Alexey Ozerov

Emmanuel Vincent

Frédéric Bimbot

A General Flexible Framework for the Handling of Prior Information in Audio Source Separation

published or not. The documents may come


Ph Delanoë

Second order variational heuristics for the Monge problem on compact manifolds *

Keywords: 53A45, 58D05, 58E99; Key-words: Monge problem, minimizing diffeomorphism, c-potential, second order variation, c-Hessian positivity, c-convexity

We consider Monge's optimal transport problem posed on compact manifolds (possibly with boundary) for a lower semi-continuous cost function c. When all data are smooth and the given measures, positive, we restrict the total cost C to diffeomorphisms. If a diffeomorphism is stationary for C, we know that it admits a potential function. If it realizes a local minimum of C, we prove that the c-Hessian of its potential function must be non-negative, positive if the cost function c is non degenerate. If c is generating non-degenerate, we reduce the existence of a local minimizer of C to that of an elliptic solution of the Monge-Ampère equation expressing the measure transport; moreover, the local minimizer is unique. It is global, thus solving Monge's problem, provided c is superdifferentiable with respect to one of its arguments.

Introduction

The solution of Monge's problem [START_REF] Monge | Mémoire sur la théorie des déblais et remblais[END_REF] in optimal transportation theory, with a general cost function, has been applied to many questions in various domains tentatively listed in the survey paper [START_REF] Guillen | Five lectures on optimal transportation: geometry, regularity and applications[END_REF], including in cosmology [START_REF] Brenier | Reconstruction of the early Universe as a convex optimization problem[END_REF]. The book [START_REF] Villani | Optimal Transport, Old and New[END_REF] offers a modern account on the theory (see also [START_REF] Caffarelli | Allocation maps with general cost functions[END_REF][START_REF] Gangbo | The geometry of optimal transportation[END_REF][START_REF] Guillen | Five lectures on optimal transportation: geometry, regularity and applications[END_REF]).

In case data are smooth, manifolds compact, measures positive, maps one-to-one and the solution of Monge's problem unique, the question of the smoothness of that solution was addressed in the landmark paper [START_REF] Ma | Regularity of potential functions of the optimal transportation problem[END_REF]. In that case, restricting Monge's problem to diffeomorphisms becomes a natural ansatz. Doing so, the use of differential geometry and the calculus of variations enables one to bypass the general optimal transportation approach and figure out directly some basic features of the solution map. Such a variational heuristics goes back to [START_REF] Appell | Mémoire sur les déblais et remblais des systèmes continus ou discontinus[END_REF] and was elaborated stepwise in [START_REF] Evans | Partial differential equations and Monge-Kantorovich mass transfer[END_REF][START_REF] Urbas | Mass transfer problems[END_REF][START_REF] Brenier | Reconstruction of the early Universe as a convex optimization problem[END_REF][START_REF] Trudinger | Optimal transportation and nonlinear partial differential equations[END_REF][START_REF] Ph | Variational heuristics for optimal transportation maps on compact manifolds[END_REF] (see also [START_REF] Brenier | Reconstruction of the early Universe as a convex optimization problem[END_REF]). In the present note, we take a new step in that Ph. Delanoë elaboration, by differentiating twice the total cost functional in order to express a minimum condition, not just a stationary one.

Specifically, working in the C ∞ category 1 , we are given a couple of compact connected diffeomorphic manifolds each equipped with a probability volume measure, (M, µ) and (P, ). The manifold M (resp. P ), either has no boundary or it is the closure of a domain contained in some larger manifold. We consider a domain Ω ⊂ M ×P such that, for each (m 0 , p 0 ) ∈ M ×P , the subset Ω P m 0 = p ∈ P, (m 0 , p) ∈ Ω is a domain of full measure in (P, ) and similarly for Ω M p 0 = m ∈ M, (m, p 0 ) ∈ Ω in (M, µ); no smoothness assumption bears on the boundary of Ω. We denote by Diff µ, (Ω) the subset of diffeomorphisms from M to P which push µ to , with graph lying in Ω (an obvious item missing in [START_REF] Ph | Variational heuristics for optimal transportation maps on compact manifolds[END_REF]). The pushing condition means for a Borel map φ : M → P that, for any continuous function h : P → R, the following equality holds:

(1)

P h d = M (h • φ) dµ,
a property commonly denoted by φ # µ = . When it is satisfied by a diffeomorphism ϕ : M → P , one may use the change of variable p = ϕ(m) in the left-hand integral of (1) and infer pointwise, in any couple (x, y) of source and target charts, the so-called Jacobian equation, namely:

(2) d dy (ϕ(x)) det ∂ϕ ∂x (x) = dµ dx (x);

here, y = ϕ(x) denotes abusively the local expression of ϕ while dµ dx stands for the Radon-Nikodym derivative of the push-forward measure x # µ with respect to the Lebesgue measure dx of the chart x, and similarly for d dy . The existence of diffeomorphisms pushing µ to is well-known [START_REF] Moser | On the volume elements on a manifold[END_REF][START_REF] Banyaga | Formes volume sur les variétés à bord[END_REF][START_REF] Dacorogna | On a partial differential equation involving the Jacobian determinant[END_REF] but we have to assume henceforth that the graph of at least one diffeomorphism of the sort actually lies in Ω. Under that assumption, Diff µ, (Ω) is non empty and so is Diff ,µ ( Ω), setting Ω = (p, m) ∈ P × M, (m, p) ∈ Ω . We view Diff µ, (Ω) and Diff ,µ ( Ω) as open manifolds respectively modeled on the Fréchet manifolds Diff (P ) := Diff , (P × P ) and Diff µ (M ) (see details in [START_REF] Ph | Variational heuristics for optimal transportation maps on compact manifolds[END_REF][START_REF] Ebin | Groups of diffeomorphisms and the motion of an incompressible fluid[END_REF]). Finally, we consider a function c : Ω → R, called the cost function 2 , together with the total cost functional

φ ∈ Diff µ, (Ω) → C(φ) = M c(m, φ(m)) dµ
and its counterpart ψ ∈ Diff ,µ ( Ω) → C(ψ) = P c(ψ(p), p) d , which satisfy the identity: C(ϕ) ≡ C(ϕ -1 ). We will occasionally require additional conditions on the cost function (anytime we do, it will be explicit) among the following ones:

c is lower semi continuous on M × P ; (3a) c(., p) is superdifferentiable on M for -almost all p ∈ P ; (3b) det(d M d P c) = 0 on Ω ; (3c) ∀m 0 ∈ M, the map p ∈ Ω P m 0 → -d M c(m 0 , p) ∈ T * m 0 M is one-to-one (3d)
and so is the map m ∈ Ω M p 0 → -d P c(m, p 0 ) ∈ T * p 0 P, ∀p 0 ∈ P .

