Régis Monneau

Jean-Michel Roquejoffre

Violaine Roussier-Michon

Travelling graphs for the forced mean curvature motion in an arbitrary space dimension Ondes progressives pour le mouvement par courbure moyenne forcé en toute dimension d'espace

Keywords: forced mean curvature movement, eikonal equation, Hamilton-Jacobi equations, viscosity solution, reaction diffusion equations, travelling fronts

We construct travelling wave graphs of the form z = -ct + φ(x), φ : x ∈ R N -1 → φ(x) ∈ R, N ≥ 2, solutions to the N -dimensional forced mean curvature motion V n = -c 0 + κ (c ≥ c 0 ) with prescribed asymptotics. For any 1-homogeneous function φ ∞ , viscosity solution to the eikonal equation |Dφ ∞ | = (c/c 0 ) 2 -1, we exhibit a smooth concave solution to the forced mean curvature motion whose asymptotics is driven by φ ∞ . We also describe φ ∞ in terms of a probability measure on S N -2 . Résumé: Nous construisons des ondes progressives sous la forme de graphes z

espace et avec un comportement asymptotique prescrit. Pour toute solution de viscosité φ ∞ , 1-homogène en espace, de l'équation eikonale |Dφ ∞ | = (c/c 0 ) 2 -1, nous mettons en évidence une solution régulière et concave du mouvement par courbure moyenne forcé dont le comportement asymptotique est donné par φ ∞ . Nous décrivons aussi φ ∞ en terme d'une mesure de probabilit à c sur la sph ère S N -2 .

1 Introduction

Setting of the problem

The question investigated here is the description of the travelling wave graph solutions to the forced mean curvature motion in any dimension N ≥ 2, that is written under the general form [START_REF] Barles | Front propagation and phase phield theory[END_REF] V n = -c 0 + κ where V n is the normal velocity of the graph, κ its local mean curvature and c 0 a given strictly positive constant to be defined later. A graph satisfying (1) can be given by the equation z = u(t, x) where u : (t, x) ∈ R + × R N -1 → u(t, x) ∈ R is a solution to the parabolic equation

(2)

u t 1 + |Du| 2 = -c 0 + div Du 1 + |Du| 2 , t > 0 , x ∈ R N -1
Indeed, at any time t > 0 fixed, the outer normal to the subgraph {(x, z) ∈ R N -1 ×R | z ≤ u(t, x)} is given by

n = 1 1 + |Du| 2 -D x u 1
its normal velocity V n by (0, ∂ t u) T • n while its mean curvature by κ =div (x,z) n, see [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]. A travelling wave to (2) is a solution of the form u(t, x) = -ct + φ(x) where φ : x ∈ R N -1 → φ(x) ∈ R is the profile of the wave and c ≥ c 0 is some given constant standing for its speed. Thus φ satisfies the following elliptic equation

(3) -div Dφ 1 + |Dφ| 2 + c 0 - c 1 + |Dφ| 2 = 0 , x ∈ R N -1

Connection with reaction diffusion equations

This work should provide us a better understanding of the multidimensional solutions to the non linear scalar reaction diffusion equation (4)

∂ t v = ∆v + f (v) , t > 0 , (x, z) ∈ R N -1 × R
where v : (t, x, z) ∈ [0, +∞) × R N -1 × R → v(t, x, z) ∈ R and, especially the case of travelling waves in dimension N. In the case of a "bistable" nonlinearity f , that is to say when f is a continuously differentiable function on R satisfying i. f (0) = f (1) = 0

ii. f ′ (0) < 0 and f ′ (1) < 0

iii. there exists θ ∈ (0, 1) such that f (v) < 0 for v ∈ (0, θ), f (v) > 0 for v ∈ (θ, 1)

iv.

1 0 f (v) dv > 0,
it is well-known [START_REF] Ya | On the stability of solutions of the cauchy problem for equations arising in the theory of combustion[END_REF] that there exists a one-dimensional travelling front v(t, z) = φ 0 (z+c 0 t) solution to [START_REF] Clutterbuck | Stability of translating solutions to mean curvature flow[END_REF] with N = 1. The speed c 0 is unique and strictly positive by [iv] while the profile φ 0 is unique up to translations. This result defines the constant c 0 > 0 that appears in equation [START_REF] Barles | Front propagation and phase phield theory[END_REF].

In the case N = 2, multidimensional solutions to (4) are well understood. Paper [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF] proves the existence of conical travelling waves solutions to (4), and paper [START_REF] Hamel | Asymptotic properties and classification of bistable fronts with Lipschitz level sets[END_REF] classifies all possible bounded non constant travelling waves solutions under rather weak conditions at infinity. In particular, it is proved in [START_REF] Hamel | Asymptotic properties and classification of bistable fronts with Lipschitz level sets[END_REF] that c ≥ c 0 and, up to a shift in x ∈ R, either u is a planar front φ 0 (±x cos α + z sin α) with α = arcsin(c 0 /c) ∈ (0, π 2 ] or u is the unique conical front found in [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF].

In higher dimensions, less is known. In [START_REF] Hamel | Asymptotic properties and classification of bistable fronts with Lipschitz level sets[END_REF], Hamel, Monneau and Roquejoffre proved the existence of conical travelling waves with cylindrical symmetry whose level sets are Lipschitz graphs moving away logarithmically from straight cones. Some special, non cylindrically symmetric pyramidal-shaped solutions (Taniguchi,[START_REF] Taniguchi | The uniqueness and the asymptotic stability of pyramidal travelling fronts in the Allen-Cahn equations[END_REF]) are also known in the particular case N = 3.

Thus, in order to get a better understanding of the mechanisms at work, we further the idea of bridging reaction-diffusion equations with geometric motions. In particular, travelling wave graph solutions to the forced mean curvature motion go back to Fife [START_REF] Fife | Dynamics of internal layers and diffusive interfaces[END_REF]. He proved (in a formal fashion) that reaction-diffusion travelling fronts propagate with normal velocity

V n = -c 0 + κ t + O 1 t 2 , t >> 1.
For a mathematically rigorous treatment of these ideas, we refer for instance to de Mottoni, Schatzman [START_REF] De Mottoni | Geometrical evolution of developed interfaces[END_REF] -small times, smooth solutions context -and Barles, Soner, Souganidis [START_REF] Barles | Front propagation and phase phield theory[END_REF] -arbitrary large times, viscosity solutions context. Related results must me mentioned in the case of a balanced bistable non-linearity f : assumption iv. is replaced by 1 0 f (v) dv = 0 and (4) is called the balanced Allen-Cahn equation. In this case, c 0 = 0 and the forced mean curvature equation is replaced by the mean curvature equation. Chen, Guo, Hamel, Ninomiya and Roquejoffre [START_REF] Chen | Traveling waves with paraboloid like interfaces for balanced bistable dynamics[END_REF] proved that there exist cylindrically symmetric traveling waves with paraboloid like interfaces solutions to (4) in dimension N ≥ 3. Precisely, they proved that those solutions' level sets are asymptotically given by the equation z = c 2(N -1) |x| 2 . On the other hand, Clutterbuck, Schnürer and Schulze [START_REF] Clutterbuck | Stability of translating solutions to mean curvature flow[END_REF] proved that there exists a unique rotationally symmetric, strictly convex, translating graph u(t, x) = -ct + φ(r) to the mean curvature motion (3) with c 0 = 0 and whose asymptotics is given by

φ(r) = c 2(N -1) r 2 -ln r + C + O 1 r
Further works have also been done in the non radial case for the mean curvature equation. For instance, Xuan Hien Nguyen [START_REF] Nguyen | Construction of complete embedded self-similar surfaces under mean curvature flow[END_REF] built non radial and non convex translating graphs solution to (3) with c 0 = 0.

Main results

Our theorem 1.1 below states that, given a 1-homogeneous solution φ ∞ to the eikonal equation derived from (3) (i.e. the equation obtained by removing the curvature term) there exists a smooth solution φ to the forced mean curvature motion equation (3) whose asymptotic behaviour is prescribed by φ ∞ . Here is the precise result.

Theorem 1.1 (Existence of solutions with prescribed asymptotics in dim. N) Let N ∈ N \ {0, 1}, α ∈ (0, π 2 ], c 0 > 0 and c = c 0 / sin α. Choose φ ∞ a 1-homogeneous viscosity solution to the eikonal equation [START_REF] Crandall | User's guide to viscosity solutions of second order PDEs[END_REF] |Dφ ∞ (x)| = cot α , x ∈ R N -1 .

Then there exists a smooth concave solution φ ∈ C ∞ (R N -1 ) to (3) such that

(6) φ(x) = φ ∞ (x) + o(|x|) as |x| → +∞.
This is the most possible general result. However, due to the possible complexity of a solution to the eikonal equation ( 5), it is useful to specialise our result to the particular case of a solution with a finite number of facets.

Theorem 1.2 (Solutions with finite number of facets in dimension N)

Let N ∈ N \ {0, 1}, α ∈ (0, π 2
], c 0 > 0 and c = c 0 / sin α. Choose φ * a viscosity solution to the eikonal equation [START_REF] Crandall | User's guide to viscosity solutions of second order PDEs[END_REF] given for any x ∈ R N -1 by

(7) φ * (x) = inf ν∈A (-(cot α) x • ν + γ ν )
where A is a finite subset of cardinal k ∈ N * of the sphere S N -2 and γ ν are given real numbers. Then there exists a unique smooth concave solution φ ∈ C ∞ (R N -1 ) to (3) such that

(8)          - 2 ln k c 0 sin α ≤ φ -φ * ≤ 0 , x ∈ R N -1 lim l→+∞ sup dist(x,E∞)≥l |φ(x) -φ * (x)| = 0
where E ∞ is the set of edges defined as

E ∞ = {x ∈ R N -1 | φ ∞ is not C 1 at x} with the 1-homogeneous function φ ∞ (x) = inf ν∈A (-(cot α) x • ν)
In space dimension N = 3, we obtain a more precise result by considering solutions having a finite number of gradient jumps. Those solutions are still more complex than the infimum of a finite number of affine forms. Here is the precise result.

