν A is s-concave on [t 1 + ε, t 2 ].
Hence we get that Φ ε (ν) ≤ 0 and it follows that V µ A is s-concave on (0, t 0 ).

We have seen that V µ

A is s-concave on [t 0 , +∞) and using Lemma 3.1 we conclude that V µ A is s-concave on (0, +∞). Finally, from the non-decreasing property of V µ A , it follows that V µ A is s-concave on R + .

Remark. The result clearly holds if we replace the interval [-1, 1] by any symmetric interval. However it is not necessarily true for an arbitrary interval. For example, let 0 < s ≤ 1 2 and take

B = [0, 1], A = [0, 1] ∪ [2, 3] and dµ(x) = x 1 γ 1 [0,3] (x) dx, with γ = s 1-s .
Then µ is an s-concave measure. For t ∈ [0, 1 2 ) we get

V µ A (t) = µ(A + tB) = γ γ + 1 (1 + t) γ+1 γ + 3 γ+1 γ -2 γ+1 γ
.

Thus,

V µ A (0)(V µ A ) ′′ (0) -(1 -s)(V µ A ) ′ (0) 2 = 1 γ + 1 3 γ+1 γ -2 γ+1 γ > 0.
Hence V µ A is not s-concave on R + . For s = 0, the same example works. For s < 0, one can take

B = [-1, 0], A = [0, 1]∪[2, 3] and dµ(x) = x 1 γ 1 [α,3] (x) dx, with γ = s
1-s and α sufficiently small.

The localization theorem (Theorem 3.2) holds only for s ≤ 1 2 . Thus this theorem could not be used to examine the case of s-concave measures, with s > 1 2 . For s ∈ ( 1 2 , 1), the answer to Problem 1 is negative in general as shown in Proposition 3.3 below, but under specific conditions regarding the support of the measure, we can show that a stronger positive answer holds. First, let us show that for s ∈ ( 1 2 , 1) the answer to Problem 1 is negative in dimension 1.

Proposition 3.3. Let s ∈ ( 1 2 , 1) and let γ = s 1-s . Let β = 10(1 -2 1-γ γ ) -1 and let µ be a measure such that dµ(x) = x 1 γ 1 [0,β] (x)dx. Let us set A = [0, 1] ∪ [2, β]. Then the function t → V µ A (t) = µ(A + t[-1, 1]) is not s-concave on R + .
Proof. For every t ∈ [0, 1 2 ),

V µ A (t) = γ γ + 1 (1 + t) 1+γ γ + β 1+γ γ -(2 -t) 1+γ γ , (V µ A ) ′ (t) = (1 + t) 1 γ + (2 -t) 1 γ , (V µ A ) ′′ (t) = 1 γ (1 + t) 1-γ γ -(2 -t) 1-γ γ .
Hence,

V µ A (0)(V µ A ) ′′ (0) -(1 -s)(V µ A ) ′ (0) 2 = 1 γ + 1 β 1+γ γ (1 -2 1-γ γ ) -2 1-γ γ -2 1+2γ γ . Since 1 -2 1-γ γ > 0 and β > (2 1+2γ γ + 2 1-γ γ )/(1 -2 1-γ γ ) γ γ+1
, it follows that

V µ A (0)(V µ A ) ′′ (0)-(1-s)(V µ A ) ′ (0) 2 > 0. We conclude that V µ A is not s-concave on R + .
We denote by supp(µ) the support of µ and by dist(A, supp(µ) c ) the distance between A and the complement of the support of µ. When the support of µ is R, the distance will be equal to +∞. Proposition 3.4. Let s ≥ 1 2 . Let µ be an s-concave measure on R which is absolutely continuous with respect to the Lebesgue measure. Let A be a compact subset of R such that dist(A, supp(µ) c )) > 0. Then the function

t → V µ A (t) = µ(A + t[-1, 1]) is concave on [0, dist(A, supp(µ) c
)). Proof. First, we assume that s = 1 2 . Hence µ admits a 1-concave density function denoted by ψ. Notice that V µ A is differentiable outside a finite number of points t 0 , . . . , t N and that for every

t ∈ [0, dist(A, supp(µ) c )) \ {t 0 , • • • , t N } one has V µ A (t) = N i=1 b i +t a i -t ψ(x) dx, (V µ A ) ′ (t) = N i=1 (ψ(b i + t) + ψ(a i -t)) .
Since ψ is concave, it follows that for every i ∈ {1, • • • , N }, the function t → ψ(b i +t)+ψ(a i -t) is non-increasing. Thus (V µ A ) ′ is piecewise non-increasing. We conclude that V µ A is piecewise concave on [0, dist(A, supp(µ) c )). From Lemma 3.1, we deduce that V µ A is concave on [0, dist(A, supp(µ) c )). Finally, if µ is s-concave with s ≥ 1 2 , then µ is 1 2 -concave and we conclude from the first part of the proof that V µ A is concave on [0, dist(A, supp(µ) c )).

