Samy Skander

Samy Skander Bahoura

A COMPACTNESS RESULT FOR AN EQUATION ON THE ANNULUS

Keywords: Mathematics Subject Classification: 35J60 35B45 35B50 blow-up, boundary, elliptic equation, a priori estimate, Lipschitz condition, boundary singularity, annulus

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A compactness result for an equation on the annulus.

INTRODUCTION AND MAIN RESULTS

Let us consider the following operator:

L ǫ := ∆ + ǫ • ǫ(x) < (x -x 0 )|∇u i >
We consider the following equation:

(P ǫ ) -∆u -ǫ • ǫ(x) < x -x 0 |∇u > = |x -x 0 | -2α V e u in Ω ⊂ R 2 , u = 0 in ∂Ω.
Here, we assume that:

We denote by C(1) and C(1/2) the unit circle and the circle of radius 1/2 respectively. Ω = A(0, 1/2, 1) is an annulus of center 0 and radii 1/2, 1, x 0 ∈ C(1/2) and, 0 < ǫ → 0, α ∈ (0, 1/2), u ∈ W 1,1 0 (Ω), |x -x 0 | -2α e u ∈ L 1 (Ω), 0 ≤ V ≤ b.

We assume that:

Ω is an annulus of exterior circle the unit circle and interior circle the circle of radius 1/2.

ǫ ≡ 1 in a neighborhood of the unit circle C(1) and ǫ ≡ -1 in a neighborhood of the circle of radius 1/2.

Important Remark: To give an example of a blow-up sequence of the previous type on the boundary. We use the counter-exemple of Brezis and Merle, this counterexample works if we replace the unit ball centered at y 0 = (1, 0) by the annulus centered at y 0 = (1, 0). After se use an inversion to have the blow-up point on the interior circle of the annulus. (do not forget to add the term ǫ • ǫ(x) < x -x 0 |∇u i > in the equation).

Here we present the result of Brezis-Merle in the regular case.

When ǫ = 0 the previous equation was studied by many authors with or without the boundary condition, also for Riemann surfaces see [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF][START_REF] Bandle | Isoperimetric Inequalities and Applications[END_REF][START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF][START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF][START_REF] Brezis | Nonlinear elliptic equations with measures revisited[END_REF][START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF][START_REF] Brezis | Semi-linear second-order elliptic equations in L1[END_REF][START_REF] Chen | A priori estimates for solutions to nonlinear elliptic equations[END_REF][START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF][START_REF] Chen | Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces[END_REF][START_REF] Chang | Scalar curvature equation on 2and 3-spheres[END_REF][START_REF] De Figueiredo | A priori Estimates and Existence of Positive Solutions of Semilinear Elliptic Equations[END_REF][START_REF] Ding | The differential equation ∆u = 8π -8πhe u on a compact Riemann surface[END_REF][START_REF] Gilbarg | Elliptic Partial Differential Equations of Second order[END_REF][START_REF] Yy | Harnack Type Inequality: the method of moving planes[END_REF][START_REF] Yy | Blow-up analysis for solutions of -∆u = V e u in dimension two[END_REF][START_REF] Ma | Convergence for a Liouville equation[END_REF][START_REF] Nagasaki | Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities[END_REF][START_REF] Shafrir | A sup+inf inequality for the equation -∆u = V e u[END_REF][START_REF] Tarantello | Multiple condensate solutions for the Chern-Simons-Higgs theory[END_REF] where one can find some existence and compactness results. Also we have a nice formulation in the sens of the distributions of this Problem in [START_REF] Brezis | Semi-linear second-order elliptic equations in L1[END_REF].

Among other results, we can see in [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF] the following important Theorem, Theorem A (Brezis-Merle [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]) If (u i ) i and (V i ) i are two sequences of functions relative to the problem (P 0 ) with ǫ = 0 and,

0 < a ≤ V i ≤ b < +∞ then it holds, sup K u i ≤ c,
with c depending on a, b, K and Ω.

We can find in [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF] an interior estimate if we assume a = 0 but we need an assumption on the integral of e u i , namely:

Theorem B(Brezis-Merle [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]).For (u i ) i and (V i ) i two sequences of functions relative to the problem (P 0 ) with,

0 ≤ V i ≤ b < +∞ and Ω e u i dy ≤ C, then it holds; sup K u i ≤ c,
with c depending on b, C, K and Ω.