Theorem 1.3 (Solutions with finite number of gradient jumps and N = 3) Let α ∈ (0, π 2 ], c 0 > 0 and c = c 0 / sin α. Choose φ ∞ a 1-homogeneous viscosity solution to the eikonal equation [START_REF] Crandall | User's guide to viscosity solutions of second order PDEs[END_REF] in dimension N = 3 with a finite number of singularities on S 1 . Then, there exist

i. a 2π-periodic continuous function ψ ∞ : θ ∈ [0, 2π] → ψ ∞ (θ) ∈ [-cot α, cot α] and a finite number k ∈ N\ {0} of angles θ 1 < • • • < θ k in [0, 2π) such that φ ∞ (r cos θ, r sin θ) = rψ ∞ (θ) , (r, θ) ∈ R + × [0, 2π)
Moreover, for any i ∈ {1, . . . , k},

a. Either ∀θ ∈ [θ i , θ i+1 ], ψ ∞ (θ) = -(cot α) and we set σ i = 1 b. Or    ∀θ ∈ θ i , θ i +θ i+1 2 , ψ ∞ (θ) = -(cot α) cos(θ -θ i ) ∀θ ∈ θ i +θ i+1 2 , θ i+1 , ψ ∞ (θ) = -(cot α) cos(θ -θ i+1 )
and we set σ i = 0

By convention, θ k+1 = 2π + θ 1 and

σ k+1 = σ 1 . If k ≥ 2, then σ i σ i+1 = 0 for any i ∈ {1, ..., k}. ii. a smooth concave function φ ∈ C ∞ (R 2 ) solution to equation (3) such that when |x| goes to infinity φ(x) = φ * (x) + O(1)
where

(9) φ * (x) = - 2 c 0 sin α ln S 1 e c 0 cos α 2 x•ν dµ(ν)
and µ is the non negative measure on S 1 with finite mass determined by ψ ∞ as follows: We set µ = k i=1 µ i where for any fixed λ 0 > 0, we set

a. If σ i = 1, then µ i = 1I (θ i ,θ i+1 ) dθ + λ 0 (δ θ i + δ θ i+1 ) (with the exception for k = 1: µ 1 = 1I (θ 1 ,θ 1 +2π) dθ). b. If σ i = 0, then µ i = λ 0 (δ θ i + δ θ i+1 )
We plan to use our travelling graphs for the forced mean curvature motion exhibited in theorems 1.1 to 1.3 in order to construct multi-dimensional travelling fronts to the reaction diffusion equation (4); we plan to do it in a forthcoming paper.

That equation ( 5) prescribes the asymptotic behaviour of (3) has nothing surprising: let ε > 0 and denote by φ ε the scaled function

φ ε (x) = εφ x ε , x ∈ R N -1
Since φ is a solution to (3), φ ε satisfies

-ε div Dφ ε 1 + |Dφ ε | 2 + c 0 - c 1 + |Dφ ε | 2 = 0 , x ∈ R N -1
Let ε go to zero. If adequate estimates for φ ε are known, (a subsequence of) (φ ε ) ε>0 converges to a function φ ∞ satisfying [START_REF] Crandall | User's guide to viscosity solutions of second order PDEs[END_REF]. The proof of Theorem 1.1 is done by a sub and super solutions argument. We first construct a family of smooth sub-solutions to (3), which will give us some better insight in the equation. This step is quite general, and works in any space dimension. Then, we will construct a Lipschitz super-solution whose rescaled asymptotics is prescribed by φ ∞ and this will give us a smooth solution whose asymptotic behaviour is not well precise. To get a better asymptotics of the super-solution prescribed by the sub-solution, this will require a more delicate matching procedure which will limit us, for the moment, to any space dimension N with a finite number of facets (theorem 1.2) or to the space dimension N = 3 and a finite number of gradient jumps (theorem 1.3).

The rest of this paper is organised as follows. In section 2, we build and characterise all 1-homogeneous solutions to the eikonal equation [START_REF] Crandall | User's guide to viscosity solutions of second order PDEs[END_REF]. In section 3, we detail Perron's method in our context, and explain why it will yield a smooth concave solution. Subsolutions are built in section 4, and super-solutions in section 5. Finally, section 6 sums up previous constructions to prove theorems 1.1 and 1.2. Section 7 presents a more precise approach in dimension N = 3 and details the proof of Theorem 1.3. An appendix is devoted to the Laplace's method that we use in our estimates.

Acknowledgments. The first author was partly supported by the ANR project MICA, the second and third ones by the ANR project PREFERED. They acknowledge a fruitful discussion with G. Barles and thank C. Imbert for enlightening discussions on his paper [START_REF] Imbert | Convexity of solutions and C 1,1 estimates for fully nonlinear elliptic equations[END_REF]. They are also indebted to H. Berestycki and CAMS Center of EHESS in Paris for their hospitality while preparing this work.

Eikonal equation

In this section, we classify the continuous viscosity solutions to the eikonal equation in any dimension N ≥ 2: [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF] |Dφ

∞ (x)| = cot α , x ∈ R N -1
where α ∈ (0, π 2 ] is some given angle. In a first subsection, we are interested in the general case. In a second one, we reduce our study to 1-homogeneous functions and give a better description of those solutions in order to use them in both sections 4 and 5.

Characterisation of solutions to (10) in any dimension N

For any unit vector ν ∈ S N -2 and γ ∈ (-∞, +∞], let us define the affine map

φ ν,γ (x) = -(cot α) ν • x + γ ∈ (-∞, +∞] , x ∈ R N -1 Proposition 2.1 (A Liouville theorem for the eikonal equation) Let φ ∞ ∈ C(R N -1 )
. Then φ ∞ is a viscosity solution to the eikonal equation [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF] if and only if there exists a lower semi-continuous map γ :

S N -2 → (-∞, +∞] such that (11) φ ∞ (x) = inf ν∈S N-2 φ ν,γ(ν) (x)
Moreover φ ∞ is 1-homogeneous if and only if for all ν ∈ S N -2 , γ(ν) ∈ {0, +∞}.

This result is most certainly known. Because we not only need the result but also an insight of the construction, we give a complete proof. Proof of Proposition 2.1. We first show the direct implication. Let φ ∞ ∈ C(R N -1 ) be a viscosity solution to [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF]. We shall prove that φ ∞ is (cot α)-Lipschitz and concave before giving its characterisation as an infimum of affine maps.

Step 1: φ ∞ is locally Lipschitz Let us consider the ball B(a, R) centered in a ∈ R N -1 with radius R > 2 and define

C := sup |x-y|≤1, (x,y)∈B(a,R) 2 |φ ∞ (x) -φ ∞ (y)| Because φ ∞ is continuous, 0 ≤ C < +∞. Denote C = max(C, cot α) ∈ (0, +∞).
Then we claim that for any (x, y) ∈ B(a, R -1) 2 such that |x -y| ≤ 1, we have

(12) |φ ∞ (x) -φ ∞ (y)| ≤ C|x -y|
which asserts that φ ∞ is locally Lipschitz. Indeed, for any point x 0 ∈ B(a, R -1), any constant C > C and any λ ≥ 0, we consider the function ψ λ defined as

ψ λ (x) := λ + φ ∞ (x 0 ) + C|x -x 0 | , x ∈ R N -1
and we set

λ * = inf λ ∈ R + | ∀µ ≥ λ , ∀x ∈ B(x 0 , 1) , ψ µ (x) ≥ φ ∞ (x)
We shall prove by contradiction that λ * = 0. If not, because ψ 0 ≥ φ ∞ on {x 0 }∪∂B(x 0 , 1), there exists a contact point z 0 between ψ λ * and φ ∞ which satisfies z 0 ∈ B(x 0 , 1)\ {x 0 }. Then ψ λ * is a test function for the viscosity subsolution φ ∞ at that point. Because |∇ψ λ * (z 0 )| = C > cot α, we get a contradiction with the viscosity subsolution inequality. Therefore λ * = 0 and ψ 0 ≥ φ ∞ on B(x 0 , 1). Because this is true for any C > C, we deduce that this is still true for C = C which implies [START_REF] Imbert | Convexity of solutions and C 1,1 estimates for fully nonlinear elliptic equations[END_REF].

Step 2: φ ∞ is (cot α)-Lipschitz We now define

L = lim sup n→+∞ L n with L n := sup φ ∞ (y) -φ ∞ (x) |y -x| , x ∈ B(a, R -2), |y -x| ≤ 1 n
Notice that for any n ∈ N * , 0 < L n ≤ C. Moreover, there exists a sequence (x n , y n

) n∈N * such that lim n→+∞ φ ∞ (y n ) -φ ∞ (x n ) |y n -x n | = L and |y n -x n | ≤ 1 n with x n ∈ B(a, R -2) Define for any x ∈ R N -1 ε n = |y n -x n | , φ n (x) = φ ∞ (x n + ε n x) -φ ∞ (x n ) ε n and ν n = y n -x n ε n ∈ S N -2
Thus (φ n (ν n )) n∈N * converges to L as n goes to infinity and for any x ∈ B(0, 1), |φ n (x)| ≤ L n |x|. Because Lip(φ n ; B(0, n)) ≤ C, we see that up to a subsequence, (φ n ) n∈N * converges locally uniformly on R N -1 to φ 0 a viscosity solution to [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF]. Moreover, (ν n ) n∈N * converges to ν 0 ∈ S N -2 with φ 0 (ν 0 ) = L and for any x ∈ R N -1 , φ 0 (x) ≤ L|x| =: ψ(x)

Because ψ touches φ 0 from above at ν 0 , we conclude from the viscosity inequality for subsolutions that L ≤ cot α Now for any ǫ > 0, there exists n ε ∈ N * such that L n ≤ L + ε for any n ≥ n ε . In particular, for any (x, y) ∈ B(a, R -2) we can split the segment

[x, y] = i=0,...,K-1 [x i , x i+1 ] with x 0 = x , x K = y and |x i+1 -x i | = |y -x| K ≤ 1 n This implies that |φ ∞ (x) -φ ∞ (y)| ≤ (L + ε)|x -y|
which is true for any ε > 0. This implies that φ is L-Lipschitz on B(a, R -2) with L ≤ cot α.

Step 3: φ ∞ is concave Because φ ∞ is a Lipschitz stationary viscosity solution to the evolution equation

u t + H(Du) = 0 , x ∈ R N -1 where H(p) = 1 2 (p 2 -cot 2 α) , p ∈ R N -1
we can apply Lemma 4 page 131 in [START_REF] Evans | Partial Differential Equations[END_REF], and get that φ ∞ satisfies for any t > 0

φ ∞ (x + x ′ ) -2φ ∞ (x) + φ ∞ (x -x ′ ) ≤ C 0 |x ′ | 2 t , for all (x, x ′ ) ∈ R 2(N -1)
and we can check that we have C 0 = 1. Letting t go to infinity shows that φ ∞ is concave in R N -1 .