Now we turn to the study of Problem 1 in dimension n ≥ 2. It was shown in [START_REF] Fradelizi | On the analogue of the concavity of entropy power in the Brunn-Minkowski theory[END_REF] that the Costa-Cover conjecture is false in dimension n ≥ 2, and thus the answer to Problem 1 is negative in general. Let us recall the counterexample.

Let n ≥ 2. Let us set A = B n 2 ∪{2e 1 } and let us denote V A (t) = |A+tB n 2 |. Then for every t ∈ [0, 1 2 ), one has

V A (t) = |B n 2 ∪ {2e 1 } + tB n 2 | = |B n 2 + tB n 2 | + |tB n 2 | = |B n 2 |((1 + t) n + t n ).
Since the 1 n -power of this function is not concave (it is strictly convex),

V A is not 1 n -concave on R + for n ≥ 2.
It could appear surprising that Problem 1 has a negative answer in dimension n ≥ 2 since we proved that this problem has a positive answer in dimension 1 with the localization theorem. This localization technique is usually used to reduce inequalities for general convex measures in dimension n to inequalities for measures for which the support is a segment, thus the problem becomes 1-dimensional (see e.g. [START_REF] Fradelizi | Concentration inequalities for s-concave measures of dilations of Borel sets and applications[END_REF] and references therein). Let us explain why one cannot reduce Problem 1 to the dimension 1. The reduction done in dimension 1 with localization works the same way in dimension n and we get the following equivalence for every compact set A of

R n : i) V µ A is s-concave for every measure µ s-concave. ii) V ν
A is s-concave for every measure ν s-affine on a segment [α, β]. However, ii) is not true in dimension n ≥ 2. Indeed, we can construct an explicit counterexample to show that the function t → |(A + tB n 2 ) ∩ [α, β]| 1 is not continuous everywhere inside its support and hence this function is not s-concave. For example, consider

A = {(0, 0)} ∪ {(3, 0)} ∪ {(x, 1); x ∈ [1, 2]} and [α, β] = {(x, 0); x ∈ [0, 3]}.
In dimension 2, it was shown in [START_REF] Fradelizi | On the analogue of the concavity of entropy power in the Brunn-Minkowski theory[END_REF]

that if A is a connected subset of R 2 then the function t → |A + tB 2 2 | is 1 2 -concave on R + .
However, the following proposition shows that this property fails in general if we replace the Lebesgue measure by an arbitrary s-concave measure. Proposition 3.5. In dimension 2, there exists a connected set A and a

1 2 -concave measure µ such that t → µ(A + tB 2 2 ) is not 1 2 -concave on R + .
Proof. We set dµ(x) = 1 B 2 1 (x)dx, where B 2 1 denotes the unit ball for the ℓ 2 1 norm. Hence µ is 1 2 -concave. We construct the points B = (-1, 0), C = (-0.5, -0.5), D = (0.5, 0.5), E = (0, 1), F = (-2, 0), G = (0, -2), H = (0, -1), I = (2, 0), J = (1, 0). We set

A = conv(BCDE) ∪ [F B] ∪ [F G] ∪ [GH] ∪ [GI] ∪ [IJ].
Then A is connected and for every t ∈ [0, 1 8 ], we get

V µ A (t) = µ(A + tB 2 2 ) = √ 2 2 + √ 2t + π 2 t 2 .
It follows that

V µ A ′′ (0) > 0. We conclude that t → µ(A + tB 2 2 ) is not 1 2 -concave on R + .
Remark. Notice that we can adapt the counterexample of Proposition 3.5 to show that there exists an s-concave measure µ on R n , n ≥ 2, such that for every r ∈ (-∞, s) there exists a compact connected set

A ⊂ R n such that t → µ(A + tB n 2 ) is not r-concave on R + .
Question. Does there exist s ≤ 1 2 such that for every s-concave measure µ on R 2 and for every compact set A ⊂ R 2 the function t → µ(A + tB 2 2 ) is s-concave for the t's so that the set supp(µ) ∩ (A + tB 2 2 ) is connected? In particular, for every convex subset K of R 2 and for every compact set A ⊂ R 2 , is the function t → |(A + tB 2

2 ) ∩ K| 1 2 -concave for the t's so that the set K ∩ (A + tB 2 2 ) is connected?