Here, dealing with a two point function, we have set d M (resp. d P ) for the exterior derivative with respect to the argument in the manifold M (resp. P ). We refer to [20, Chapter 10] (see also [14, p.598]) for an account on the notion of superdifferentiability. As it will be clear from the proof of Proposition 1.2 below, the results of this paper would hold as well with condition (3b) replaced by the symmetric one for c(m, .) instead. Condition (3c) is a non-degeneracy condition labelled as (A2) in [START_REF] Ma | Regularity of potential functions of the optimal transportation problem[END_REF], while (3d) is a generating condition often called bi-twist [START_REF] Kim | Continuity, curvature, and the general covariance of optimal transportation[END_REF][START_REF] Villani | Optimal Transport, Old and New[END_REF] (when the smoothness of the inverse maps is further assumed, it becomes condition (A1) of [START_REF] Ma | Regularity of potential functions of the optimal transportation problem[END_REF]). A typical example of cost function for which all the conditions (3) are fulfilled is given by M = P equipped with a Riemannian metric and c is (half) the squared distance (the so-called Brenier-McCann cost function) [START_REF] Mccann | Polar factorization of maps on Riemannian manifolds[END_REF]; if so, p ∈ Ω P m means that p is not a cut point of m and the inverse maps determined by (3d) are given by the exponential map.

Assuming (3a), we consider the restricted Monge problem, namely the question: can we find ϕ ∈ Diff µ, (Ω) satisfying C(ϕ) = inf φ∈Diffµ, (Ω)

C(φ) ?

Extending the total cost functional C to the set B µ, of Borel maps from M to P pushing µ to , Monge's problem itself reads: can we find φ ∈ B µ, realizing the inf Bµ,

C ?

The outline of the paper is as follows: in Section 1, we recall what was obtained in [START_REF] Ph | Variational heuristics for optimal transportation maps on compact manifolds[END_REF] by writing the stationary condition for the total cost C and we relate it to Monge's problem; in Section 2, we state the new results which can be obtained by expressing the minimum condition for C; except for an elementary one, the corresponding proofs are deferred till Section 3.

Preliminary results

In [START_REF] Ph | Variational heuristics for optimal transportation maps on compact manifolds[END_REF], writing down the Euler equation of the functional C, we obtained the following result: Proposition 1.1 If ϕ ∈ Diff µ, (Ω) is stationary for C, so is ϕ -1 for C, and there exists two functions f : M → R, f : P → R, defined up to addition of constants, such that each point of the graph of ϕ is stationary for the two Ph. Delanoë point real function:

(m, p) ∈ Ω → F (m, p) = c(m, p) + f (m) + f (p).
Let us call the function f (resp. f ) so determined (up to constant addition), the c-potential of the diffeomorphism ϕ (resp. ϕ -1 ). In [START_REF] Ph | Variational heuristics for optimal transportation maps on compact manifolds[END_REF], we assumed condition (3d), but it is not required for the proof of Proposition 1.1, indeed solely based on the Helmholtz lemma. For the reader's convenience, let us indicate the argument (see [START_REF] Ph | Variational heuristics for optimal transportation maps on compact manifolds[END_REF] for details). We write δC = 0 with δC = M d P c(m, ϕ(m))(δϕ(ϕ(m)) dµ, where δϕ stands for a variation of the transporting diffeomorphism ϕ which keeps it on the manifold Diff µ, (Ω), that is, a vector field of a special kind on P , evaluated at the image point ϕ(m). Specifically, such a vector field V on P should be: first of all tangential to the boundary of P , if any, so that its flow send P to itself (without crossing ∂P ); moreover, its flow should preserve the volume measure or, equivalently, V should satisfy: div V = 0. In other words, the tangent space to Diff µ, (Ω) at ϕ is spanned by the tangential vectors of the form V • ϕ with V ∈ ker div . Here, the symbol div denotes the divergence operator defined by the identity: P h div V d ≡ P dh(V ) d valid for each function h : P → R and each vector field V on P (tangential, as said 3 ). Recalling (1), we thus find:

∀V ∈ ker div , P d P c(ϕ -1 (p), p)(V (p)) d = 0.
Arguing likewise on C, we further get:

∀U ∈ ker div µ , M d M c(m, ϕ(m))(U (m)) dµ = 0.
The conclusion of Proposition 1.1 now readily follows from Helmholtz lemma, which we recall (for a proof, see [START_REF] Ph | Variational heuristics for optimal transportation maps on compact manifolds[END_REF]Appendix]):

Lemma 1.1 (Helmholtz) Let (N, ν) be a measured manifold as above. A 1-form α on N satisfies: N α(Z) dν = 0 for each vector field Z ∈ ker div ν (tangential to ∂N if ∂N = ∅) if and only if α is exact.
The outcome of Proposition 1.1 for the Monge problem goes as follows:

Proposition 1.2 If (3a) (3b) (3d) hold, a diffeomorphism ϕ ∈ Diff µ, (Ω) is stationary for C if and only if it solves Monge's problem; moreover, if so, the c-potential of ϕ is c-convex.
Proof. The 'if' part is obvious, let us prove the 'only if' one with an argument of [START_REF] Mccann | Polar factorization of maps on Riemannian manifolds[END_REF]; the meaning of the last statement of the proposition will be cleared up on the way. Letting ϕ ∈ Diff µ, (Ω) be stationary for C and using Proposition 1.1, consider the function f c given by:

∀p ∈ P, f c (p) = F ϕ -1 (p), p -inf M c(m, p) + f (m) ,
called the c-transform [START_REF] Villani | Optimal Transport, Old and New[END_REF] (or supremal convolution [START_REF] Mccann | Polar factorization of maps on Riemannian manifolds[END_REF]) of f , up to the addition of the constant term F ϕ -1 (p), p . From (3a), for each p 0 ∈ P , the infimum appearing in the right-hand side is assumed at some point m 0 ∈ M . The latter satisfies:

∀m ∈ M, c(m, p 0 )+f (m)+f c (p 0 ) c(m 0 , p 0 )+f (m 0 )+ f c (p 0 ), or else: ∀m ∈ M, c(m, p 0 ) c(m 0 , p 0 )-f (m)-f (m 0 ) , which shows that the function m ∈ M → c(m, p 0 ) is subdifferentiable at m 0 . By (3b), it is thus differentiable at m 0 with d M c(m 0 , p 0 ) = -df (m 0 ). From (3d) combined with Proposition 1.1, we get m 0 = ϕ -1 (p 0 ) hence (p 0 , m 0 ) ∈ Ω and f c (p 0 ) = f (p 0 )
; since p 0 is arbitrary, we obtain: f c = f . From the latter and the definition of f c , we infer:

∀m ∈ M, F (m, .) F (m, ϕ(m))
, from what we readily conclude that f = ( f ) c with ( f ) c given by: [START_REF] Villani | Optimal Transport, Old and New[END_REF][START_REF] Guillen | Five lectures on optimal transportation: geometry, regularity and applications[END_REF]. Besides, for each map φ ∈ B µ, , integrating on M the inequality F (m, φ(m)) F (m, ϕ(m)) satisfied µ-almost everywhere yields C(φ) C(ϕ) by using [START_REF] Appell | Mémoire sur les déblais et remblais des systèmes continus ou discontinus[END_REF], which shows that ϕ solves, indeed, the Monge problem Remark 1.1 If, in Proposition 1.2, the manifold M has no boundary and we strengthen (3b) by assuming that the map m ∈ M → c(m, p) is differentiable for -almost all p ∈ P , then either C has no stationary point or Monge's problem is trivial. Indeed, if so, letting ϕ ∈ Diff µ, (Ω) be stationary for C, condition (3d) implies that, for -almost all p ∈ P , the equation p = ϕ(m) holds at any stationary point m of the function F (., p). In particular, it holds at the extrema of that function. By Proposition 1.1, the function F (., p) must be constant, equal to F ϕ -1 (p), p which is independent of p ∈ P . We infer that F is constant on M × P hence, recalling [START_REF] Appell | Mémoire sur les déblais et remblais des systèmes continus ou discontinus[END_REF], that the total cost C itself must be constant.

∀m ∈ M, ( f ) c (m) = F (m, ϕ(m)) -inf P c(m, p) + f (p) . So f = (f c ) c , a property of f called c-convexity

Second order results

When the cost function c does not fulfill the assumption of Proposition 1.2, it is worth observing that the variational heuristics presented so far is incomplete since no local minimum condition is expressed yet for the total cost C. It is our aim in the present note to write down that minimum condition and to derive from it further properties of minimizing diffeomorphisms. Before stating our results, we require a notion of c-Hessian.

Hess c,Φ (h)(x 0 ) := ∂ 2 ∂x i ∂x j [c(x, y 0 ) + h(x)] at x = x 0 ,
where, if x 0 = x(m 0 ), we have set y 0 = y(Φ(m 0 )).

If ϕ ∈ Diff µ, (Ω) is stationary for the total cost C, Proposition 1.1 shows that the couple (ϕ, f ), with f the c-potential of ϕ, fulfills the assumption of Definition 2.1. In that case, for simplicity, we will simply speak of the c-Hessian of f and denote it by Hess c (f ). We would define likewise the c-Hessian of the c-potential f of ϕ -1 by the local expression (sticking to the notations used in the preceding definition):

Hess c ( f )(y 0 ) := ∂ 2 ∂y i ∂y j c(x 0 , y) + f (y) at y = y 0 ,
where, if y 0 = y(p 0 ), x 0 = x ϕ -1 (p 0 ) . We are in position to state our first result: Theorem 2.1 Let ϕ ∈ Diff µ, (Ω) be stationary for the total cost C. The following properties are equivalent (still setting f for the c-potential of ϕ and f for that of ϕ -1 ):

(i) the second variation of C at ϕ is non negative;

(ii) ∀U ∈ ker div µ , M Hess c (f )(U, U ) dµ 0; (iii) ∀V ∈ ker div , P Hess c ( f )(V, V ) d 0.
Assuming that C admits a local minimum at ϕ, we will infer the pointwise non negativity of the c-Hessians of f and f from the conclusions (ii)(iii) of Theorem 2.1, arguing by contradiction with suitable localized divergence free vector fields. The resulting statement goes as follows:

Corollary 2.1 If the total cost C admits a local minimum at ϕ ∈ Diff µ, (Ω), the c-Hessian of the c-potential of ϕ is non negative, and so is that of ϕ -1 .

At this stage, let us record a related result, usually derived from the optimality of the mass transfer plan (I × ϕ) # µ by means of Kantorovich relaxation and duality [START_REF] Guillen | Five lectures on optimal transportation: geometry, regularity and applications[END_REF], according to which the Hessian of the function F considered in Proposition 1.1 is non negative at each point of the graph of ϕ.

Specifically, since F is stationary on that graph, its Hessian is a quadratic form field intrinsically defined there [15, pp.4-5], and it is elementary to derive from Corollary 2.1 the Corollary 2.2 If the total cost C admits a local minimum at ϕ ∈ Diff µ, (Ω), the Hessian of F is non negative at each point of the graph of ϕ.

Proof. Indeed, Hess(F ) vanishes along the graph Γ ϕ = (m, ϕ(m)), m ∈ M ⊂ Ω while transversally, in the direction of M , it is given by Hess c (f ). Specifically, let (m, p) ∈ M × P lie in Γ ϕ , take a basis (e 1 , . . . , e n ) of T m M and another one (f 1 , . . . , f n ) of T p P , set dϕ(e i ) = n j=1 J j i f j and:

J = J j i 1 i,j n , C = (d M d P c)(e i , f j ) 1 i,j n , H = Hess c (f )(e i , e j ) 1 i,j n , H = Hess c ( f )(f i , f j ) 1 i,j n
.

The matrix of Hess(F )(m, p) in the basis B 0 = (e 1 , . . . , e n , f 1 , . . . , f n ) is the block matrix Hess(F )

B 0 = H C t C e
H (where t C stands for the transpose of C). Besides, the following matrices identities hold 5 : H = -CJ, H = -t CJ -1 . We thus find:

Hess(F ) B 0 = C 0 0 t C -J I I -J -1 ,
from what we infer that the matrix of Hess(F )(m, p) in the new basis B 1 = (e 1 , . . . , e n , Je 1 , . . . , Je n ) is equal to:

(4a) Hess(F ) B 1 = I 0 I t J Hess(F ) B 0 I I 0 J = H 0 0 0 .
Alternatively, using at (m, p) the new basis B 2 = (J -1 f 1 , . . . , J -1 f n , f 1 , . . . , f n ), we would find:

(4b) Hess(F ) B 2 = t J -1 I 0 I Hess(F ) B 0 J -1 0 I I = 0 0 0 H .