Step 4: Tangent cone Since φ ∞ is Lipschitz continuous, it is differentiable almost everywhere by Rademacher's theorem. Let D ⊂ R N -1 be the set of differentiability of φ ∞ and fix x 0 ∈ D. Since φ ∞ is concave, for any x ∈ R N -1 , we have

φ ∞ (x) ≤ φ ∞ (x 0 ) + Dφ ∞ (x 0 ) • (x -x 0 )
Passing to the infimum on D, we get for any

x ∈ R N -1 , φ ∞ (x) ≤ ψ(x) := inf x 0 ∈D φ ∞ (x 0 ) + Dφ ∞ (x 0 ) • (x -x 0 )
Thus, ψ and φ ∞ are (cot α)-Lipschitz functions that coincide on D which is a dense set on R N -1 . Therefore, they are in fact equal on R N -1 . Using equation [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF], we finally have

φ ∞ (x) = inf x 0 ∈D -(cot α) ν(x 0 ) • x + g(x 0 )
where for any

x 0 ∈ D, ν(x 0 ) = -Dφ ∞ (x 0 )/ cot α ∈ S N -2 and g(x 0 ) = φ ∞ (x 0 ) -x 0 • Dφ ∞ (x 0 ) ∈ R. Defining γ as (13) γ : S N -2 → (-∞, +∞] ν → inf x 0 ∈A g(x 0 ) if A := {x 0 ∈ D | ν(x 0 ) = ν} = ∅ +∞ otherwise
we get the desired characterisation [START_REF] Hamel | Asymptotic properties and classification of bistable fronts with Lipschitz level sets[END_REF]. Since φ ∞ is continuous, we also deduce from [START_REF] Hamel | Asymptotic properties and classification of bistable fronts with Lipschitz level sets[END_REF] that γ is lower semi-continuous.

Step 5: The 1-homogeneous case We assume that φ ∞ is a 1-homogeneous continuous viscosity solution to [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF]. Then for any x 0 ∈ R N -1 , there exists p ∈ R N -1 with |p| = cot α such that by ( 11)

∀x ∈ R N -1 , φ ∞ (x) ≤ φ ∞ (x 0 ) + p • (x -x 0 )
On the one hand, considering x = 0, we get

p • x 0 ≤ φ ∞ (x 0 )
because φ ∞ is 1-homogeneous. On the other hand considering λx instead of x and taking the limit λ → +∞, we get

ψ(x) := p • x ≥ φ ∞ (x) with equality at x = x 0 .
Therefore if we call L φ∞ the set of linear functions

ψ satisfying ψ ≥ φ ∞ such that |∇ψ| = cot α, we have φ ∞ = inf ψ∈L φ∞ ψ because this is true at any point x 0 ∈ R N -1 .
Step 6: Conclusion Conversely, if a function φ ∞ is given by ( 11), then it is straightforward to check that φ ∞ is a viscosity solution to [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF].

Remark 2.2 In dimension N = 2, the previous proposition simply reads:

If N = 2 and α ∈ (0, π 2 ], φ ∞ is a viscosity solution to (10) if and only if φ ∞ is affine or if there exists (x 0 , y 0 ) ∈ R 2 such that (14) φ ∞ (x) = -(cot α) |x -x 0 | + y 0 , x ∈ R Moreover, φ ∞ is 1-homogeneous if and only if y 0 = 0.
The proof of this proposition can also be done directly from definitions of viscosity solutions and we omit the details. Notice however the link with [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF]: two-dimensional reaction diffusion waves are either planar fronts or the unique (up to translations) conical front whose level sets are asymptotics to the graph of φ ∞ just described.

The 1-homogeneous case

As stressed is theorem 1.1, we only build solutions to the forced mean curvature motion equation (3) whose asymptotics is prescribed by a 1-homogeneous solution to the eikonal equation [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF]. Therefore, it is worth emphasising this particular case.

Notice however that there exist viscosity solutions to the eikonal equation ( 10) defined in R N -1 that are not homogeneous of order 1. For instance, consider solutions given by ( 14) with x ∈ R N -1 and y 0 = 0. We can also consider any translation of a 1-homogeneous solution. Another example is for instance given in dimension N = 3 by a function φ ∞ = inf i=1...4 φ i where (φ i ) i∈{1...4} are four planar solutions defined for

x = (x 1 , x 2 ) ∈ R 2 by φ 1 (x) = -(cot α) x 1 + 2 φ 2 (x) = (cot α) x 1 + 2 φ 3 (x) = -(cot α) x 2 φ 4 (x) = (cot α) x 2
It is straightforward to check that φ ∞ satisfies |Dφ ∞ | = cot α in the viscosity sense and that it is not homogeneous of order 1 since there exists

λ > 0 such that φ ∞ (λ, 0) = λφ ∞ (1, 0).
In any case, a solution φ ∞ to the eikonal equation ( 10) is concave (see the proof of proposition 2.1, step 3). Therefore the function g :

λ ∈ R + * → g(λ) = φ ∞ (λx)/(λ|x|) ∈ R is decreasing in λ > 0. Since φ ∞ is (cot α)-Lipschitz, g is bounded from below and for any x ∈ S N -2 , the limit lim λ→+∞ φ ∞ (λx)
|λ| exists and φ ∞ is asymptotically homogeneous. Thus we have a fairly general understanding of what is going on by restricting ourselves to homogeneous solutions to equation [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF].

Proposition 2.3 (A countable characterisation of homogeneous solutions)

Let φ ∞ ∈ C(R N -1 ). Then φ ∞ is a 1-homogeneous viscosity solution to the eikonal equation [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF] if and only if there exists a sequence

(ν i ) i∈N of S N -2 such that (15) φ ∞ (x) = inf i∈N -(cot α) ν i • x Proof of Proposition 2.3. Let φ ∞ ∈ C(R N -1
) be a 1-homogeneous viscosity solution to [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF]. According to proposition 2.1, there exists a lower semi continuous function γ defined from

S N -2 to {0, +∞} such that φ ∞ (x) = inf ν∈S N-2 φ ν,γ(ν) (x) , x ∈ R N -1 Then K = {ν ∈ S N -2 | γ(ν) = 0} is a compact set of S N -2 .
We claim (see lemma 2.4 and corollary 2.5 below) that there exists a sequence (ν i ) i∈N of S N -2 such that

K = i∈N {ν i }
Thus, φ ∞ (x) can be described as the infimum over ν ∈ K of the linear functions

-(cot α) x• ν. Since ∪ i∈N {ν i } is dense in K, φ ∞ (x)
can also be written as the infimum over i ∈ N of the linear functions -(cot α) x • ν i . This ends the proof of proposition 2.3 since the converse implication is straightforward.

Lemma 2.4 (Decomposition of a compact set of S N -2 in cubes) For any compact set K of S N -2 , there exists a countable family

(Q i ) i∈N of closed cubes of R N -1 such that (16)          ∀n ∈ N , K ⊂ i≥n Q i ∀i ∈ N , Q i ∩ K = ∅ lim sup i→+∞ diam(Q i ) = 0 Proof of Lemma 2.4.
We built this decomposition in cubes by induction. Let

C 0 = [-1, 1] N -1 be the first cube of width 2. Thus K ⊂ C 0 . Since C 0 ∩ K is not empty, we divide C 0 in 2 N -1 smaller cubes of width 2 0 = 1. We call C 1,i for i = 1 . . . n 1 those whose intersection with K is not empty. Then, 1 ≤ n 1 ≤ 2 N -1 and K ⊂ n 1 i=1 C 1,i
In the same way, for i = 1, ..., n 1 , we divide each cube C 1,i in 2 N -1 smaller cubes of width 2 -1 and keep only those whose intersection with K is not empty. We call them

C 2,k for k = 1 . . . n 2 and 1 ≤ n 2 ≤ 2 N -1 n 1 . Then, one can easily verify that K ⊂ ∪ k=1...n 2 C 2,k . Assume the cubes C j,i are built for j ∈ N, i = 1 . . . n j and 1 ≤ n j ≤ 2 j(N -1) such that    K ⊂ n j i=1 C j,i ∀i = 1 . . . n j , C j,i ∩ K = ∅ diam(C j,i ) = 2 -j+1
Then we construct the cubes C j+1,i as follows. We divide each cube C j,i into 2 N -1 smaller cubes of width 2 -j and keep only those whose intersection with K is not empty. We call them C j+1,i for i = 1 . . . n j+1 and 1 1) . By construction, it is easy to verify that

≤ n j+1 ≤ 2 N -1 n j ≤ 2 (j+1)(N -
   K ⊂ n j+1 i=1 C j+1,i ∀i = 1 . . . n j+1 , C j+1,i ∩ K = ∅ diam(C j+1,i ) = 2 -j
The induction is then proved. We thus construct a countable family of cubes that we recall (Q j ) j∈N for convenience with the desired assumptions [START_REF] Taniguchi | The uniqueness and the asymptotic stability of pyramidal travelling fronts in the Allen-Cahn equations[END_REF]. This ends the proof of lemma 2.4.

Corollary 2.5 (Representation of a compact set of S N -2 ) For any compact set K of S N -2 , there exists a sequence (ν j ) j∈N of S N -2 such that K = j∈N {ν j }
Proof of Corollary 2.5. For K a compact set of S N -2 , we define (Q j ) j∈N a family of cubes as proposed in lemma 2.4. For any j ∈ N, we choose ν j ∈ K ∩ Q j . Then, it is straightforward to check that ∪ j∈N {ν j } ⊂ K. Regarding the converse inclusion, we fix x 0 ∈ K and ε > 0. By [START_REF] Taniguchi | The uniqueness and the asymptotic stability of pyramidal travelling fronts in the Allen-Cahn equations[END_REF], there exists n ε ∈ N such that the width of cube Q i is smaller than

ε provided i ≥ n ε . Since K ⊂ ∪ i≥nε Q i , there exists i ε ≥ n ε such that x 0 ∈ Q iε and |x 0 -ν iε | ≤ ε √ N -1
This shows the density of ∪ j∈N {ν j } in K and ends the proof of corollary 2.5.

Perron's method and comparison principle

In this section, we are concerned with the forced mean curvature motion equation

(17) -div Dφ 1 + |Dφ| 2 + c 0 - c 1 + |Dφ| 2 = 0 , x ∈ R N -1
with the condition at infinity

(18) φ(x) = φ ∞ (x) + o(|x|) , x ∈ R N -1
where φ ∞ is a homogeneous viscosity solution to |Dφ ∞ | = cot α found in section 2 with α = arcsin(c 0 /c) ∈ (0, π 2 ]. We choose to solve (17) using Perron's method with sub and super-solutions (see [START_REF] Crandall | User's guide to viscosity solutions of second order PDEs[END_REF] or [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]). Let us first recall the existence process and clarify the regularity of the solution in the following

Proposition 3.1 (Existence of a solution to (17) in dimension N) Let N ∈ N \ {0, 1}, (c 0 , c) ∈ R 2 such that c ≥ c 0 > 0.
Assume that φ * is a viscosity sub-solution and φ * a viscosity super-solution to (17) such that φ * ≤ φ * on R N -1 . Then, i) there exists a function φ ∈ [φ * , φ * ] viscosity solution to (17). ii) Moreover, if φ * is concave, and satisfies the following technical condition:

(19) there exists p ∈ R N -1 such that lim sup |x|→+∞ φ * (x) -p • x |x| < 0,
then φ can be chosen concave and smooth.