Functional version

In this section, we give a partial answer to Problem 2. As in section 3 where Problem 1 was solved for convex sets, the next proposition shows that Problem 2 is solved for γ-concave functions, as is expected.

Proposition 4.1. Let γ ≥ -1 n . Let f, g : R n → R + be two γ-concave functions. Then the function t → R n h (γ) t (z) dz is γ 1+γn -concave on R + , where h (γ) t (z) = sup z=x+ty f (x)>0; g(y)>0 (f (x) γ + tg(y) γ ) 1 γ and h (0) t (z) = sup z=x+ty f (x)g(y) t .
Proof. We examine the case γ = 0, the case γ = 0 can be proved with the same argument.

For convenience, let us denote h t = h (γ) t . Let λ ∈ [0, 1] and let t 1 , t 2 ∈ R + . We want to show that

R n h (1-λ)t 1 +λt 2 ≥ (1 -λ) R n h t 1 γ 1+γn + λ R n h t 2 γ 1+γn 1+γn γ
.

From the Borell-Brascamp-Lieb inequality [START_REF] Borell | Convex set functions in d-space[END_REF], [START_REF] Brascamp | On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation[END_REF] (dimensional Prékopa's inequality), it is sufficient to show that for every z 1 , z 2 ∈ R n one has

h (1-λ)t 1 +λt 2 ((1 -λ)z 1 + λz 2 ) ≥ ((1 -λ)h t 1 (z 1 ) γ + λh t 2 (z 2 ) γ ) 1 γ . Let z 1 , z 2 ∈ R n . Let x 1 , x 2 ∈ R n such that ∀i ∈ {1, 2}, h t i (z i ) = f (x i ) γ + t i g z i -x i t i γ 1 γ . Let us denote h = h (1-λ)t 1 +λt 2 ((1 -λ)z 1 + λz 2 ) and t = (1 -λ)t 1 + λt 2 . We get, h = sup x∈R n f (x) γ + tg (1 -λ)z 1 + λz 2 -x t γ 1 γ ≥ f ((1 -λ)x 1 + λx 2 ) γ + tg (1 -λ)z 1 + λz 2 -((1 -λ)x 1 + λx 2 ) t γ 1 γ ≥ (1 -λ)f (x 1 ) γ +λf (x 2 ) γ + (1 -λ)t 1 g z 1 -x 1 t 1 γ +λt 2 g z 2 -x 2 t 2 γ 1 γ = ((1 -λ)h t 1 (z 1 ) γ + λh t 2 (z 2 ) γ ) 1 γ .
As a consequence of the Hölder inequality, if f : R n → R + is β-concave and if g : R n → R + is γ-concave, then f g is α-concave for every α, β, γ ∈ R ∪ {+∞} such that β + γ ≥ 0 and 1 β + 1 γ = 1 α (see e.g. [START_REF] Hardy | Inequalities[END_REF]). A generalized form of Proposition 4.1 follows:

Proposition 4.2. Let γ ≥ -1
n . If a measure µ has a β-concave density, with β ≥ -γ, and if f, g : R n → R + are two γ-concave functions, then

t → R n h (γ) t (z) dµ(z) is α 1+αn -concave on R + , where 1 β + 1 γ = 1 α .
In Problem 2, for γ < 0 the function V = g γ is convex by assumption and one can naturally connect the function h (γ) t with the Hopf-Lax solution of the Hamilton-Jacobi equation:

h (γ) t (z) = sup x∈R n f (x) γ + tV z -x t 1 γ = Q (V ) t f γ (z) 1 γ ,
where for arbitrary convex function V and for arbitrary function u,

Q (V ) t u(z) = inf x∈R n u(x) + tV z -x t .
Let us recall basic properties of the Hopf-Lax solution of the Hamilton-Jacobi equation. For a convex function V such that lim |z|→+∞ V (z)/|z| = +∞ and for a Lipschitz function u, it is known that Q

(V )
t u is the solution, called Hopf-Lax solution, of the following partial differential equation, called Hamilton-Jacobi equation (see e.g. [START_REF] Evans | Partial differential equations[END_REF]).