From Corollary 2.1, the non negativity of Hess(F )(m, p) appears obvious in either new basis

Remark 2.1 The formulas (4) show that the following equivalence hold at each point of the graph of a diffeomorphism ϕ ∈ Diff µ, (Ω) stationary for the total cost C:

Hess(F ) 0 ⇐⇒ Hess c (f ) 0 ⇐⇒ Hess c ( f 0 .

5 easily checked by differentiating at (m, p) the identity: dF ≡ 0 on Γϕ

Ph. Delanoë

From now on, condition (3c) is assumed; let us figure out its impact on the preceding results. If (3c) holds, the implicit function theorem provides, for each (m 0 , p 0 ) ∈ Ω, setting α 0 = -d M c(m 0 , p 0 ), neighborhoods U 0 of p 0 in P and W 0 of (m 0 , α 0 ) in T * M , and a map E (m 0 ,p 0 ) c : W 0 → U 0 such that:

α ≡ -d M c m, E (m 0 ,p 0 ) c (m, α) .
Following [START_REF] Kim | Continuity, curvature, and the general covariance of optimal transportation[END_REF], maps like E (m 0 ,p 0 ) c may be called local c-exponential maps. Observe that, for any (m 1 , p 1 ) close enough to (m 0 , p 0 ) in Ω such that, setting α 1 = -d M c(m 1 , p 1 ), the point (m 1 , α 1 ) lies in W 0 , the maps E (m 0 ,p 0 ) c and E (m 1 ,p 1 ) c coincide on W 0 ∩W 1 . If M is simply connected, one can thus extend uniquely the map E (m 0 ,p 0 ) c to a neighborhood W (m 0 ,α 0 ) of a global section of T * M passing through α 0 at m 0 and call it the (m 0 , p 0 )-determination of the c-exponential map on M . With these notions at hand, Corollary 2.1 can be strengthened as follows:

Corollary 2.3 If condition (3c) holds and C admits a local minimum at ϕ ∈ Diff µ, (Ω), the c-Hessian of the c-potential f of ϕ must be positive definite, and so must be that of ϕ -1 ; in particular, the local minimum of C must be strict. Moreover, the Jacobian equation (2) satisfied by ϕ, written locally in terms of f via a local exponential map, reads as a Monge-Ampère equation of elliptic type.

Remark 2.2 From Corollary 2.3 combined with (4a), or (4b), using the compactness of M , we infer the existence of a neighborhood N ϕ of Γ ϕ in Ω such that F > F Γϕ on N ϕ \ Γ ϕ . This local result, obtained under (3c), should be compared to the global one, namely F Γϕ = inf M ×P F , derived from the conditions (3) but (3c), in the course of the proof of Proposition 1.2.

Finally, if the conditions (3c) (3d) hold, all local c-exponential maps coincide with the one (just denoted by E c ) globally defined by the generating condition. If so, we can improve the preceding results and, ultimately, reduce the solution of Monge's problem to the construction of an elliptic solution of a Monge-Ampère equation. Specifically, we will prove: 3 Proofs of the second order results

Proof of Theorem 2.1

Let ϕ ∈ Diff µ, (Ω) be stationary for C and, for t a small real parameter, let t → ϕ t ∈ Diff µ, (Ω) be an arbitrary path such that ϕ 0 = ϕ. We can uniquely write ϕ t = ξ t • ϕ with t → ξ t ∈ Diff (P ) such that ξ 0 = the identity of P . In particular, ξ0 is a tangential vector field on P lying in ker div (setting as usual ξ0 (p) = ∂ ∂t ξ t (p) t=0 ). We will prove the equivalence (i) ⇐⇒ (ii) by establishing the equality:

(6) d 2 dt 2 C(ϕ t ) t=0 = P Hess c ( f )( ξ0 , ξ0 ) d .
Using arbitrary source and target charts x, y, and the Einstein summation convention, one routinely finds:

d dt C(ϕ t ) = M ∂c ∂y i (x, ϕ t (x))
∂ϕ i t ∂t (x) dµ(x) where dµ(x) = dµ dx (x)dx, then the second variation expression:

d 2 dt 2 C(ϕ t ) = M ∂c ∂y i (x, ϕ t (x)) ∂ 2 ϕ i t ∂t 2 (x) + ∂ 2 c ∂y i ∂y j (x, ϕ t (x)) ∂ϕ i t ∂t (x) ∂ϕ i t ∂t (x) dµ(x).
Here, to spare the unfamiliar reader, we did not use a global linear connection (on P ) to compute the second derivatives which occur in the integrand. Doing so, we must be careful, as explained in the following remark.

Remark 3.1 Taken separately, the local scalar terms ∂c ∂y i (x, ϕ t (x))

∂ 2 ϕ i t ∂t 2 (x) and ∂ 2 c
∂y i ∂y j (x, ϕ t (x)) ∂t 2 c(m, ϕ t (m)). Therefore these terms cannot be integrated separately (unless the deformation ξ t is supported in the domain of a single chart of P , of Ph. Delanoë course). Still, in each couple of source and target charts, splitting the local expression ∂ 2 ∂t 2 c(x, ϕ t (x)) of the global function into the above non invariant terms can be done in a unique way, by using the canonical flat connection of the P chart. Anytime we will have to integrate on a manifold a global real function splitting uniquely into a sum of non invariant local terms, we will stress that the integral should not be split by putting the sum between braces, as done above. As long as they are written in the same charts, the addition of two such sums {a 1 + a 2 } + {b 1 + b 2 } may, of course, be written between a single pair of braces

∂ϕ i t ∂t (x)
{a 1 + a 2 + b 1 + b 2 }.
Back to our proof, using ϕ t = ξ t • ϕ and recalling (1), we infer at t = 0:

(7) d 2 dt 2 C(ϕ t ) t=0 = P ξi 0 ∂c ∂y i (ϕ -1 (y), y) + ξi 0 ξj 0 ∂ 2 c ∂y i ∂y j (ϕ -1 (y), y) d (y),
where d (y) = d dy (y)dy and ξi

0 = ∂ 2 ∂t 2 ξ i t (y)| t=0 .
To proceed further, we note that the integral P f (ξ t (y)) d (y) is independent of t. Differentiating it twice with respect to t at t = 0, we get:

0 = P ξi 0 ∂ f ∂y i (y) + ξi 0 ξj 0 ∂ 2 f ∂y i ∂y j (y) d (y).
Adding this vanishing integral to the right-hand side of (7), applying the last part of Remark 3.1 and recalling the stationary point equation:

∂c ∂y i (ϕ -1 (y), y) + ∂ f ∂y i (y) ≡ 0,
derived at once from Proposition 1.1, we obtain (6) as desired.