The condition Ω e u i dy ≤ C is a necessary condition in the Problem (P ǫ ) as showed by the following counterexample for ǫ = 0: Theorem C (Brezis-Merle [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]).There are two sequences (u i ) i and (V i ) i of the problem (P 0 ) with;

0 ≤ V i ≤ b < +∞, Ω e u i dy ≤ C, such that, sup Ω u i → +∞.
To obtain the two first previous results (Theorems A and B) Brezis and Merle used an inequality (Theorem 1 of [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]) obtained by an approximation argument and they used Fatou's lemma and applied the maximum principle in W 1,1 0 (Ω) which arises from Kato's inequality. Also this weak form of the maximum principle is used to prove the local uniform boundedness result by comparing a certain function and the Newtonian potential. We refer to [START_REF] Brezis | Nonlinear elliptic equations with measures revisited[END_REF] for a topic about the weak form of the maximum principle.

Note that for the problem (P 0 ), by using the Pohozaev identity, we can prove that Ω e u i is uniformly bounded when 0 < a ≤ V i ≤ b < +∞ and ||∇V i || L ∞ ≤ A and Ω starshaped, when a = 0 and ∇ log V i is uniformly bounded, we can bound uniformly Ω V i e u i . In [START_REF] Ma | Convergence for a Liouville equation[END_REF] Ma-Wei have proved that those results stay true for all open sets not necessarily starshaped in the case a > 0.

In [START_REF] Chen | A priori estimates for solutions to nonlinear elliptic equations[END_REF] Chen-Li have proved that if a = 0 and Ω e u i is uniformly bounded and ∇ log V i is uniformly bounded then (u i ) i is bounded near the boundary and we have directly the compactness result for the problem (P 0 ). Ma-Wei in [START_REF] Ma | Convergence for a Liouville equation[END_REF] extend this result in the case where a > 0.

When ǫ = 0 and if we assume V more regular we can have another type of estimates called sup + inf type inequalities. It was proved by Shafrir see [START_REF] Shafrir | A sup+inf inequality for the equation -∆u = V e u[END_REF] that, if (u i ) i , (V i ) i are two sequences of functions solutions of the Problem (P 0 ) without assumption on the boundary and 0 < a ≤ V i ≤ b < +∞ then it holds:

C a b sup K u i + inf Ω u i ≤ c = c(a, b, K, Ω).
We can see in [START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF] an explicit value of C a b = a b . In his proof, Shafrir has used the blow-up function,

the Stokes formula and an isoperimetric inequality see [START_REF] Bandle | Isoperimetric Inequalities and Applications[END_REF]. For Chen-Lin, they have used the blow-up analysis combined with some geometric type inequality for the integral curvature see [START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF]. Now, if we suppose (V i ) i uniformly Lipschitzian with A its Lipschitz constant then C(a/b) = 1 and c = c(a, b, A, K, Ω) see Brezis-Li-Shafrir [START_REF] Brezis | A sup+inf inequality for some nonlinear elliptic equations involving exponential nonlinearities[END_REF]. This result was extended for Hölderian sequences (V i ) i by Chen-Lin see [START_REF] Chen | A sharp sup+inf inequality for a nonlinear elliptic equation in R 2[END_REF]. Also have in [START_REF] Yy | Harnack Type Inequality: the method of moving planes[END_REF], an extension of the Brezis-Li-Shafrir result to compact Riemannian surfaces without boundary. One can see in [START_REF] Yy | Blow-up analysis for solutions of -∆u = V e u in dimension two[END_REF] explicit form, (8πm, m ∈ N * exactly), for the numbers in front of the Dirac masses when the solutions blow-up. Here the notion of isolated blow-up point is used. Also one can see in [START_REF] Chen | Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces[END_REF] refined estimates near the isolated blow-up points and the bubbling behavior of the blow-up sequences.

Here we give the behavior of the blow-up points on the boundary and a proof of a compactness result with Lipschitz condition. Note that our problem is an extension of the Brezis-Merle Problem.