∂ ∂t h(t, z) + V * (∇ z h(t, z)) = 0 on (0, +∞) × R n h(t, z) = u(z) on {t = 0} × R n ,
where V * is the Legendre transform of V defined on R n by

V * (y) = sup x∈R n ( x, y -V (x)). It is shown in [16] that if u is Lipschitz on R n then Q (V ) t u is Lipschitz on [0, +∞) × R n . However, for an arbitrary convex function V , t → Q (V ) t u is not necessarily continuous at 0.
Proof of Theorem 2. We examine the case γ = 0, the case γ = 0 can be proved with the same argument. Since γ ∈ [-1, 0), one has

h (γ) t (z) = sup x∈R f (x) γ + tV z -x t 1 γ = Q (V ) t f γ (z) 1 γ .
We denote for t ∈ R + ,

F (t) = R h (γ) t (z) dz = R Q (V ) t f γ (z) 1 γ dz.
For p = 1, the function F is constant. We then consider p > 1. For convenience, we set φ(t, z) = Q (V ) t f γ (z) and φ ′ = ∂φ ∂z . By assumption, there exists t 0 > 0 such that for almost every z ∈ R, t → h (γ) t (z) is twice differentiable on [0, t 0 ] with derivatives that are bounded by an integrable function, hence we get for every t ∈ [0, t 0 ],

F ′ (t) = - 1 γ R V * φ ′ φ 1-γ γ , F ′′ (t) = 1 γ R φ ′′ (V * ) ′ (φ ′ ) 2 φ 1-γ γ + 1 -γ γ 2 R (V * (φ ′ )) 2 φ 1-2γ γ .
We assumed that V (u) = |u| p p . Hence V * (u) = |u| q q , with 1 p + 1 q = 1. It follows that

F ′′ (t) = 1 γ R φ ′′ (φ ′ ) 2q-2 φ 1-γ γ + 1 -γ γ 2 R (φ ′ ) 2q q 2 φ 1-2γ γ .
By assumption, lim z→±∞ ∂h (γ) t ∂z (z) = 0, thus integration by parts gives

2q -1 γ R φ ′′ (φ ′ ) 2q-2 φ 1-γ γ = - 1 -γ γ 2 R (φ ′ ) 2q φ 1-2γ γ .
Finally,

F ′′ (t) = - 1 -γ γ 2 (q -1) 2 q 2 (2q -1) R (φ ′ ) 2q φ 1-2γ γ ≤ 0. We conclude that t → R h (γ) t (z) dz is concave on [0, t 0 ].
Open problem. Problem 2 is open in dimension 1 for arbitrary γ-concave function g.

Links with weighted Brascamp-Lieb-type inequalities

In this section, we express the s-concavity of the function t → R n h

∈ R n , h (γ) t (z) = sup z=x+ty f (x)>0; V (y)>0 (f (x) γ + tV (y)) 1 γ .
If there exists t 0 > 0 such that for almost every z ∈ R, t → h (γ) t (z) is twice differentiable on [0, t 0 ] with derivatives that are bounded by an integrable function and if G ∈ L 2 (µ), where dµ(z) = (Q

(V ) t f γ (z)) 1 γ / (Q (V ) t f γ ) 1 γ dz and G = V * (∇ z Q (V ) t f γ )/Q (V ) t f γ , then the function t → R n h (γ) t (z) dz is s-concave on [0, t 0 ] if and only if Var µ (G) ≤ - γ 1 -γ (Hess Q (V ) t f γ )(∇ z V * )(∇ z Q (V ) t f γ ), (∇ z V * )(∇ z Q (V ) t f γ ) Q (V ) t f γ dµ + γ -s 1 -γ G dµ 2 ,
Proof. Recall that h For convenience, let us denote φ = f γ and

Q t = Q (V )
t . We get

∂h (γ) t ∂t = - 1 γ V * (∇ z Q t φ)(Q t φ) 1-γ γ , ∂ 2 h (γ) t ∂t 2 = 1 γ (Hess Q t φ)(∇ z V * )(∇ z Q t φ), (∇ z V * )(∇ z Q t φ) (Q t φ) 1-γ γ + 1 -γ γ 2 (V * (∇ z Q t φ)) 2 (Q t φ) 1-2γ γ .
Thus the function F is s-concave if and only if F (t)F ′′ (t) ≤ (1s)F ′ (t) 2 if and only if

Var µ (G) ≤ - γ 1 -γ (Hess Q t φ)(∇ z V * )(∇ z Q t φ), (∇ z V * )(∇ z Q t φ) Q t φ dµ + γ -s 1 -γ G dµ 2 ,
where dµ(z) = (Q t φ(z))

1 γ / (Q t φ) 1 γ
dz and G = (V * (∇ z Q t φ)/Q t φ).