A similar argument would yield:

d 2 dt 2 C(ϕ -1 t ) t=0 = M Hess c (f )( ζ0 , ζ0 ) dµ,
with the vector field ζ0 ∈ ker div µ obtained by writing ϕ -1

t = ζ t • ϕ -1 for a unique path t → ζ t ∈ Diff µ (M ).
It would imply the other equivalence (i) ⇐⇒ (iii), since C(ϕ -1 ) ≡ C(ϕ) is a local minimum of C as well. The proof of Theorem 2.1 is complete.

Proof of Corollary 2.1

Strategy Let ϕ ∈ Diff µ, (Ω) realize a local minimum of the total cost C and let f : M → R denote its c-potential, as provided by Proposition 1.1. Arguing by contradiction, we suppose the existence of a point m 0 ∈ M such that the quadratic form associated to the symmetric bilinear one Hess c (f )(m 0 ) : T m 0 M × T m 0 M → R can take negative values. We will contradict property (ii) of Theorem 2.1 by constructing a vector field U ∈ ker div µ supported near m 0 such that M Hess c (f )(U, U ) dµ < 0. A similar argument would hold for ϕ -1 , of course. We will proceed stepwise, choosing a good chart at m 0 , constructing the vector field U in that chart and evaluating the above integral; we set n = dim M .

Choice of a chart We pick any chart y of P at ϕ(m 0 ) but a special chart x of M centered at m 0 , namely a chart which pushes the measure dµ to the canonical Lebesgue measure dx. The existence of such µ-adapted charts, to call them so, is well-known [START_REF] Banyaga | Formes volume sur les variétés à bord[END_REF][START_REF] Dacorogna | On a partial differential equation involving the Jacobian determinant[END_REF] and timely, here, to transform the div µ operator on M into the usual div operator of R n (up to sign) that is, the divergence operator associated to the measure dx (simply denoted by div below). Since the orthogonal group O(n) preserves the measure dx, we may further choose the chart x such that the matrix H ij (0) of Hess c (f )(m 0 ) is diagonal, with eigenvalues λ 1 λ 2 . . . λ n (each repeated with its multiplicity). Under our assumption: λ 1 < 0. Since the unimodular group SL(n, R) preserves the measure dx, we may rescale the chart x in order to have: λ 1 -3 and, ∀i ∈ {2, . . . , n}, λ i 1 2 . Let the chart x be fixed so and let x → H ij (f )(x) denote the local expression of the map m → Hess c (f )(m). The inequality: 2 , combined with the continuity of the map (x, v) → H ij (f )(x)v i v j as (x, v) varies near x = 0 with v of length 1 (say), implies the existence of a real ε > 0 such that:

∀v ∈ R n , H ij (f )(0)v i v j -3(v 1 ) 2 + 1 2 n i=2 (v i )
(8) ∀(x, v) ∈ R n ×R n , max 1 i n |x i | ε ⇒ H ij (f )(x)v i v j -2(v 1 ) 2 + n i=2 (v i ) 2 .
Construction of a divergence free vector field The vector field on R n given by w(

x) = x 1 ∂ ∂x 2 -x 2 ∂ ∂x 1 satisfies div(w) = 0. The flow of w preserves any function h : R n → R factoring through a function H : [0, ∞)×R n-2 → R as: h(x) = H( (x 1 ) 2 + (x 2 ) 2 , x 3 , . . . , x n ). Any such function h thus satisfies div(hw) = 0. Let us fix a cut-off function α : [0, ∞) → [0, 1] equal to 1 on [0, ε/2], vanishing on [ε/ √ 2, ∞)
, decreasing in-between, and consider the function h : R n → R given by:

h(x) = α (x 1 ) 2 + (x 2 ) 2 n i=3 α |x i | .
We know that div(hw) = 0 and that the vector field hw is supported in the open box

B n ε = x ∈ R n , max 1 i n |x i | < ε . Since the chart x is µ-adapted,
Ph. Delanoë we may view hw as the expression in that chart of a div µ free vector field U in M supported in the inverse image x -1 (B n ε ) ⊂ M .

Calculation of an integral Let us consider the integral M Hess c (f )(U, U )dµ which is equal to:

B n ε h 2 (x)H ij (f )(x)w i w j dx. From (8), it is bounded above by: 2 ∞ 0 α 2 (ρ)dρ n-2 B 2 ε α 2 (x 1 ) 2 + (x 2 ) 2 -2(x 2 ) 2 + (x 1 ) 2 dx 1 dx 2 .
Note that the function (x 1 , x 2 ) → α (x 1 ) 2 + (x 2 ) 2 vanishes outside the Euclidean ball of radius ε centered at 0; using polar coordinates (r, θ) in R 2 \ {0}, we thus find that the last integral is equal to:

ε 0 α 2 (r)r 3 dr × 2π 0 cos 2 θ -2 sin 2 θ dθ ≡ -π ε 0 α 2 (r)r 3 dr.
We conclude that M Hess c (f )(U, U )dµ is bounded above by a negative real, namely by -π 2 ∞ 0 α 2 (ρ)dρ n-2 ε 0 α 2 (r)r 3 dr, contradicting property (ii) of Theorem 2.1, as desired.

Proof of Corollary 2.3

Let again ϕ ∈ Diff µ, (Ω) realize a local minimum of the total cost C. Fix an arbitrary point m 0 ∈ M and take a µ-adapted chart x of M centered at m 0 and a -adapted chart y of P centered at p 0 = ϕ(m 0 ). From (3c) and the definition of E (m 0 ,p 0 ) c , the map x → y = ϕ(x) is defined near x = 0 by the equation:

(9) ∂f ∂x i (x) + ∂c ∂x i (x, y) y=ϕ(x) = 0.
Differentiating the latter yields (sticking to the notation Hess

c (f ) = H ij (f )dx i ⊗ dx j ): H ij (f )(x) = - ∂ 2 c ∂y k ∂x i (x, y) y=ϕ(x) ∂ϕ k ∂x j (x).
Taking determinants, recalling that the symmetric matrix H ij (f )(x) is non negative (by Corollary 2.1) and using (2), we obtain a local Monge-Ampère equation satisfied by f , namely:

det (H ij (f )(x)) = det ∂ 2 c ∂y k ∂x i (x, y) y=ϕ(x)
, where y = ϕ(x) is given by [START_REF] Evans | Partial differential equations and Monge-Kantorovich mass transfer[END_REF]. From (3c), the right-hand side of this equation nowhere vanishes; so the matrix H ij (f )(x) must be positive definite. So must be the c-Hessian of f throughout the manifold M , since the point m 0 is arbitrary. Finally, as is well-known, the positive definiteness just obtained implies, indeed, the ellipticity of the Monge-Ampère equation; from [START_REF] Dacorogna | On a partial differential equation involving the Jacobian determinant[END_REF], it also shows that C(ψ) > C(ϕ) for any ψ = ϕ close enough to ϕ in Diff µ, (Ω). A similar argument would hold for ϕ -1 .