The Brezis-Merle Problem (see [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF]) is:

Problem. Suppose that V i → V in C 0 ( Ω) with 0 ≤ V i . Also, we consider a sequence of solutions (u i ) of (P 0 ) relative to (V i ) such that, Ω e u i dx ≤ C, is it possible to have: ||u i || L ∞ ≤ C = C(b, C, V, Ω)?
Here we give blow-up analysis on the boundary when V (similar to the prescribed curvature when ǫ = 0) are nonegative and bounded, and on the other hand, if we add the assumption that these functions (similar to the prescribed cruvature) are uniformly Lipschitzian, we have a compactness of the solutions of the problem (P ǫ ) for ǫ small enough. (In particular we can take a sequence of ǫ i tending to 0):

For the behavior of the blow-up points on the boundary, the following condition is sufficient,

0 ≤ V i ≤ b,
The condition V i → V in C 0 ( Ω) is not necessary. But for the compactness of the solutions we add the following condition:

||∇V i || L ∞ ≤ A i → 0.
Our main results are:

Theorem 1.1. Assume that α ∈ (0, 1/2) and max Ω u i → +∞, where (u i ) are solutions of the probleme (P ǫ i ) with:

0 ≤ V i ≤ b, and Ω |x -x 0 | -2α e u i dx ≤ C,
then, after passing to a subsequence, there is a finction u, there is a number N ∈ N and N points x 1 , . . . , x N ∈ ∂Ω, such that: x 1 = x 0 and x 2 , . . . , x N ∈ ∂Ω -{x 0 } and,

∂ ν u i → ∂ ν u + N j=1 α j δ x j , α 1 ≥ 4π(1 -α) and α j ≥ 4π, in the sens of measures on ∂Ω. u i → u in C 1 loc ( Ω -{x 1 , . . . , x N }), or, x 1 , . . . , x N ∈ ∂Ω -{x 0 } and, ∂ ν u i → ∂ ν u + N j=1
α j δ x j , α 1 ≥ 4π and α j ≥ 4π, in the sens of measures on ∂Ω.

u i → u in C 1 loc ( Ω -{x 1 , . . . , x N }), Theorem 1.2. Assume that (u i ) are solutions of (P ǫ i ) relative to (V i ) with α ∈ (0, 1/2) and the following conditions: 0 ≤ V i ≤ b, ||∇V i || L ∞ ≤ A i → 0 and Ω |x -x 0 | -2α e u i ≤ C.
Then we have:

||u i || L ∞ ≤ c(b, α, ((A) i ), C, x 0 , Ω),

PROOF OF THE THEOREMS

Proof of theorem 1.1:

First remark that:

-∆u i = ǫ i (x 1 ∂ 1 u i + x 2 ∂ 2 u i ) + |x -x 0 | -2α V i e u i ∈ L 1 (Ω) in Ω ⊂ R 2 , u i = 0 in ∂Ω.
and,

u i ∈ W 1,1 0 (Ω).
By the corollary 1 of Brezis-Merle see [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF] we have e u i ∈ L k (Ω) for all k > 2 and the elliptic estimates of Agmon and the Sobolev embedding see [START_REF] Aubin | Some Nonlinear Problems in Riemannian Geometry[END_REF] and α ∈ (0, 1/2) imply that:

u i ∈ W 2,k (Ω) ∩ C 1,ǫ ( Ω).
Also remark that, we have for two positive constants C q = C(q, Ω) and C 1 = C 1 (Ω) (see [START_REF] Brezis | Semi-linear second-order elliptic equations in L1[END_REF]) :

||∇u i || L q ≤ C q ||∆u i || L 1 ≤ (C ′ q + ǫC 1 ||∇u i || L 1 ), ∀ i and 1 < q < 2.
Thus, if ǫ > 0 is small enough and by the Holder inequality, we have the following estimate:

||∇u i || L q ≤ C ′′ q , ∀ i and 1 < q < 2.
Step 1: interior estimate First remark that, if we consider the following equation:

-∆w i = ǫ i (x 1 ∂ 1 u i + x 2 ∂ 2 u i ) ∈ L q , 1 < q < 2 in Ω ⊂ R 2 , w i = 0 in ∂Ω.
If we consider v i the Newtonnian potential of

ǫ i (x 1 ∂ 1 u i + x 2 ∂ 2 u i ),
we have:

v i ∈ C 0 ( Ω), ∆(w i -v i ) = 0.
By the maximum principle w i -v i ∈ C 0 ( Ω) and thus w i ∈ C 0 ( Ω).