Proof of Proposition 3.1.

We build the solution φ using Perron's method directly in the framework of viscosity solutions to (17), that is to say φ is chosen as the maximal sub-solution to (17) (see the user's guide to viscosity solutions [START_REF] Crandall | User's guide to viscosity solutions of second order PDEs[END_REF]).

Step 1: Concavity We apply a result due to Imbert (see [START_REF] Imbert | Convexity of solutions and C 1,1 estimates for fully nonlinear elliptic equations[END_REF]) that we first recall. Denote F the following Hamiltonian

F (p, M) = - trM 1 + |p| 2 + tr(M • (p ⊗ p)) (1 + |p| 2 ) 3/2 +c 0 - c 1 + |p| 2 , (p, M) ∈ R N -1 ×R (N -1)×(N -1) sym where R (N -1)×(N -1) sym
is the set of (N -1)-square symmetric matrices.

Proposition 3.2 (Imbert's proposition 5 in [START_REF] Imbert | Convexity of solutions and C 1,1 estimates for fully nonlinear elliptic equations[END_REF]) Let u be a lower semi-continuous and epi-pointed function. If u is a supersolution to

F (Du(x), D 2 u(x)) = 0 , x ∈ R N -1
then, so is its convex envelope.

In our context, -φ * is epi-pointed because of the technical condition (19). Thus, the maximal sub-solution φ to (17) is concave (otherwise Imbert's result stated in proposition 3.2 contradicts the maximal property of φ). Then, φ is a concave viscosity solution to (17).

Step 2: Regularity Once concavity is at hand, a Lipschitz bound is automatically available from the equation (17

) itself: |Dφ(x)| ≤ cot α , x ∈ R N -1
where α ∈ (0, π 2 ] is such that c 0 = c sin α. Then F becomes uniformly elliptic, thus allowing for C 1,1 estimates (see Theorem 4 in [START_REF] Imbert | Convexity of solutions and C 1,1 estimates for fully nonlinear elliptic equations[END_REF]). A bootstrap argument then shows that the solution is C ∞ . This concludes the proof of proposition 3.1.

Remark 3.3 Notice that the condition (19) is hidden in the statement of Proposition 9 in [START_REF] Imbert | Convexity of solutions and C 1,1 estimates for fully nonlinear elliptic equations[END_REF]. Thus, the proof of Proposition 5 in [START_REF] Imbert | Convexity of solutions and C 1,1 estimates for fully nonlinear elliptic equations[END_REF] uses Proposition 9.

It now remains to find sub and super-solutions to (17).

Sub-solution

In this section we build smooth sub-solutions to the forced mean curvature equation (17) as global solutions to a viscous eikonal equation and we do believe that they are really close to the desired solutions.

Sub-solutions as solutions to a viscous eikonal equation

We have the following

(20) -∆φ * = c 0 sin α 2 cot 2 α -|Dφ * | 2 , x ∈ R N -1 such that (21) |Dφ * (x)| ≤ cot α , x ∈ R N -1
Then φ * is a smooth sub-solution to equation (17).

Proof of Lemma 4.1. Let φ * be any concave function verifying (20) and (21). Since φ * is smooth and concave, we have

N[φ * ] := -div Dφ * 1 + |Dφ * | 2 + c 0 - c 1 + |Dφ * | 2 = - ∆φ * 1 + |Dφ * | 2 + D 2 φ * (Dφ * , Dφ * ) (1 + |Dφ * | 2 ) 3 2 + c 0 - c 1 + |Dφ * | 2 ≤ 1 1 + |Dφ| 2 -∆φ * + c 0 1 + |Dφ * | 2 -c
From (21) and c 0 = c sin α, we deduce that

cot 2 α -|Dφ * | 2 = c c 0 2 - 1 + |Dφ * | 2 2 ≤ 2c c 0 c c 0 -1 + |Dφ * | 2
Using equation (20) satisfied by φ * , we get

N[φ * ] ≤ 1 1 + |Dφ * | 2 c c 0 -1 + |Dφ * | 2 2c c 0 c 2 0 2c -c 0 = 0.
Thus, φ * is a sub-solution to (17).

As it is well-known, equation ( 20) is readily transformed into a linear one by the Hopf-Cole transform

φ * (x) = exp - c 0 sin α 2 φ * 2x c 0 cos α , x ∈ R N -1 , α = π 2
where φ * is a positive solution to

(22) -∆ φ * (x) + φ * (x) = 0 , x ∈ R N -1
From [START_REF] Caffarelli | Representation formulas for solutions to ∆u-u = 0 in R n , Studies in Partial Differential Equations[END_REF], a positive solution φ * to (22) has the form

φ * (x) = S N-2 e ν•x dµ(ν) , x ∈ R N -1 ,
where µ is a non negative measure on S N -2 with finite mass. Now, for any non negative measure µ on the sphere S N -2 , let us define

(23) φ * (x) = - 2 c 0 sin α ln S N-2 e c 0 cos α 2 x•ν dµ(ν) , x ∈ R N -1 , α ∈ 0, π 2
By construction, φ * is a smooth solution to (20). Let us now prove that φ * is a sub-solution to equation ( 17), with all the requirements.

Lemma 4.2 (Inequalities for the derivatives of φ * ) Let µ be a non negative measure on S N -2 with finite mass, α ∈ (0, π 2 ], c 0 > 0 and c = c 0 / sin α. Define φ * as in (23). Then φ * is a smooth concave solution to (20) and its gradient is uniformly bounded, that is to say, for any

(x, ξ) ∈ R N -1 × R N -1 , (24) |Dφ * (x)| ≤ cot α , D 2 φ * (x)(ξ, ξ) ≤ 0 Proof of Lemma 4.2.
Let µ and φ * be so defined. We have

Dφ * (x) = -(cot α) F ν (x) F 1 (x) , x ∈ R N -1
where for any continuous (scalar or vector) function f defined on S N -2

F f (x) = S N-2 e c 0 cos α 2 x•ν f (ν) dµ(ν)
Remark that if we define for some fixed

x ∈ R N -1 , S N-2 f (ν) dµ x (ν) = F f (x) F 1 (x)
, then µ x is a probability measure on S N -2 . We can then apply Jensen's inequality to the convex function y → |y| 2 . This gives (25)

F f (x) F 1 (x) 2 = S N-2 f (ν) dµ x 2 ≤ S N-2 |f (ν)| 2 dµ x = F |f | 2 (x) F 1 (x)
for any continuous function f defined on S N -2 . Applying this inequality to f (ν) = ν, we get the desired bound on the gradient of φ * :

|Dφ * (x)| ≤ cot α.
Regarding the concavity property of φ * , we use the same type of arguments. Indeed, for any ξ ∈ R N -1 and x ∈ R N -1 , we have

D 2 φ * (x)(ξ, ξ) = - c 0 cos 2 α 2 sin α F f 2 (x) F 1 (x) - F f (x) F 1 (x)
2
where f is the continuous function defined on S N -2 by f (ν) = ν • ξ. Applying again Jensen's inequality (25), we conclude that D 2 φ * (x)(ξ, ξ) ≤ 0 for any ξ ∈ R N -1 and x ∈ R N -1 which shows that φ * is concave. Finally, we proved the following proposition:

Proposition 4.3 (Existence of a sub-solution to (17)) Fix α ∈ (0, π 2
], c 0 > 0 and c = c 0 / sin α. Let µ be a non negative measure on S N -2 with finite mass. Define φ * as in (23). Then, φ * is a smooth concave sub-solution to (17).

Remark 4.4

The way we choose the measure µ is decisive in the asymptotic behaviour of the sub-solution φ * built as in proposition 4.3. Indeed, if we want the subsolution (and hence the solution) to the mean curvature equation (17) to follow asymptotically some given solution φ ∞ to the eikonal equation (10), we will have to choose the measure µ carefully. In that procedure, information collected in section 2 will help.

Of course, it will be also very interesting to assess whether each sub-solution built with a general probability measure gives rise to a solution to the mean curvature equation (17).

Super-solution

A natural super-solution to the forced mean curvature equation ( 17) is a viscosity solution φ ∞ to the eikonal equation [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF]. Indeed, φ ∞ satisfies (in the distributional and viscosity sense)

-div Dφ ∞ 1 + |Dφ ∞ | 2 + c 0 - c 1 + |Dφ ∞ | 2 ≥ 0 , x ∈ R N -1
However, this super-solution does not satisfy the right comparison with the previous subsolution φ * . For instance, if N = 3, φ ∞ is the radially symmetric viscosity solution to the eikonal equation [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF] and φ * the sub-solution associated with the Lebesgue measure µ = dθ on [0, 2π] as in ( 23), then we can compute the asymptotic behaviour of both functions φ ∞ and φ * for x ∈ R 2 with |x| large enough using Laplace's method (see appendix 8). We then observe that the sub-solution φ * is above the super-solution φ ∞ in this area. This contradicts the crucial assumption φ * ≤ φ * on R 2 in the Perron's method (see proposition 3.1).

Super-solutions as infimum of hyperplanes

Since we do believe that the sub-solution is close to the viscosity solution to the forced mean curvature equation ( 17) at infinity, we prefer to change the super-solution. In the general case of dimension N, we use the countable characterisation of the solution φ ∞ to the eikonal equation ( 10) that we want to approach (see proposition 2.3).

Proposition 5.1 (Existence of a super-solution to (17)) Fix α ∈ (0, π 2 ], c 0 > 0 and c = c 0 / sin α. Choose φ ∞ a 1-homogeneous solution to the eikonal equation [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF]. Define (ν i ) i∈N the sequence of S N -2 given by its countable characterisation in proposition 2.3.

For any sequence (λ i ) i∈N such that λ i > 0 and i∈N λ i < +∞, we set

φ i (x) = -(cot α) x • ν i - 2 c 0 sin α ln λ i , i ∈ N , x ∈ R N -1 and φ * (x) = inf i∈N φ i (x) , x ∈ R N -1
Then, φ * is a concave continuous super-solution to (17).

Proof of Proposition 5.1.

Since φ i are exact solutions to the forced mean curvature equation (17), it is clear that φ * is a super-solution to that equation. As the infimum of affine functions, it is concave and continuous.