Remark. For γ = 0 and V (u) = |u| 2 2 , one may use the same argument to get that t → R n h (0) t (z) dz is log-concave if and only if

Var µ (|∇ z Q t φ| 2 ) ≤ 4 (Hess Q t φ)∇ z Q t φ, ∇ z Q t φ dµ,
where φ =log f and dµ(z) = e -Qtφ(z) / e -Qtφ dz.

Question. For which function u does the following inequality holds?

Var µ (|∇ z u| 2 ) ≤ 4 (Hess u)∇ z u, ∇ z u dµ, where dµ(z) = e -u(z) / e -u dz.

From Proposition 4.1, if f is γ-concave with γ ∈ [-1 n , 0), then one may apply Proposition 5.1 to get the following weighted Brascamp-Lieb-type inequality by letting t go to 0: Corollary 5.2. Let γ ∈ [-1 n , 0) and let s = γ 1+γn . For every V, φ : R n → R + convex such that lim |z|→+∞ V (z)/z = +∞, one has

Var µ (G) ≤ - γ 1 -γ (Hessφ) -1 ∇ z Gφ, ∇ z Gφ φ dµ + γ -s 1 -γ G dµ 2 , (3)
where dµ(z) = φ dz and G = V (∇ z φ)/φ.

We have reproved Bobkov-Ledoux's result [START_REF] Bobkov | Weighted Poincaré-type inequalities for Cauchy and other convex measures[END_REF] (for a smaller class of functions G) who used the same idea since inequality (3) is derived from the Borell-Brascamp-Lieb inequality (dimensional Prékopa's inequality). Bobkov and Ledoux have already seen in [START_REF] Bobkov | From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities[END_REF] that one can deduce the classical Brascamp-Lieb inequality from the classical Prékopa inequality (corresponding to the log-concave case). This idea has been explored by Cordero-Erausquin and Klartag in [START_REF] Cordero-Erausquin | Interpolations, convexity and geometric inequalities[END_REF] where they showed that the converse is true, i.e. one can derive the Prékopa inequality from the Brascamp-Lieb inequality. Thereafter, Nguyen [START_REF] Nguyen | Dimensional variance inequalities of Brascamp-Lieb type and a local approach to dimensional Prékopa's theorem[END_REF] generalized the work by Cordero-Erausquin and Klartag to the case of s-concave measures (even for s ≥ 0) and improved Bobkov-Ledoux's Brascamp-Lieb-Type inequality (inequality (3)). Recently, Kolesnikov and Milman [START_REF] Kolesnikov | Poincaré and Brunn-Minkowski inequalities on weighted Riemannian manifolds with boundary[END_REF] generalized the weighted Brascamp-Lieb-type inequalities obtained by Bobkov, Ledoux and Nguyen to the setting of Riemannian manifolds.

2

 2 norm | • |. The closed unit ball is denoted by B n 2 and the canonical basis by (e 1 , • • • , e n ). We also denote by | • | the Lebesgue measure on R n . For non-empty sets A, B ⊂ R n we define the Minkowski sum A + B = {a + b; a ∈ A, b ∈ B}.

Proposition 5 . 1 .

 51 in term of a weighted Brascamp-Lieb-type inequality. Let γ ∈ [-1 n , 0) and let s ∈ R. Let f : R n → R + be such that f γ is a bounded Lipschitz function. Let V : R n → R + be a convex function such that lim |z|→+∞ V (z)/|z| = +∞. Let us define for every z

1 γ

 1 , where for arbitrary convex function V and arbitrary function u,Q (V ) t u(z) = inf x∈R n u(x) + tV zx t .

1 γ

 1 (z)/ φ 1 γ

Acknowledgements.

I would like to thank my advisor Matthieu Fradelizi, Pierre Youssef and the referee for their valuable comments and suggestions.

The author was supported in part by the Agence Nationale de la Recherche, project GeMeCoD (ANR 2011 BS01 007 01).