Also we have by the elliptic estimates that w i ∈ W 2,1+ǫ ⊂ L ∞ , and we can write the equation of the Problem as:

-∆(u i -w i ) = Ṽi e u i -w i in Ω ⊂ R 2 , u i -w i = 0 in ∂Ω.
with,

0 ≤ Ṽi = V i e w i ≤ b, Ω |x -x 0 | -2α e u i -w i ≤ C.
We apply the Brezis-Merle theorem to u i -w i to have:

u i -w i ∈ L ∞ loc (Ω),
and, thus:

u i ∈ L ∞ loc (Ω).

Step2: boundary estimate

The boundary contains two coonected components. Set ∂ ν u i the inner derivative of u i . We consider one of the two components and without loss of generality, by the maximum principle one can assume that ∂ ν u i ≥ 0.

We have:

∂Ω ∂ ν u i dσ ≤ C.
We have the existence of a nonnegative Radon measure µ such that,

∂Ω ∂ ν u i ϕdσ → µ(ϕ), ∀ ϕ ∈ C 0 (∂Ω). Case 1: µ({x 0 }) ≥ 4π(1 -α).
This means that x 0 is non-regular point.

We take an

x 1 ∈ ∂Ω such that, µ(x 1 ) < 4π. Set B(x 1 , ǫ) ∩ ∂Ω := I ǫ .
We choose a function η ǫ such that,

           η ǫ ≡ 1, on I ǫ , 0 < ǫ < δ/2, η ǫ ≡ 0, outside I 2ǫ , 0 ≤ η ǫ ≤ 1, ||∇η ǫ || L ∞ (I 2ǫ ) ≤ C 0 (Ω, x 1 ) ǫ .
We take a ηǫ such that,

-∆η ǫ = 0 in Ω ⊂ R 2 , ηǫ = η ǫ in ∂Ω.
Remark: We use the following steps in the construction of ηǫ :

We take a cutoff function η 0 in B(0, 2) or B(x 1 , 2):

1-We set η ǫ (x) = η 0 (|x -x 1 |/ǫ) in the case of the unit disk it is sufficient.
2-Or, in the general case: we use a chart (f, Ω) with f (0) = x 1 and we take µ ǫ (x) = η 0 (f (|x|/ǫ)) to have connected sets I ǫ and we take η ǫ (y

) = µ ǫ (f -1 (y)). Because f, f -1 are Lipschitz, |f (x) -x 0 | ≤ k 2 |x| ≤ 1 for |x| ≤ 1/k 2 and |f (x) -x 0 | ≥ k 1 |x| ≥ 2 for |x| ≥ 2/k 1 > 1/k 2 , the support of η is in I (2/k 1 )ǫ .            η ǫ ≡ 1, on f (I (1/k 2 )ǫ ), 0 < ǫ < δ/2, η ǫ ≡ 0, outside f (I (2/k 1 )ǫ ), 0 ≤ η ǫ ≤ 1, ||∇η ǫ || L ∞ (I (2/k 1 )ǫ ) ≤ C 0 (Ω, x 1 ) ǫ .
3-Also, we can take: µ ǫ (x) = η 0 (|x|/ǫ) and η ǫ (y) = µ ǫ (f -1 (y)), we extend it by 0 outside f (B 1 (0)). We have

f (B 1 (0)) = D 1 (x 1 ), f (B ǫ (0)) = D ǫ (x 1 ) and f (B + ǫ ) = D + ǫ (x 1 ) with f and f -1 smooth diffeo- morphism.            η ǫ ≡ 1, on a the connected set J ǫ = f (I ǫ ), 0 < ǫ < δ/2, η ǫ ≡ 0, outside J ′ ǫ = f (I 2ǫ ), 0 ≤ η ǫ ≤ 1, ||∇η ǫ || L ∞ (J ′ ǫ ) ≤ C 0 (Ω, x 1 ) ǫ .
And,

H 1 (J ′ ǫ ) ≤ C 1 H 1 (I 2ǫ ) = C 1 4ǫ, because f is Lipschitz.
Here H 1 is the Hausdorff measure.