Remark 5.2 This construction is very easy. However, it is not clear whether the technical condition (19) is satisfied or not. It is even clear that when the set of {ν i } i∈N is finite of cardinal less or equal to N -1, this condition is NOT verified. We will see later (see Step 4 of the proof of theorem 1.1) how to modify the sub-and super-solutions in order to satisfy condition (19) and then pass to the limit to recover the general case.

Remark 5.3 In the case when the set {ν i } i∈N is infinite, the convergence of λ i forces (λ i ) i∈N to go to zero and the sequence (-ln λ i ) i∈N grows as i goes to infinity.

General existence results

Now equipped with sub and super-solutions as well as a Perron's method, we are able to prove existence results. The general case in dimension N ≥ 2 is the easiest one since the asymptotics is less precise. Let us explain our ideas in details depending on the degree of precision we want to obtain in our construction.

Let N ∈ N \ {0, 1}, α ∈ (0, π 2 ], c 0 > 0 and c = c 0 / sin α. It is worth noticing that some of our constructions do not work for α = π/2. However, this case is obvious and leads to planar fronts. Therefore, we restrict ourselves to α ∈ (0, π 2 ).

6.1 Proof of Theorem 1.1

Step 1: Sub and super-solutions Choose φ ∞ a 1-homogeneous continuous viscosity solution to the eikonal equation ( 10) in R N -1 . By proposition 2.3, there exists a sequence (ν i ) i∈N of S N -2 such that

φ ∞ (x) = inf i∈N -(cot α)ν i • x
Let µ be the probability measure on S N -2 be defined as

µ = i∈N λ i δ ν i
where (λ i ) i∈N are chosen so that λ i > 0 and +∞ i=0 λ i = 1. Build the sub-solution φ * as in (23) with the above measure µ. Then by proposition 4.3, φ * is a smooth concave sub-solution to the mean curvature motion equation (17). Build a concave continuous super-solution φ * by proposition 5.1 as the infimum of hyperplanes where the (λ i ) i∈N and (ν i ) i∈N are defined by the choice of µ. For any

x ∈ R N -1 , φ * (x) = - 2 c 0 sin α ln +∞ i=0 λ i e c 0 cos α 2 ν i •x ≤ - 2 c 0 sin α ln λ i e c 0 cos α 2 ν i •x
Since the last inequality holds for any i ∈ N, we have

φ * (x) ≤ inf i∈N -(cot α)ν i • x - 2 c 0 sin α ln λ i = inf i∈N φ i (x) = φ * (x)
and the super-solution φ * is above the sub-solution φ * .

Step 2: Asymptotics of sub and super-solutions Let us now precise their asymptotics: we claim that as |x| goes to infinity

(26) φ * (x) = φ ∞ (x) + o(|x|) and φ * (x) = φ ∞ (x) + o(|x|)
To prove such a claim, the idea is to compare φ ∞ (x) with the limits as ε goes to zero of εφ * (x/ε) and εφ * (x/ε). In particular, we will prove the sequence of three inequalities: for any

x ∈ R N -1 (27) φ ∞ (x) ≤ lim ε→0 εφ * (x/ε) ≤ lim ε→0 εφ * (x/ε) ≤ φ ∞ (x)
which proves the desired claim (26). The first step of the present proof leads easily to the second inequality in (27) since φ * ≤ φ * on R N -1 . As far as the first inequality is concerned, we have for any i ∈ N and

x ∈ R N -1 , |x • ν i | = |x| cos(θ x -θ i ) ≤ |x| cos δ x
where δ x is the angular distance between x/|x| and K := ∪ i∈N {ν i } . Thus,

φ * (x) ≥ - 2 c 0 sin α ln e c 0 sin α 2 |x| cos δx µ(S N -2 ) = -(cot α)|x| cos δ x = inf ν∈K -(cot α) x • ν = φ ∞ (x)
where µ(S N -2 ) = i λ i = 1. Thus φ ∞ ≤ φ * on R N -1 and the homogeneity of φ ∞ gives the first inequality of (27).

Regarding the last inequality in (27), we know that for any

x ∈ R N -1 , φ * (x) ≤ φ i (x) = -(cot α) x • ν i - 2 c 0 sin α ln λ i Since lim ε→0 εφ i (x/ε) = -(cot α) x • ν i , it is clear that lim ε→0 εφ * x ε ≤ inf i∈N -(cot α) x • ν i = φ ∞ (x)
This ends the proof of the three inequalities (27) and hence of (26).

Step 3: Existence of a solution By proposition 3.1, there exists a function φ ∈ [φ * , φ * ] viscosity solution to (17) and by the previous step, φ verifies the right asymptotics

φ(x) = φ ∞ (x) + o(|x|)
However, in the statement of theorem 1.1, we claim that there exists a smooth concave solution to (17) and the above construction does not provide such information. By proposition 3.1, the regularity and concavity of the solution are at hand if the super-solution φ * satisfies the technical assumption (19). If it does not, we will first modify the sub and the super-solutions in order to satisfy (19), then get a concave solution, and in a last step pass to the limit to find a solution (still concave) between φ * and φ * .

Step 4: Regularity and concavity Let us consider for any ε > 0

(28) φ ε * (x) = - 2 c 0 sin α ln S N-2 e c 0 cos α 2 x•ν dµ ε (ν) , x ∈ R N -1
with µ ε = µ + εµ 1 where

µ 1 = ± N -1 j=1 δ ±e j
denoting (e i ) i∈{1,...,N -1} as the canonical orthonormal basis of R N -1 . In the same way, we define

φ ε * (x) = inf i∈N, j=1...N -1, ± -(cot α) x • ν i - 2 c 0 sin α ln λ i , -(cot α) x • (±e j ) - 2 c 0 sin α ln ε
Then, φ ε * is a sub-solution, φ ε * is a super-solution and φ ε * ≤ φ ε * . It satisfies (19) for any ε > 0 and for p = 0. By proposition 3.1, there exists a concave smooth solution φ ε satisfying equation ( 17), with φ ε being (cot α) -Lipschitz such that

φ ε * (x) ≤ φ ε (x) ≤ φ ε * (x) , x ∈ R N -1
Finally, we take the limit as ε goes to zero. The sub-solutions φ ε * go to φ * . The supersolutions φ ε * converge to φ * . This follows from the expression of super-solutions as an infimum of hyperplanes, those associated to the ε weights going to +∞. Moreover by Ascoli's theorem, (φ ε ) ε>0 converges (up to a subsequence) to some concave and (cot α) -Lipschitz function φ 0 solution to (17) and satisfying

φ * ≤ φ 0 ≤ φ *
Again a bootstrap argument shows that φ 0 is smooth. Therefore, φ 0 is the intended solution to the mean curvature equation (17).

Proof of Theorem 1.2

Step 1: Existence of a solution Choose φ * the viscosity solution to the eikonal equation [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF] given by (29)

φ * (x) = inf ν∈A (-(cot α) x • ν + γ ν ) , x ∈ R N -1
where A = {ν 1 , . . . , ν k } is a finite subset of the sphere S N -2 , k ∈ N * and γ ν are any given real numbers. We build a sub-solution φ * as in proposition 4.3

(30) φ * (x) = - 2 c 0 sin α ln k i=1 λ i e c 0 cos α 2 x•ν i , x ∈ R N -1
where λ i is determined by the relation γ ν i = -2 c 0 sin α ln λ i for i = 1 . . . k. Let us notice that in the particular case when A is finite, the super-solution built in proposition 5.1 coincides with the solution φ * to the eikonal equation. As in section 6.1, φ * ≤ φ * and the assumptions of proposition 3.1 i) are satisfied. Thus, there exists a function φ ∈ [φ * , φ * ] viscosity solution to (17). Dealing as in section 6.1 step 4, we can even find a smooth concave solution still denoted φ ∈ [φ * , φ * ]. It now remains to study φ * -φ * to get a precise asymptotics of the solution φ.

Step 2: Asymptotics (first line of ( 8)) Setting

φ i (x) = -(cot α) x • ν i - 2 c 0 sin α ln λ i we have φ * (x) = - 2 c 0 sin α ln k i=1 e -c 0 sin α 2 φ i (x) ≥ - 2 c 0 sin α ln   ke - c 0 sin α 2 min i=1,...,k φ i (x)   = - 2 c 0 sin α ln ke -c 0 sin α 2 φ * (x) = φ * (x) - 2 ln k c 0 sin α
This implies in particular that

(31) - 2 ln k c 0 sin α ≤ φ * -φ * ≤ 0
which shows the first line of (8).

Step 3: Asymptotics (second line of ( 8))

We now notice that the set E ∞ of edges (where φ ∞ is not C 1 ) is characterized by

E ∞ = x ∈ R N -1 , max ν∈A x • ν = x • ν i 0 = x • ν i 1 , with ν i 0 = ν i 1 and (ν i 0 , ν i 1 ) ∈ A 2
For each index i 0 ∈ {1, ..., k}, let us denote the convex set

K i 0 = x ∈ R N -1 , x • ν i 0 = max ν∈A x • ν Then ∂K i 0 ⊂ j =i 0 (ν i 0 -ν j ) ⊥ For x ∈ Int(K i 0 ), let x i 1 ∈ ∂K i 0 ⊂ E ∞ such that dist(x, E ∞ ) = |x -x i 1 | with x i 1 ∈ (ν i 0 -ν i 1 ) ⊥ .
For j = i 0 , we define the orthogonal projection of x on (ν i 0ν j ) ⊥ as

x j = Proj |(ν i 0 -ν j ) ⊥ (x)
In particular |x -

x j | ≥ |x -x i 1 |. Moreover x • ν j = (x -x j ) • ν j + x j • ν j = (x -x j ) • ν j + x j • ν i 0 = (x -x j ) • (ν j -ν i 0 ) + x • ν i 0 = x • ν i 0 -|ν j -ν i 0 ||x -x j | ≤ x • ν i 0 -δ dist(x, E ∞ ) with δ = min ν =ν ′ , ν,ν ′ ∈A |ν -ν ′ | > 0 Therefore φ * (x) = - 2 c 0 sin α ln k i=1 λ i e c 0 cos α 2 x•ν i ≥ - 2 c 0 sin α ln k i=1 λ i e c 0 cos α 2 (x•νi 0 -δ dist(x,E∞))
and then for x ∈ K i 0 , we have

φ * (x) ≥ φ * (x) ≥ φ * (x) - 2 c 0 sin α ln 1 + i =i 0 λ i λ i 0 e -c 0 cos α 2 δ dist(x,E∞)
This shows that lim

l→+∞ sup dist(x,E∞)≥l |φ * (x) -φ * (x)| = 0
which implies the second line of (8).