Proof of Theorem 2.2

Let ϕ ∈ Diff µ, (Ω) be a local minimizer of C. By Proposition 1.1 and (2), it must admit a c-potential f : M → R solving equation [START_REF] Caffarelli | Allocation maps with general cost functions[END_REF]. Furthermore, from Corollary 2.3 and its proof, the c-Hessian of f must be positive definite, so f is an elliptic solution of [START_REF] Caffarelli | Allocation maps with general cost functions[END_REF]. As such, it is unique up to addition of a constant, as readily shown by a maximum principle argument [2, p.97].

Conversely, let f solve (5) with ϕ(m) ≡ E c (m, df (m)), the graph of ϕ lying in Ω and Hess c (f ) non negative at some point m 0 ∈ M . From (5) read at m 0 and (3c), we see that Hess c (f )(m 0 ) must be positive definite; the positive definiteness of Hess c (f ) now spreads to the whole of M due to [START_REF] Caffarelli | Allocation maps with general cost functions[END_REF]. At each m ∈ M , using adapted charts at m and ϕ(m) as above, the definition of the map ϕ yields ( 9) from what we infer, by differentiation and combination with [START_REF] Caffarelli | Allocation maps with general cost functions[END_REF], that (2) holds. Integrating (2) over M shows that ϕ(M ) has full measure in (P, ); so [START_REF] Aubin | Nonlinear Analysis on Manifolds. Monge-Ampère equations[END_REF] implies that ϕ pushes µ to . Since the latter are volume measures, the map ϕ : M → P must be onto7 as it is easy to check using (1) and, by the inverse function theorem, it must be a local diffeomorphism, thus a covering map. If M is orientable, since µ and have equal total mass, a degree argument shows that ϕ must be 1-sheeted hence, indeed, a diffeomorphism. In the non orientable case, one can argue similarly using the orientation covers of M and P , and infer again that ϕ must be a diffeomorphism (exercise). So ϕ ∈ Diff µ, (Ω). Finally, from the definition of ϕ and the proof of Proposition 1.1, we know that ϕ -1 ∈ Diff ,µ ( Ω) is stationary for C. From the positive definiteness of Hess c (f ) combined with the analogue of ( 6) for C at ϕ -1 , we see that C admits at ϕ -1 a strict local minimum. Since C(ϕ) ≡ C(ϕ -1 ), so does C at ϕ.

Definition 2 . 1

 21 Let Φ : M → P be a map whose graph lies in Ω and h : M → R a function related to Φ by the equation d M c(m, Φ(m)) + dh(m) = 0 on M . The (c, Φ)-Hessian of h is the covariant symmetric 2-tensor on M , denoted by Hess c,Φ (f ), intrinsically 4 defined, in any couple of source and target charts (x, y), by:

Theorem 2 . 2

 22 If the cost function satisfies (3c) (3d), there exists at most one local minimizer ϕ ∈ Diff µ, (Ω) of the total cost functional C. If it exists, it should read m → ϕ(m) = E c (m, df (m)) for some c-potential f with Hess c (f ) positive definite on M and f elliptic solution of the Monge-Ampère equation: (5) d dy (ϕ(x)) det Hess c (f )(x) = det ∂ 2 c ∂y k ∂x i (x, y) y=ϕ(x) dµ dx (x) Conversely, if f solves (5) with 6 , ∀m ∈ M, ϕ(m) = E c (m, df (m)) ∈ Ω P m and Hess c (f ) non negative at one point, the map ϕ : M → P must lie in Diff µ, (Ω) and be a local minimizer of the total cost C. Furthermore, if (3a) (3b) hold, f is c-convex and the diffeomorphism ϕ actually solves the Monge problem associated to C. The next section contains successively the proofs of Theorem 2.1, Corollary 2.1, Corollary 2.3, and Theorem 2.2 (except for its last part, straightforward here from Proposition 1.2).

  are not invariant under a change of charts, unlike their sum, indeed equal to the global real function: m ∈ M → ∂ 2

the Hessian of a function at a stationary point is intrinsic[15, pp.4-5]

the first condition is sometimes called stay-away (away, from the singular locus of c) and, if M has a boundary, it is also part of what yields the second boundary value condition (see the last footnote of the paper)

in particular, if ∂M is non empty, ϕ sends ∂M to ∂P (expressed in terms of f , this is sometimes called the second boundary value condition)

Acknowledgments

The author got the idea of this note soon after returning from a visit at the Fields Institute during the "Concentration period on Partial Differential Equations and Geometric Analysis" (October 25-29, 2010). He wishes to thank the Institute for partial support and Robert McCann for his invitation and a stimulating discussion. He is also grateful to the Referee for pointing out a mistake in the original proof of Corollary 2.2.

† Supported by the CNRS (INSMI) at UMR 6621


Beilei Du
email: beilei.du@ep5.rub.de

Mitsutoshi Aramaki

Sebastian Mohr

Yusuf Celik

Dirk Luggenhölscher

Uwe Czarnetzki

Spatially and temporally resolved optical spectroscopic investigations inside a self-pulsing micro thin-cathode discharge

A fiber probe is developed to investigate the temporal development and the spatial distribution of the population density of argon 1s 5 state and the emission in the afterglow in a self-pulsing micro thin-cathode discharge (µTCD) at atmospheric pressure. The fiber probe with a diameter (125 µm) of about half the discharge diameter (200 µm) is moved along the discharge axis. A tunable diode laser beam at 696.73 nm at resonance with the argon 1s 5 → 2p 2 transition is coupled into the far end of the fiber probe. The spatially differentiated absorbance yields the axial distribution of the 1s 5 state. In addition, the fiber is used to determine similarly the spatially resolved optical emission of various argon lines. Both the emission and the absorption profiles confirm that, in the afterglow, the plasma is strongly localized inside the hole of the discharge. Qualitatively, the measurements agree very well with the results of a zero-dimensional simulation. The long living afterglow is related to the high population density of the metastable atoms.