We solve the Dirichlet Problem:

∆η ǫ = ∆η ǫ in Ω ⊂ R 2 , ηǫ = 0 in ∂Ω.
and finaly we set ηǫ = -η ǫ + η ǫ . Also, by the maximum principle and the elliptic estimates we have :

||∇η ǫ || L ∞ ≤ C(||η ǫ || L ∞ + ||∇η ǫ || L ∞ + ||∆η ǫ || L ∞ ) ≤ C 1 ǫ 2 ,
with C 1 depends on Ω.

As we said in the beguening, see also [START_REF] Boccardo | Nonlinear elliptic and parabolic equations involving measure data[END_REF][START_REF] Brezis | Semi-linear second-order elliptic equations in L1[END_REF][START_REF] Ding | The differential equation ∆u = 8π -8πhe u on a compact Riemann surface[END_REF][START_REF] Tarantello | Multiple condensate solutions for the Chern-Simons-Higgs theory[END_REF], we have:

||∇u i || L q ≤ C q , ∀ i and 1 < q < 2.
We deduce from the last estimate that, (u i ) converge weakly in W 1,q 0 (Ω), almost everywhere to a function u ≥ 0 and Ω |x -x 0 | -2α e u < +∞ (by Fatou lemma). Also, V i weakly converge to a nonnegative function V in L ∞ . The function u is in W 1,q 0 (Ω) solution of :

-∆u = |x -x 0 | -2α V e u ∈ L 1 (Ω) in Ω ⊂ R 2 , u = 0 in ∂Ω.
According to the corollary 1 of Brezis-Merle result, see [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF], we have e ku ∈ L 1 (Ω), k > 1. By the elliptic estimates, we have, because α ∈ (0, 1/2), u ∈ W 2,k (Ω) ∩ C 1,ǫ ( Ω).

We denote by f • g the inner product of any two vectors f and g of R 2 .

We can write,

-∆((u i -u)η ǫ ) = |x -x 0 | -2α (V i e u i -V e u )η ǫ -2∇(u i -u) • ∇η ǫ + ǫ i (∇u i • (x -x 0 ))η ǫ . (1)
We use the interior esimate of Brezis-Merle, see [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF],

Step 1: Estimate of the integral of the first term of the right hand side of (1).

We use the Green formula between ηǫ and u, we obtain,

Ω |x -x 0 | -2α V e u ηǫ dx = ∂Ω ∂ ν uη ǫ ≤ Cǫ = O(ǫ) (2)
We have,

-∆u i -ǫ i ∇u i • (x -x 0 ) = |x -x 0 | -2α V i e u i in Ω ⊂ R 2 , u = 0 in ∂Ω.
We use the Green formula between u i and ηǫ to have:

Ω |x -x 0 | -2α V i e u i ηǫ dx = ∂Ω ∂ ν u i η ǫ dσ -ǫ i Ω (∇u i • (x -x 0 ))η ǫ = = ∂Ω ∂ ν u i η ǫ dσ + o(1) → µ(η ǫ ) ≤ µ(J ′ ǫ ) ≤ 4π -ǫ 0 , ǫ 0 > 0 (3)
From ( 2) and (3) we have for all ǫ > 0 there is i 0 such that, for i ≥ i 0 ,

Ω ||x -x 0 | -2α (V i e u i -V e u )η ǫ |dx ≤ 4π -ǫ 0 + Cǫ (4)
Step 2.1: Estimate of integral of the second term of the right hand side of (1).

Let Σ ǫ = {x ∈ Ω, d(x, ∂Ω) = ǫ 3 } and Ω ǫ 3 = {x ∈ Ω, d(x, ∂Ω) ≥ ǫ 3 }, ǫ > 0. Then, for ǫ small enough, Σ ǫ is an hypersurface. The measure of Ω -Ω ǫ 3 is k 2 ǫ 3 ≤ meas(Ω -Ω ǫ 3 ) = µ L (Ω -Ω ǫ 3 ) ≤ k 1 ǫ 3 .
Remark: for the unit ball B(0, 1), our new manifold is B(0, 1 -ǫ 3 ).