Step 4: Uniqueness To end the proof of theorem 1.2, it only remains to prove uniqueness of the above smooth solution φ to the mean curvature equation ( 3) with the prescribed asymptotics given by φ * . Let φ and φ be two solutions to (3) with the asymptotics [START_REF] Fife | Dynamics of internal layers and diffusive interfaces[END_REF]. Let

ε := inf ε ′ > 0 | ∀x ∈ R N -1 , φ(x) + ε ′ ≥ φ(x) then for any x ∈ R N -1 , φ(x) + ε ≥ φ(x)
and there exists a sequence of points (x n ) n such that

φ(x n ) + ε -φ(x n ) → 0 as n goes to infinity Let us define for any x ∈ R N -1    φ n (x) = φ(x + x n ) -φ(x n ), φ n (x) = φ(x + x n ) -φ(x n )
Then, up to the extraction of a subsequence, we have as n goes to infinity

φ n → φ ∞ and φ n → φ ∞
with a uniform convergence on any compact sets of R N -1 . Moreover φ ∞ and φ ∞ solve equation ( 3) and satisfy

φ ∞ + ε ≥ φ ∞ with equality at x = 0
From the strong maximum principle, we deduce that for any

x ∈ R N -1 , (32) φ ∞ (x) + ε = φ ∞ (x)
Let us now assume that ε > 0. Because we have

E ∞ ⊂ ν =ν ′ , (ν,ν ′ )∈A 2 (ν -ν ′ ) ⊥ =: Ê∞
we deduce that there exists C > 0 such that for any R ≥ 1 and any x ∈ R N -1 , we have

(33) sup y∈B R (x) dist(y, E ∞ ) ≥ sup y∈B R (x) dist(y, Ê∞ ) = R sup y∈B 1 (x/R) dist(y, Ê∞ ) ≥ CR with C = inf z∈R N-1 sup y∈B 1 (z) dist(y, Ê∞ )
We easily check by contradiction that C > 0. Therefore by ( 8), let us choose R large enough such that

sup dist(y,E∞)≥CR |φ(y) -φ * (y)| ≤ ε 4 for φ = φ, φ
Then using (33), we get for some

y n ∈ B R (x n ) with dist(y n , E ∞ ) ≥ CR, inf y∈B R (xn) |φ(y) -φ(y)| ≤ |φ(y n ) -φ(y n )| ≤ |φ(y n ) -φ * (y n )| + |φ * (y n ) -φ(y n )| ≤ ε 2 This implies that inf y∈B R (0) |φ ∞ (y) -φ ∞ (y)| ≤ ε 2
which is in contradiction with (32). Therefore ε = 0 and we get φ ≥ φ. By symmetry, we also get φ ≥ φ, which implies φ = φ and shows the uniqueness of the solution. This ends the proof of the theorem 1.2.

Proof of further results in dimension N = 3

In this section, the space dimension is N = 3 and we denote any x ∈ R 2 with its polar coordinates (r, θ x ) ∈ R + × [0, 2π) such that x = r(cos θ x , sin θ x ).

Classification in dimension N = 3 of solutions to the eikonal equation with a finite number of singularities

This subsection gives alternative statement and proof of proposition 2.3 in dimension N = 3, in the special case of a finite number of singularities (i.e. gradient jumps).

Proposition 7.1 (Classification with a finite number of singularities, N = 3) Let α ∈ (0, π 2 ], c 0 > 0 and c = c 0 / sin α. Choose φ ∞ a 1-homogeneous viscosity solution to the eikonal equation [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF] in dimension N = 3 with a finite number of singularities on S 1 . Then the i. of Theorem 1.3 holds.

Proof of Proposition 7.1. From Proposition 2.1, we know that there exists a (non empty) compact set

K = γ -1 ({0}) ⊂ S 1 , such that (34) φ ∞ (x) = inf ν∈K (-(cot α) ν • x) Thus, for any θ ∈ [0, 2π), ψ ∞ (θ) = φ ∞ (cos θ, sin θ) defines a continuous function with values in [-cot α, cot α]. Firstly ψ ∞|K = -cot α.
Moreover for any maximal interval (a, b) contained in S 1 \K, we necessarily have

ψ ∞ (θ) =    -(cot α) cos(θ -a) if θ ∈ a, a+b 2
,

-(cot α) cos(θ -b) if θ ∈ a+b 2 , b .
Therefore φ ∞ has a singularity (gradient jump) at θ = a+b 2 . If ψ ∞ only has a finite number of singularities, then we get the characterization of ψ ∞ given in the i. of Theorem 1.3. This ends the proof of proposition 7.1.

Remark 7.2 Notice that without assuming that φ ∞ has a finite number of singularities on S 1 , the set K could be a Cantor set in (34).

Remark 7.3 Notice that the particular function φ ∞ (x) = -(cot α)|x| is the analogue (at the level of the eikonal equation) of the level sets of cylindrically symmetric solutions to reaction diffusion equation, constructed in [START_REF] Hamel | Asymptotic properties and classification of bistable fronts with Lipschitz level sets[END_REF] by Hamel, Monneau and Roquejoffre. Similarly, the particular case where the graph of φ ∞ is a pyramid is also the analogue of solutions contructed by Taniguchi in [START_REF] Taniguchi | The uniqueness and the asymptotic stability of pyramidal travelling fronts in the Allen-Cahn equations[END_REF].

Explicit construction of super-solutions in dimension N = 3

In the particular case N = 3, we construct super-solutions by hand and try to be more precise than in section 5, above all when ψ ∞ is constant and equal to -(cot α) on some interval I. In that case, we construct our super-solution by hand. We explain our ideas on different elementary pieces that we bring together in the proof of theorem 1.3 to build a global super-solution φ * . Those different elementary pieces are: a cone, an edge or an arc .

The cone case

Lemma 7.4 (Radially symmetric solutions) Let φ ∞ be the viscosity solution to eikonal equation [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF] whose graph is the straight cone i.e. φ ∞ (x) = -(cot α)|x| for x ∈ R 2 . Then, there exists a unique radially symmetric solution φ c (unique up to an additive constant) to the forced mean curvature equation (17), satisfying φ ′ c (0) = 0 and φ c (x) = φ ∞ (x) + o(|x|) Moreover φ c is concave and |Dφ c | ≤ cot α. In the case α = π/2, φ c is zero (up to an additive constant). Otherwise, as |x| goes to infinity, its asymptotics is more precisely given (up to a constant C ∈ R) by

(35) φ c (x) = -(cot α)|x| + 1 c 0 sin α ln |x| + C + 2 -3 sin 2 α c 2 0 sin(2α)|x| + O 1 |x| 2 , α = π 2
Moreover, let φ * be the sub-solution defined by (23) with µ = dθ 2π and N = 3. Fix φ c such that C = C 0 := ln(πc 0 cos α) c 0 sin α , then for any x ∈ R 2 , φ c (x) ≥ φ * (x) and as |x| goes to infinity

(36) φ c (x) = φ * (x) + O 1 |x| .
Proof of Lemma 7.4. This result is proved using quite classical methods. The proof is sketched for the reader's convenience. With a slight misuse of notation, we denote in the case of radially symmetric solutions φ c (x) by φ c (|x|) = φ c (r) with r = |x| ≥ 0. Then, equation ( 17) reads

- φ ′ c r - φ c " 1 + φ ′ 2 c + c 0 1 + φ ′ 2 c -c = 0 , r > 0
Thus, φ c satisfies an ODE involving only its first two derivatives and it can only be defined up to constants. Setting v = φ ′ c , we get (37)

v ′ = (1 + v 2 ) c 0 √ 1 + v 2 -c - v r := (1 + v 2 )g(v, r) , r > 0
The proof of lemma 7.4 now reduces to the study of this ODE (existence, uniqueness and asymptotics).

Step 1: Existence Since for any r > 0, g(0, r) ≤ 0 and g(v 0 (r), r) = 0 where

v 0 (r) = - c 2 -c 2 0 c r + c 0 1 r 2 + c 2 -c 2 0 ≤ 0 , v =
0 is a super-solution and v = v 0 is a negative decreasing sub-solution to the ODE (37). Thus, for every r 1 > 0, there exist

r 2 > r 1 and a solution v ∈ C ∞ ((r 1 , r 2 ), R) to the ODE v ′ = (1 + v 2 )g(v, r) satisfying v 0 ≤ v ≤ 0 for any r ∈ (r 1 , r 2 ). Moreover we have g ′ v (v, r) ≤ 0 for v ∈ [v 0 , 0]
, and then we conclude that

(38) v ′ ≤ 0 for r ∈ (r 1 , r 2 )
Step 2: Qualitative properties Since for any r > 0, v 0 (r) ∈ (-cot α, 0), the bounds of v(r) by v 0 (r) and zero force v to exist globally for r > 0. Moreover, as lim r→0 v 0 (r) = 0, v satisfies the same limit and we can extend v to 0 by continuity as v(0) = 0. This proves that v is a global smooth solution to (37) with initial condition v(0) = 0. Thus it is easy to check that any primitive function φ c to v is a smooth radially symmetric solution to (17) satisfying φ ′ c (0) = 0. From (38), we conclude that φ c is concave.

Step 3: Asymptotics Since v is strictly decreasing on R + and bounded from below, it converges to a finite limit cot α ≤ l < 0 as r goes to infinity. Since v is uniformly bounded in [-cot α, 0], l must satisfy g(l, +∞) = 0 which leads to l =cot α.

Linearising equation (37) aroundcot α, we set w = v + cot α. As w is uniformly bounded on R + and goes to zero at infinity, equation (37) reads w ′ (r) = -c (cot α)w + g(w, r) , r > 0 where g(w, r) = O(w 2 ) + O(1/r) as r goes to infinity. By the Duhamel's formula, w follows exponentially fast the behaviour of the slowest term of g. Thus w ∼ C/r as r goes to infinity and a straight calculation gives C = 1/(c 0 sin α). Repeating this method up to order 2, one gets

v(r) = -cot α + 1 c 0 (sin α)r + 3 sin 2 α -2 c 2 0 sin(2α)r 2 + O 1 r 3
This gives the desired asymptotics for φ c up to constants.

Step 4: Uniqueness Let φ 1 c and φ 2 c be two smooth radially symmetric solutions to (17). From step 3, we know that they satisfy the same asymptotic expansion as r goes to infinity and we assume the constants C are the same. Since φ 1 c -φ 2 c solves an elliptic equation with smooth coefficients and no zero order term, the classical maximum principle applies. Hence φ 1 cφ 2 c = 0 because lim r→∞ (φ 1 cφ 2 c )(r) = 0. This proves the uniqueness of φ c up to constants.