In recent years, micro-plasmas have become an emerging field of both plasma research and application. In research, due to the extreme conditions of the microdischarges, spatially resolved investigations are mostly restricted. In micro-cavity discharges, the radial distributions can be obtained by magnification of the discharge cross section emission image [1,2], but the axial distributions are often unknown. Here, a new technique is developed and applied to the investigation of the axial distributions in a micro thin-cathode discharge (µTCD) operated in the self-pulsing mode. In this mode, a repeated ignition of a nanosecond spark discharge occurs [2] with a surprisingly long lifetime of the afterglow of several hundred nanoseconds. A kinetic model was developed to identify the mechanisms for the long living afterglow [3]. The model indicates that the long living afterglow is a result of the high population density of the excited states, especially the metastable 1s states, the formation of excimers and the high gas temperature which effectively results in a low recombination rate. Here, the axial distribution and the temporal development of the metastable atoms of the 1s 5 state in the afterglow are determined from tunable diode laser absorption spectroscopic (TDLAS) measurements using an optical fiber probe. The tunable diode laser beam at 696.73 nm in resonance with the argon 1s 5 → 2p 2 transition is coupled into the far end of the fiber probe. The spatially differentiated absorbance yields the axial density distribution of the 1s 5 state. The measurement result confirms the high population density of the metastable state indicated by the simulation. In addition, the fiber probe is used to determine the spatially resolved optical emission of various argon lines. Both the emission profiles and the distribution of the metastable atoms confirm a strong localization of the plasma in the afterglow around the dielectric isolator between the electrodes.

The µTCD has a sandwich structure. A thin dielectric layer (here: 100 µm) separates two electrodes (Figure 1). One electrode, which is used as the anode, is much thicker (here: 1 mm) than the cathode. The cathode has the same thickness as the dielectric. The discharge develops in a hole with a diameter of 200 µm penetrating the sandwich structure. Molybdenum and alumina were used for the electrodes and the dielectric layer, respectively. To provide a pure argon environment, the µTCD was placed in a chamber. The cathode side of the discharge faces a mirror, which reflects the emission light of the discharge under 90 degrees to an achromatic lens fixed on the side wall of the chamber. The achromatic lens images the emission light onto an optical fiber, which can be coupled with an optical detector, e. g. a photomultiplier or a spectrograph, etc. The fiber probe was moved from the anode side into the chamber through a small hole with a diameter of 1.5 mm. The fiber probe is made of a multimode optical fiber. The bare fiber (including the core, the cladding and the coating) has a diameter of 250 µm. On one end of the fiber the coating is removed for about 10 mm. The cladding with the core with a diameter of 125 µm is moved into the discharge along discharge axis. The bare fiber is fixed in a ceramic tube, which has a length of 20 cm, an inner diameter of 300 µm, and an outer diameter of 1 mm. The rest of the bare fiber is protected by a jacket tube. A holder fixes the fiber probe with a motorized translation stage with a step resolution of 0.1 µm. The remaining gap between the ceramic tube and the hole has a cross section of about 1 mm 2 . Due to the small gap, the air in the chamber was not evacuated by a vacuum pump but pushed through the gap out of the chamber by a argon flux of 900 sccm. The µTCD was operated at atmospheric pressure. The gas purity was checked via the emission spectrum of the discharge.

The µTCD was charged via a 500 kΩ series resistor. A 100 Ω resistor was connected in series with the anode to monitor the current. In the self-pulsing mode, the discharge current pulse cannot be directly measured. More details about the current measurement are described in [2]. A 700 V dc voltage with negative polarity was applied. The discharge operates in the spark self-pulsing mode with a self-pulsing frequency of about 10 kHz. To avoid an overheating of the electrodes, the applied voltage was pulsed with 10 Hz and with 0.3% duty cycle. The density of the argon metastable atoms in the 1s 5 state was deduced from the TDLAS measurement. The laser system is described in detail in a recent work [4]. To avoid saturation of the absorption, the laser power is reduced to about 1.5 µW by a neutral density filter. The laser beam is coupled into the far end of the fiber probe, then passes through the discharge, and is imaged onto the optical fiber connecting to a photomultiplier (Hamamatsu R928). An interference filter (FWHM = 10 nm) is installed in front of the detector to discriminate the investigated spectral line. The signal of the photomultiplier is read out by an oscilloscope (LeCroy WaveSurfer 424), triggered by the discharge voltage pulse. The maximal output current of the photomultiplier is only 0.1 mA. To intensify the voltage signal at the oscilloscope, while at the same time providing the temporal resolution, a 1 kΩ terminal resistor was chosen, which lead to a response time of about τ = 200 ns. The real signal S real is recovered from the output signal S by deconvolution:

S real = S + τ Ṡ.
(1)

While moving the fiber probe along the discharge axis, the intensities of the laser radiation I 0,ν , the plasma emission I pl,ν , and the laser radiation through the plasma emission I l+pl,ν are recorded at each position and laser frequency with averaging over 200 plasma pulses. The intensity of the transmission I ν is deduced by the difference of I l+pl,ν and I pl,ν :

I ν = I l+pl,ν -I pl,ν , (2)
and the absorbance

+∞ -∞ ln I 0,ν
Iν dν with respect of the probe position is obtained.

The difference in I ν depends on the absorption coefficient α ν and the absorption length dx:

dI ν = -α ν I ν dx =⇒ α ν = - dI ν I ν dx = d dx ln I 0,ν I ν (3)
The absorption coefficient α ν depends on the density of the metastable atoms n Ar * [START_REF] Cornay | Atomic and Laser Spectroscopy[END_REF]:

α ν = 1 4πε 0 πe 2 m e c f n Ar * g ν , (4)
where f = 0.0313 is the oscillator strength of the 1s 5 → 2p 2 transition [START_REF] Zatsarinny | [END_REF] and g ν is the normalized line profile. Equation (3) and Equation ( 4) lead to:

d dx ln I 0,ν I ν = 1 4πε 0 πe 2 m e c f n Ar * g ν . (5)
Integrating Equation ( 5) over the frequency, the spatially resolved metastable atom density is obtained by a simple differentiation of the measured absorbance:

n Ar * = 4πε 0 m e c πe 2 1 f d dx +∞ -∞ ln I 0,ν I ν dν. (6)
Instead of coupling the laser into the fiber probe, the far end can also be connected to a 2m-spectrograph equipped with an intensified diode array with a gate-width of 33 ms, much longer than the duration of the discharge. In this way, the fiber probe was used to perform time integrated and spatially resolved optical emission spectroscopic (OES) measurement, similar to an optical probe for spatially resolved OES described in [7]. Here, at the probe position x, the measured intensity S(x) is given as an integral over the local emissivity I(x) of the plasma:

S(x) = +∞ x I(x ) dx . (7)
By a simple differentiation of the measured profile with respect to the probe position the local emissivity is deduced: In order to investigate a possible perturbance of the discharge by the presence of the fiber probe, various additional measurements were performed. Firstly, V -I curves were recorded which show no influence by the presence of the fiber probe at all (breakdown voltage: 600 V, current peak FWHM: 2 ns, current amplitude: 36 A). Secondly, the optical wave guide collecting the emitted radiation is replaced by an ICCD camera equipped with a microscope lens. Spatially resolved images are taken from the top of the discharge (Figure 2) with the integration time of 300 µs covering the first two selfpulsing sparks in a burst series. Clearly, the discharge is located as a tiny spot in the vicinity of the wall independent of the fiber probe position. The fiber image is blurred due to the speckle interference of the coherent diode laser beam. The strong localization of the discharge allows moving of the fiber probe along the axis without a significant disturbance of the plasma. On the other hand, in the absorption measurement, the main absorption may not be caused by the metastables within the localized plasma located at the wall. In fact, up to 80% absorption is observed which is only possible by the metastables covering the majority of the hole cross section. We will discuss this aspect in more detail in the later part of this paper.

I(x) = - dS(x) dx . (8)
The spectral absorption profiles were recorded at each position and time. They exhibit a Lorentz profile with a substantial broadening of about 11 pm. The FWHM of the pressure broadening is given by [9]:

w pressure = 2γN, ( 9
)
where N is the neutral gas density and γ is the collision broadening coefficient given in [10]. For the 1s 5 → 2p 2 transition, 2γ = 0.910 • 10 -21 nm/atom cm 3 . Assuming the line broadening is only contributed by the pressure broadening, the neutral gas density of Ar is calculated to be 1.2 • 10 19 cm -3 relating to a gas temperature of about 600 K at atmospheric pressure. In the small spark created by the nanosecond current pulse, a higher gas temperature is expected. This indicates that the assumption of a gas temperature around 1000 K in the zero-dimensional simulation is reasonable.

To determine the absorption profiles, the cross section integrated plasma emission is recorded in parallel by the PMT as described above. The temporal development of the emission and absorption behavior at various fiber probe position is displayed in Figure 3. Apparently, emission decreases when the fiber is moved beyond the inner edge of the anode, which indicates shadowing. Further, the temporal development of the absorption profile changes. The absorption data are processed according to the procedure described above, and the result is summarized in Figure 4. The temporal development of the metastable density coincides with the temporal decay of the optical emission. An interpretation might be that the small spark created by the nanosecond current pulse at the edge is emitting high intensity VUV radiation which then leads to formation of the excited states over the full cross section of the discharge hole. Propagation of this VUV radiation is probably slowed down by radiation diffusion due to the very high optical density for ground state absorption at atmospheric pressure. These excited states relax to the metastable states which then causes the large volume diode laser beam absorption. In this model, the fiber will shadow the VUV radiation when moved beyond the cathode edge. i. e. causing a certain perturbation of the measured upper metastable densities. Nevertheless the measured 1s 5 state density of 6.3•10 14 cm -3 coincides well with he value calculated by the zero-dimensional model [3]. However, one has to be careful in the interpretation, since most likely the source of metastable production in the measurement is radiation and not the plasma directly.

In addition, two of the 2p 2 → 1s 5 emission curves are displayed in Figure 4. The lower left curve (the dashed line) shows the emission from the entire discharge without spatial resolution, recorded in parallel with the absorption. Connecting the fiber probe with the 2m-spectrograph, the axially resolved but time integrated emission is obtained. This curve is shown on the right part in Figure 4 (the dotted line). Clearly, metastables are localized mainly close to the inner edge of the cathode with a small peak close to the anode. The metastables appear with a substantial delay of about 0.5 µs and then last for several microseconds. The emission is localized between the anode and the cathode over a region of about 300 µm. The emission spectrum exhibits lines only after a certain temporal interval of about 200 ns after the nanosecond current pulse. At early times, the emission is dominated by a strong continuum which might be caused by bremsstrahlung. This continuum is also visible in the time integrated emission shown in Figure 5.

The power density in the discharge can be estimated from the measured volume of the emission (cross section diameter: 50 µm; length: 200 µm) and the stored energy of the capacitor (120 pF) at the breakdown voltage of 600 V. The corresponding power density in the 2 ns current pulse is 3•10 10 W/cm 3 . These extreme values most likely cause a shock wave transversing the discharge in a similar time as the build up of the metastables. However, this shock wave can not be expected to transport metastables from the small spark through the highly collisional argon gas. However, it might be possible that the increased collisionality in the shock wave front might lead to quenching of the excited states and thereby causes metastable population.

In conclusion, in this work a simple optical fiber probe was applied to perform spatially and temporally resolved optical absorption and emission spectroscopic measurements along the discharge axis in the afterglow of a self-pulsing argon µTCD. Both the emission profiles and the distribution of the metastable atoms indicate that in the afterglow the plasma is strongly localized around the dielectric isolator. Qualitatively, the measurement results agree very well with the zero-dimensional simulation results [3], where the long living afterglow is related to the high population density of the metastable atoms.

Figure 1 .

 1 Figure 1. Experimental setup.

  The fiber probe is on the cathode surface.

Figure 2 .

 2 Figure 2. ICCD images of the plasma emission and the laser beam from the fiber probe. White circles indicate the orifice of the discharge hole, the plasma volume and the beam out of the fiber probe.

Figure 3 .

 3 Figure 3. Measured plasma emission I pl (solid line) and the measured absorption (dashed line) with different fiber probe position along the discharge axis. The fiber probe position corresponds to the distance to the anode outer edge. 1.20 mm is at the cathode outer edge.

Figure 4 . 5 .

 45 Figure 4. Color map: spatially (y-axis) and temporally (x-axis) resolved 1s 5 density. ----: Temporal development of the 2p 2 → 1s 5 emission (696.54 nm) measured from the cathode side. • • • • • •: Time integrated spatial distribution of the 2p 2 → 1s 5 emission measured by the fiber probe.

Acknowledgments

This work was supported by the "German Research Foundation" (DFG) in the frame of the "Research Unit FOR 1123 -Physics of Microplasmas", the Research Department "Plasmas with Complex Interaction" of the Ruhr-University Bochum, the Research School of the Ruhr-University Bochum, and the JSPS ITP program. The authors would also like to thank Professor Igor Adamovich for the valuable suggestions.