(Proof of this fact; let's consider d(x, ∂Ω) = d(x, z 0 ), z 0 ∈ ∂Ω, this imply that (d(x, z 0 )) 2 ≤ (d(x, z)) 2 for all z ∈ ∂Ω which it is equivalent to (z -z 0 ) • (2x -z -z 0 ) ≤ 0 for all z ∈ ∂Ω, let's consider a chart around z 0 and γ(t) a curve in ∂Ω, we have;

(γ(t) -γ(t 0 ) • (2x -γ(t) -γ(t 0 )) ≤ 0 if we divide by (t -t 0 ) (with the sign and tend t to t 0 ), we have γ ′ (t 0 ) • (x -γ(t 0 )) = 0, this imply that x = z 0 -sν 0 where ν 0 is the outward normal of ∂Ω at z 0 ))

With this fact, we can say that S = {x, d(x, ∂Ω) ≤ ǫ} = {x = z 0 -sν z 0 , z 0 ∈ ∂Ω, -ǫ ≤ s ≤ ǫ}. It is sufficient to work on ∂Ω. Let's consider a charts (z, D = B(z, 4ǫ z ), γ z ) with z ∈ ∂Ω such that ∪ z B(z, ǫ z ) is cover of ∂Ω . One can extract a finite cover (B(z k , ǫ k )), k = 1, ..., m, by the area formula the measure of S ∩ B(z k , ǫ k ) is less than a kǫ (a ǫ-rectangle). For the reverse inequality, it is sufficient to consider one chart around one point of the boundary).

We write,

Ω |∇(u i -u) • ∇η ǫ |dx = Ω ǫ 3 |∇(u i -u) • ∇η ǫ |dx + Ω-Ω ǫ 3 |∇(u i -u) • ∇η ǫ |dx. (5)
Step 2.1.1:

Estimate of Ω-Ω ǫ 3 |∇(u i -u) • ∇η ǫ |dx.
First, we know from the elliptic estimates that

||∇η ǫ || L ∞ ≤ C 1 /ǫ 2 , C 1 depends on Ω We know that (|∇u i |) i is bounded in L q , 1 < q < 2,
we can extract from this sequence a subsequence which converge weakly to h ∈ L q . But, we know that we have locally the uniform convergence to |∇u| (by the Brezis-Merle's theorem), then, h = |∇u| a.e. Let q ′ be the conjugate of q.

We have, ∀f ∈ L q ′ (Ω)

Ω |∇u i |f dx → Ω |∇u|f dx
If we take f = 1 Ω-Ω ǫ 3 , we have:

for ǫ > 0 ∃ i 1 = i 1 (ǫ) ∈ N, i ≥ i 1 , Ω-Ω ǫ 3 |∇u i | ≤ Ω-Ω ǫ 3 |∇u| + ǫ 3 . Then, for i ≥ i 1 (ǫ), Ω-Ω ǫ 3 |∇u i | ≤ meas(Ω -Ω ǫ 3 )||∇u|| L ∞ + ǫ 3 = ǫ 3 (k 1 ||∇u|| L ∞ + 1) = O(ǫ 3 ).
Thus, we obtain,

Ω-Ω ǫ 3 |∇(u i -u) • ∇η ǫ |dx ≤ ǫC 1 (2k 1 ||∇u|| L ∞ + 1) = O(ǫ) (6)
The constant C 1 does not depend on ǫ but on Ω.

Step 2.1.2: Estimate of

Ω ǫ 3 |∇(u i -u) • ∇η ǫ |dx.
We know that, Ω ǫ ⊂⊂ Ω, and ( because of Brezis-Merle's interior estimates)

u i → u in C 1 (Ω ǫ 3 ). We have, ||∇(u i -u)|| L ∞ (Ω ǫ 3 ) ≤ ǫ 3 , for i ≥ i 3 .
We write,

Ω ǫ 3 |∇(u i -u) • ∇η ǫ |dx ≤ ||∇(u i -u)|| L ∞ (Ω ǫ 3 ) ||∇η ǫ || L ∞ = C 1 ǫ = O(ǫ) for i ≥ i 3 , For ǫ > 0, we have for i ∈ N, i ≥ i ′ , Ω |∇(u i -u) • ∇η ǫ |dx ≤ ǫC 1 (2k 1 ||∇u|| L ∞ + 2) = O(ǫ) (7)
From ( 4) and (7), we have, for ǫ > 0, there is

i ′′ such that, i ≥ i ′′ , Ω |∆[(u i -u)η ǫ ]|dx ≤ 4π -ǫ 0 + ǫ2C 1 (2k 1 ||∇u|| L ∞ + 2 + C) = 4π -ǫ 0 + O(ǫ) (8)
We choose ǫ > 0 small enough to have a good estimate of (1).