Step 5: Comparison with φ * Using Lemma 8.1, we can check (36) with the suitable value of the constant C = C 0 (see for instance the computation (44) with N 0 (x) ≃ 1/ √ π). Finally, using the comparison principle (as in Step 4), we deduce that φ * ≤ φ c . This ends the proof of lemma 7.4.

The edge case

Lemma 7.5 (Edge super-solution) Assume φ * is given by (23) where the measure µ is the sum of two Dirac masses

µ = µ {θ 1 ,θ 2 } = λ 1 δ θ 1 + λ 2 δ θ 2
with λ i > 0, θ i ∈ [0, 2π) for i = 1, 2 such that θ 1 < θ 2 and δ θ i the Dirac mass in θ i . In the case α = π/2, define φ e for any x ∈ R 2 by

(39) φ e (x) = min(p 1 (x), p 2 (x)) with p i (x) = -(cot α) x • ν i - 2 c 0 sin α ln λ i
where ν i = (cos θ i , sin θ i ). Then, φ e is a Lipschitz and piecewise smooth global supersolution to (17) verifying φ * ≤ φ e on R 2 . Moreover, as |x| goes to infinity,

(40) φ e (x) = φ * (x) + O(1) , x ∈ R 2
Proof of Lemma 7.5.

Notice first that φ e = φ * with φ * defined as a special case of Proposition 5.1. This shows that φ e is a concave (Lipschitz) supersolution. Finally (40) follows from (31). This ends the proof of the lemma.

The arc case

Here we wish to describe a super-solution to (17) which, from above, looks like an arc, i.e. is made up of two non parallel straight lines connected by a circle.

Lemma 7.6 (Arc super-solution) Assume φ * is given by (23) where the measure µ is the sum of two Dirac masses and a Lebesgue measure

µ = µ [θ 1 ,θ 2 ] = λδ θ 1 + λδ θ 2 + 1I (θ 1 ,θ 2 ) dθ,
where λ > 0, θ i ∈ [0, 2π) for i = 1, 2 and θ 1 < θ 2 .

Define φ e as in the edge case (39) with λ 1 = λ 2 = λ. Define φ c as in the cone case (lemma 7.4) where the constant C ∈ R in (35) is chosen such that φ c (0) = φ e (0) = -2 c 0 sin α ln λ.

Finally, define φ a on R 2 by

(41) ∀x ∈ R 2 , φ a (x) =      - 2 c 0 sin α ln λ if x = 0 min(φ c (x), φ e (x)) if θ x ∈ (θ 1 , θ 2 ) φ e (x) otherwise,
Then φ a is a Lipschitz continuous global super-solution to (17). Moreover, as |x| goes to infinity φ a (x) = φ * (x) + O(1)

The shape of φ a is sketched on Figures 1,2, 3.

Proof of Lemma 7.6

Step 1: φ a is a global continuous super-solution By definition and lemmas 7.4 and 7.5, φ a is a super-solution to (17) where it is locally the minimum of supersolutions, i.e. everywhere except on the two half lines θ x = θ i for i = 1, 2. However, we have φ c (0) = φ e (0) and Dφ e (x) = -(cot α)ν i while φ ′ c (r) ∈ (-cot α, 0] for any x ∈ R 2 with θ x = θ i , i = 1 or 2. Thus, φ e (x) ≤ φ c (x) on a neighborhood N (not containing the origin) of the two half lines θ x = θ i for i = 1, 2. This implies φ a = φ e on N and then φ a is at least a supersolution on R 2 \ {0}. Moreover, φ a is a supersolution on the whole R 2 . Indeed, φ a = φ e for θ x ∈ [θ 1 , θ 2 ], then φ a has a gradient jump along the edge θ = (θ 1 + θ 2 )/2 + π up to the origin. And this gradient jump implies that there is no C 2 test function touching φ a from below at x = 0. where θ(r) ∈ (0, π/2) for r > 0 (see Figure 4). Notice that from the concavity of φ c , we deduce that the set

{x, p 1 (x) ≤ φ c (x)} = x, θ x ∈ [θ 1 -θ(r), θ 1 + θ(r)]
is a convex set. Therefore we deduce that

φ e (x) ≤ φ c (x) ⇔ θ x ∈ [θ 1 -θ(r), θ 1 + θ(r)] ∪ [θ 2 -θ(r), θ 2 + θ(r)] and φ a (x) = φ c (x) if and only if θ x ∈ I r = [θ 1 + θ(r), θ 2 -θ(r)
]. Since we choose φ c such that φ ′ c (0) = 0, we get lim r→0 θ(r) = + π 2 This forces both curves θ x = θ 1 + θ(r) and θ x = θ 2 -θ(r) to intersect at some point (x, z) = (0, φ c (0)) as soon as θ 2θ 1 < π. In that case, it is worth noticing that the above interval I r is empty for sufficiently small r (see Figures 5,6). On the other hand, using the asymptotics (35) of φ c found in lemma 7.4, we get as r goes to infinity

(42) cos θ(r) = 1 - 1 c 0 cos α ln r r + φ c (0) -C r cot α + O 1 r 2 , α = π 2
where C is the constant given by (35) and fixed by the choice φ c (0) = φ e (0). To prove that φ a is above the sub-solution φ * up to an additive constant, it remains to compare φ * and φ c when θ x ∈ I r = [θ 1 + θ(r), θ 2 -θ(r)] and r sufficiently large. According to lemma 8.1, one gets that for any x ∈ R 2 for r = |x| sufficiently large and uniformly in θ

x ∈ [θ 1 , θ 2 ] φ * (x) = -(cot α)r + ln r c 0 sin α - 2 c 0 sin α ln(Φ(x)) (44) with Φ(x) := 2πN 0 (x) √ b + λ √ re br 2 (cos(θx-θ 1 )-1) + λ √ re br 2 (cos(θx-θ 2 )-1) + O 1 √ r where N 0 (x) = √ rg(θ 2 -θx) √ rg(θ 1 -θx) e -u 2 4 du 2π
as defined in lemma 8.1. Since g is odd and θ x ∈ [θ 1 , θ 2 ], we see that

N 0 (x) ≥ √ rg( θ 2 -θ 1 2 ) 0 e -u 2 4 du 2π = 1 2 √ π + o r (1)
We deduce in particular that for r large enough and uniformly in θ

x ∈ [θ 1 , θ 2 ]: Φ(x) ≥ √ π 2 √ b
Therefore, from the asymptotics (35) of φ c , we deduce that there exist

r 1 > 0, C 1 > 0 such that (45) ∀r ≥ 0 , ∀θ x ∈ [θ 1 , θ 2 ] , r ≥ r 1 ⇒ φ * (x) ≤ φ c (x) + C 1
Now from ( 43) and (45), we deduce that (up to increasing the constant C 1 ),

∀r ≥ 0 , ∀θ x ∈ [θ 1 , θ 2 ] , r ≥ r 1 ⇒ φ * (x) ≤ φ a (x) + C 1 Step 4: φ * ≥ φ a -constant Case 1: θ x ∈ I r
We start with the asymptotics (44). Using (42), we see that there exist

r 2 > 0, C 2 > 0 such that ∀i = 1, 2 , ∀r ≥ 0 , ∀θ x ∈ I r , r ≥ r 2 ⇒ √ re br 2 (cos(θx-θ i )-1) ≤ √ re br 2 (cos(θ(r))-1) ≤ C 2
Using also the fact that N 0 (x) ≤ 1/(2 √ π), we deduce that Φ is bounded for r large enough and then (up to increasing r 2 and C 2 )

(46) ∀r ≥ 0 , ∀θ x ∈ I r , r ≥ r 2 ⇒ φ * (x) ≥ φ c (x) -C 2 Case 2: θ x ∈ [θ 1 , θ 2 ]\I r
Let us assume that θ x ∈ [θ 1 , θ 1 + θ(r)) (the symmetric case is similar). Then there exist

r 3 > 0, C 3 > 0 such that ∀r ≥ 0 , ∀θ x ∈ [θ 1 , θ 1 +θ(r)) , r ≥ r 3 ⇒ √ re br 2 (cos(θx-θ 1 )-1) ≥ √ re br 2 (cos(θ(r))-1) ≥ C 3 > 0
Therefore,( up to increasing the constants r 3 and C 3 ) for any r ≥ 0 and any θ

x ∈ [θ 1 , θ 1 + θ(r)), r ≥ r 3 ⇒ Φ(x) ≤ C 3 λ √ re br 2 (cos(θx-θ 1 )-1) + λ √ re br 2 (cos(θx-θ 2 )-1)
and then

φ * (x) ≥ - 2 c 0 sin α ln λe br 2 cos(θx-θ 1 ) + λe br 2 cos(θx-θ 2 ) - 2 ln C 3 c 0 sin α ≥ - 2 c 0 sin α ln 2e c 0 sin α 2 φe(x) - 2 ln C 3 c 0 sin α = φ e (x) - 2 ln(2C 3 ) c 0 sin α (47) Case 3: θ x ∈ S 1 \[θ 1 , θ 2 ] Notice that the set S 1 \[θ 1 , θ 2 ] is not empty because θ 2 -θ 1 < 2π (as a consequence of θ 1 , θ 2 ∈ [0, 2π)). In that case, we define θ ′ m = θ m -π with θ m = θ 1 + θ 2 2 .
Then, it satisfies θ 2 -2π < θ ′ m < θ 1 . Let us assume that (the other case is similar):

(48)

θ x ∈ [θ ′ m , θ 1 ).
We also define θ ′ x = θ x + π Then we have

θ 2 θ 1 e br 2 cos(θx-θ) dθ = [θ 1 ,θ 2 ]∩[θx,θ ′ x ] (...) dθ + [θ 1 ,θ 2 ]\[θx,θ ′ x ]
(...) dθ

We have Using (48), we also see that 47) and (49) together, we get that there exists a constant C > 0 such that for r large enough and uniformly in θ

[θ 1 ,θ 2 ]∩[θx,θ ′ x ]
[θ 1 ,θ 2 ]\[θx,θ ′ x ]
x ∈ S 1 . φ * (x) ≥ φ a (x) -C
The functions φ * and φ a being continuous, the result still holds for any r ≥ 0 (up to increasing the constant C). This concludes the proof of lemma 7.6.

Proof of Theorem 1.3

Step 1: Existence of a solution Let α ∈ (0, π 2 ], c 0 > 0 and c = c 0 / sin α. The case α = π/2 is obvious and we omit it. Choose φ ∞ a 1-homogeneous viscosity solution to the eikonal equation [START_REF] Hamel | Stability of travelling waves in a model for conical flames in two space dimensions[END_REF] in dimension N = 3 with a finite number m of singularities. By proposition 7.1, the i. of Theorem 1.3 is already established, and we can consider the measure µ given in the ii. of Theorem 1.3. Then Proposition 4.3 implies that the function φ * given by ( 9) is a smooth concave subsolution of (17).

If k = 1, φ ∞ has no gradient jump and the corresponding measure is µ = dθ or µ = λ 0 (δ θ 1 + δ θ 1 +2π ). In the first case, we saw in lemma 7.4 that φ c , to which a suitable constant is added, is a smooth solution to (17) with the right asymptotics at infinity. In the second one, φ = φ ∞ is a suitable solution to (17).

We now turn to the case k ≥ 2. For any i ∈ {1, . . . , k}, choose λ 0 > 0 a given positive constant. We have µ

= j µ j ≥ μi with (50) μi = 2λ 0 (δ θ i + δ θ i+1 ) + σ i 1I (θ i ,θ i+1 ) dθ
Let φi * be the subsolution defined in (23) with the measure μi . If μi corresponds to an arc (σ i = 1), denote φ * i the global supersolution defined in lemma 7.6 with λ = 2λ 0 . Notice that there is a constant C > 0 (that can be chosen independently of the index i) such that

(51) φ * i -C ≤ φi * ≤ φ * i + C
If μi corresponds to an edge (σ i = 0), denote φ * i the global supersolution defined in lemma 7.5 with λ 1 = λ 2 = 2λ 0 , which satisfies in particular (51). Finally, define on R 2 the function φ * as the infimum over i ∈ {1, . . . , k} of φ * i . Notice that, by construction, we have

(52) φ * (x) = φ * i (x) if θ x ∈ [θ i , θ i+1 ]
We also have in particular

φ * ≤ φi * ≤ φ * i + C ≤ φ * + C =: φ *
We claim that at infinity (53) φ * (x) = φ * (x) + O(1).

We shall first finish the proof of theorem 1.3 and come back to the proof of that claim in a second step. Thus φ * is a global supersolution above the subsolution φ * . Moreover, either there exists σ i = 1 and then we have (see in particular Step 2: Proof of (53) in the case k ≥ 2 Let x ∈ R 2 , then there exists i ∈ {1, . . . , k} such that θ x ∈ [θ i , θ i+1 ]. We can write: µ = μi + μi where μi is defined by (50). So 8 Appendix: Laplace's method

For the reader's convenience, we reproduce here Laplace's method. It investigates asymptotics as r goes to infinity of integrals involving expressions of the form e -rJ , J denoting some given function. Our interest is to find uniform estimates as x = r(cos θ x , sin θ x ) lies in a given angle sector [θ 1 , θ 2 ]. The proof develops ideas that can be found for a simpler case in [START_REF] Evans | Partial Differential Equations[END_REF], chapter 4.

θ * = θ 1 -θ x if θ 1 -θ x ≥ -π + δ -π + δ otherwise θ * = θ 2 -θ x if θ 2 -θ x ≤ π -δ π -δ otherwise
We then cut the integral I into three parts, integrating between θ 1θ x and θ * , between θ * and θ * and finally between θ * and θ 2θ x . We call those three integrals I 1 , I 2 and I 3 respectively.

Regarding I 1 and I 3 , cos θ can be bounded in both cases by cos(πδ) and f by its L ∞ norm on the compact set [0, 2π]. Thus, there exists a constant C > 0 such that for any x ∈ R 2 with θ x ∈ [θ 1 , θ 2 ], I 1 + I 3 ≤ Ce where N * 0 is defined as in lemma 8.1 with θ * or θ * when needed, but it only changes the desired asymptotics with an exponentially small correction as above. The remainder term R is defined as

R(x) = r

Lemma 4 . 1 (

 41 Sufficient condition for a sub-solution to (17)) Fix α ∈ (0, π 2 ], c 0 > 0 and c = c 0 / sin α. Let φ * be a concave smooth solution to

θ θ 2 1Figure 1 : 1 θ 2 Figure 2 :

 21122 Figure 1: Sketch of φ a for θ 2θ 1 < π

θ θ 1 2Figure 3 :Figure 4 :

 134 Figure 3: Sketch of φ a for θ 2θ 1 > π

Step 3 :Figure 5 :Figure 6 :

 356 Figure 5: The set I r if θ 2θ 1 < π

e br 2

 2 cos(θx-θ) dθ ≤ π e br 2 cos(θx-θ 1 )

e br 2

 2 cos(θx-θ) dθ ≤ π e br 2 cos(θx-θ 1 ) Therefore, we conclude that in this third case, φ * (x) ≥ -2 c 0 sin α ln (λ + 2π)e br 2 cos(θx-θ 1 ) + (λ + 2π)e br 2 cos(θx-θ 2 )≥ φ e (x) -

Figures

  there exists p ∈ R N -1 such that lim sup|x|→+∞ φ * (x)p • x |x| < 0.Or σ i = 0 for any i, and condition (54) is satisfied if k ≥ 3. The special case k = 2 and σ 1 = σ 2 = 0 corresponds to an edge for which we already know the existence of a smooth concave solution, by Theorem 1.2. In the other cases, condition (54) and Proposition 3.1 imply the existence of a smooth concave solution φ ∈ [φ * , φ * ].

1 e br 2 S 1 e br 2 1 e br 2

 121212 i ) ⊂ S 1 \(θ i , θ i+1 ) = ∅ Then k ≥ 2 implies that l x := min {θ xθ i , θ i+1θ x } < π and S cos(θx-θ) dμ i (θ) ≤ μi (S 1 )e br 2 cos lx ≤ μi (S 1 ) 2λ 0 cos(θx-θ) dμ i (θ) with S cos(θx-θ) dμ i (θ) = 2λ 0 e br 2 cos(θx-θ i ) + 2λ 0 e br 2 cos(θx-θ i+1 ) + σ i (x) -C ′where we have used (51) in the last line. Using (52), we see that this implies(56) φ * (x) ≥ φ * (x) -C ′′ for θ x ∈ [θ i , θ i+1 ]Finally, this implies (53) and ends the proof of theorem 1.3.

5 . 2 6 . 8 . 1 ( 2 θ 1 e br 2 2 f 2

 5268121222 page 204. Lemma 8.1 below is only used in Step 3 and 4 of the proof of Lemma 7.Lemma Uniform asymptotics in a sector [θ 1 , θ 2 ]) Define for any x = r(cos θ x , sin θ x ) ∈ R 2 with θ x ∈ [0, 2π)F (x) = λ 1 e br 2 cos(θ 1 -θx) + λ 2 e br 2 cos(θ 2 -θx) + θ cos(θ-θx) f (θ) dθ 2πwherei. b = c 0 cos α > 0, λ i ∈ R, θ i ∈ [0, 2π] for i = 1, 2 and θ 1 < θ 2 ii. f ∈ C 1 ([0, 2π], C) is 2π-periodic.As r goes to infinity, we have the following asymptotics uniform in the angular sectorθ x ∈ [θ 1 , θ 2 ] F (x) = λ 1 e br 2 cos(θ 1 -θx) + λ 2 e br 2 cos(θ 2 -θx) + e br 2b(1cos θ) for θ ∈ [-π, π] sign(θ)2 √ b for θ ∈ R\[-π, π]Moreover, there exists a constant C > 0 such that for anyx ∈ R 2 , if r > 1 and θ x ∈ [θ 1 , θ 2 ] then |R(x)| ≤ C.Proof of Lemma 8.1.It is straightforward to check that g defined by (57) is an oddC 3 -diffeomorphism from [-π, π] to [-2 √ b, 2 √ b] satisfying g(0) = 0, g ′ (0) = √b and g"(0) = 0. We have also chosen to extend g to the real line by continuity. However, when we speak about g -1 , it has to be understood as the inverse of g on [-π, π].Afterwards, for any x ∈ R 2 , we define cos θ f (θ + θ x ) dθ 2πAssume θ x ∈ [θ 1 , θ 2 ]. In order to get a bound on I uniform in the angle θ x , we fix some δ > 0 and set

br 2 I 2 ( 2 f

 222 cos(π-δ)For sufficiently small δ > 0, the right hand term decreases exponentially fast and the contribution of I 1 and I 3 in I is exponentially small as r goes to infinity uniformly in θ x .Using the change of variables u = √ rg(θ), we rewriteI 2 as t) = f (θ x + g -1 (t))/ 1 -(t 2 /(4b)) . Since h(t) = h(0) + t 0 h ′(s) ds, we haveI 2 (x) = e br

1 2 √I 1 ,

 121 rg(θ * ) √ rg(θ * ) e -h is smooth, h ′ is uniformly bounded on [g(θ * ), g(θ *)] and the bound only depends on δ. A straight calculation then shows that there exists C > 0 such that R(x) ≤ C √ rg(θ * ) I 2 and I 3 together, we get the desired asymptotics.


Laurence Amalric

Noura Meklat

Occurrence of synthetic musks in the aquatic environment of the Loire-Bretagne basin

OCCURRENCE of SYNTHETIC MUSKS in the AQUATIC ENVIRONMENT of the LOIRE-BRETAGNE BASIN

 2004).

Since there is no occurrence data about musks in France, this study intended to investigate the presence of synthetic musks in surface water from the Loire-Bretagne basin. A liquid liquid extraction method was applied to extract nitro musks (musk xylene and musk ketone) and polycyclic musks (AHTN, HHCB, phantolide (AHMI), traseolide (ATII), and celestolide (ADBI)) from water sample; analysis was performed by gas chromatography mass spectrometry. The quantification limit was 30 ng.l -1 for each analyte. The recoveries ranged from 76 to 92 % in spiked deionised water. Analysis of 59 water samples from the Loire-Bretagne basin revealed the presence of HHCB and AHTN in most of the samples. None of the other synthetic musks was detected. This can confirm the steps taken in recent years in Europe on the restriction of nitro musks (MX and MK) for the benefit of polycyclic musks (HHCB and AHTN) in the formulation of personal care products (Reiner et al., 2007). HHCB values are higher than concentrations of AHTN, this reflects the higher utilisation of HHCB compared to AHTN in Europe (Dsikowitzky et al., 2002)

  . Concentrations of AHTN and HHCB observed in this French study are comparable to those found in previously in Europe. Identification of musk xylene and musk ketone in freshwater fish collected from the TamaRiver, Tokyo Bull Environ Contam Toxicol 26: 656-662 (1981)

	acknowledgements -References
	We acknowledge the Carnot project for his financial support and the Loire-Bretagne agency
	for the samples.
	Eschke, H.-D. Synthetic musks in different water matrices. In Synthetic musk fragrances in
	the environment G. Rimkus, ed., Springer, Berlin, Germany, pp. 17-28. (2004)
	Dsikowitzky L. et al. Distribution of polycyclic musks in water and particulate matter of the
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