Indeed, we have:

-∆[(u i -u)η ǫ ] = g i,ǫ in Ω ⊂ R 2 , (u i -u)η ǫ = 0 in ∂Ω. with ||g i,ǫ || L 1 (Ω) ≤ 4π -ǫ 0 /2.
We can use Theorem 1 of [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF] to conclude that there are q ≥ q > 1 such that:

Vǫ(x 0 ) e q|u i -u| dx ≤ Ω e q|u i -u|ηǫ dx ≤ C(ǫ, Ω).
where, V ǫ (x 0 ) is a neighberhood of x 1 in Ω. Here we have used that in a neighborhood of x 1 by the elliptic estimates, 1 -Cǫ ≤ ηǫ ≤ 1.

Thus, for each x 1 ∈ ∂Ω -{x 1 , . . . , xm } there is ǫ 0 > 0, q 0 > 1 such that:

B(x 1 ,ǫ 0 ) e q 0 u i dx ≤ C, ∀ i.
By the elliptic estimate see [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second order[END_REF] we have:

||u i || C 1,θ [B(x 1 ,ǫ)] ≤ c 3 ∀ i. Case 2: µ({x 0 }) < 4π(1 -α).
This means that x 0 is a regular point for the measure µ.

Let us consider B ǫ (x 0 ), a ball of center 0 and radius ǫ > 0. As in the previous case, we use the uniform estimate in W 1,q 0 (Ω), (1 ≤ q < 2) and Brezis-Merle's method, see [START_REF] Brezis | Uniform estimates and Blow-up behavior for solutions of -∆u = V (x)e u in two dimension[END_REF], to have

e u i ∈ L (1-ǫ ′ )/(1-α-ǫ ′ ) (B ǫ (x 0 )).
with a uniform bound.

Thus, by the Hölder inequality we have

u i ∈ L ∞ (B ǫ (x 0 )).
If we take µ({x 0 }) < 4π, by the Brezis-Merle estimate we have e u i ∈ L r (B ǫ (x 0 )) with r > 1, but this r may not be large enough to ensure u i ∈ L ∞ (B ǫ (x 0 )), because we have the term |x -x 0 | -2α in the equation.

Then, by the elliptic estimates, for α ∈ (0, 1)

u i ∈ W 2,1+ǫ ′ (B ǫ (x 0 )) ∩ C 0,ǫ ′ (B ǫ (x 0 )) ∩ C 2,ǫ ′ loc (Ω -{x 0 , x 1 , x 2 , . . . , x N }), (9)
and, for α ∈ (0, 1/2), we have

u i ∈ W 2,1+ǫ ′ (B ǫ (x 0 )) ∩ C 1,ǫ ′ (B ǫ (x 0 )) ∩ C 2,ǫ ′ loc (Ω -{x 0 , x 1 , x 2 , . . . , x N }). (10)
We have proved that, there is a finite number of points x1 , . . . , xm such that the squence (u i ) i is locally uniformly bounded in C 1,θ , (θ > 0) on Ω -{x 1 , . . . , xm }.

Proof of theorem 1.2:

We have:

u i ∈ W 2,k ∩ C 1,ǫ ( Ω), Thus, ∂ j u i ∈ W 1,k ∩ C 0 ( Ω),
and,

∂ j u i • ∂ k u i ∈ W 1,p ∩ C 0 ( Ω),
Thus, we can use integration by parts. The Pohozaev identity gives locally around each blow-up : a) When x goes in the neighborhood of the unit circle, the Pohozaev identity gives

∂Ω (∂ ν u i ) 2 dx ≤ c 0 (b, A, C, Ω). (11)
Thus we can use the weak convergence in L 2 (∂Ω) to have a subsequence ∂ ν u i , such that:

∂Ω ∂ ν u i ϕdx → ∂Ω ∂ ν uϕdx, ∀ ϕ ∈ L 2 (∂Ω),
Thus, α j = 0, j = 1, . . . , N and (u i ) is uniformly bounded.

b) When x goes in the neighborhood of x 0 or y 0 ∈ C(1/2), and the circle of radius 1/2, we use again the Pohozaev identity, by multplying by < x -x 0 |∇u i >. Here we use ∇V i → 0 (becasue we do not multiply by < x|∇u i > but by < x -x 0 |∇u i >) and the radius ǫ of the neighborhood, ǫ → 0 in the Pohozaev identity.

Indeed, we have,

-∆u i -ǫ • ǫ(x) < x -x 0 |∇u i >= |x -x 0 | -2α V e u i (12)
Let Ω 1 ǫ be a neighborhood of C(1) or C(1/2); first, we have ǫ ≡ 1 in a neighborhood of C(1), we multiply by < x -x 0 |∇u i > the previous equation and we integrate by parts, we obtain around the blow-up y 0 , and, also, we consider blow-up from C(1/2), in the neighborhood of C(1/2), ǫ ≡ -1, we multiply the equation of u i , (12), by < x -x 0 |∇u i > and we integrate by parts, Thus,

-∆u i (< x -x 0 |∇u i >) -ǫ • ǫ(x)(< x -x 0 |∇u i >) 2 = |x -x 0 | -2α V i (x) < x -x 0 |∇(e u i ) >
We have:

Ω 1 ǫ ∆u i (< x -x 0 |∇u i >) = ∂ + Ω 1 ǫ < x -x 0 |∇u i >< ν|∇u i > - < x -x 0 |ν > 2 |∇u i | 2 + + ∂Ω 1 2 < x -x 0 |ν > (∂ ν u i ) 2 dσ,
and,

-|x -x 0 | -2α V i (x) < x -x 0 |∇(e u i ) >= 2(1 -α) |x -x 0 | -2α V i (x)e u i + + |x -x 0 | -2α < x -x 0 |∇V i > e u i - ∂Ω 1 ǫ |x -x 0 | -2α V i (x) < x -x 0 |ν > e u i .
We have:

(∂ ν u i ) 2 ≤ C
Now, we consider blow-up from C(1/2), in the neighborhood of C(1/2), ǫ ≡ -1, we multiply the equation of u i , (12), by < x -x 0 |∇u i > and we integrate by parts, we obtain (here ν = -2x on C(1/2)): We tend i → +∞ and then ǫ → 0, ∇V i → 0, to obtain:

2(1 -α) Ω 1 ǫ |x -x 0 | -2α V i e u i + C(1/2)∩Bǫ (y 0 ) -(-||x|| 2 + < x 0 |x >)(∂ ν u i ) 2 dσ+ + C(1/2)∩Bǫ(y 0 ) -2(-||x|| 2 + < x 0 |x >)|x -x 0 | -2α V i + Ω 1 ǫ (< x -x 0 |∇u i >) 2 dx = = Ω 1 ǫ < x -x 0 |∇V i > |x -x 0 | -2α e u i + O(ǫ).
lim ǫ→0 lim i→+∞ Ω 1 ǫ |x -x 0 | -2α
V i e u i = 0, however:

Ω 1 ǫ |x -x 0 | -2α V i e u i dx = ∂Ω 1 ǫ ∂ ν u i dσ + O(ǫ) + o(1) → α 1 > 0,
it is a contradiction.

C( 1 ) 2 (||x|| 2 - 1 ǫ(

 1221 ∩Bǫ(y 0 )1 < x 0 |x >)(∂ ν u i ) 2 dσ + C(1)∩Bǫ(y 0 ) (||x|| 2 -< x 0 |x >)|x -x 0 | -2α V i + + Ω < x -x 0 |∇u i >) 2 dx = O(ǫ) + O(1), but, ||x 0 || = 1/2 and ||x|| = 1, thus ||x|| 2 -< x 0 |x >≥ 1/2,and thus: C(1)

  The previous left hand side is non-negative because||x 0 || = 1/2 and ||x|| = 1/2 and | < x 0 |x > | ≤ ||x 0 || × ||x|| = ||x|| 2 .


Remus Romulus

Tarquinius Priscus

Servius Tullius

Patrice Montzamir Romulus

P Montzamir Halshs-02275960

Keywords: Rome, kings, Servius Tullius, Tarquinius, Vibenna, François Tomb

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

In 985, H. Zehnacker proposed to identify this hidden name to Volupia, goddess of pleasure , whose sanctuary was near the porta Romulana, «kind of eponymous gate» of the city, and hosted the statue of Angerona, whose mouth is covered with a bandage, and Pliny links this statue to the story of Valerius Soranus. H. Zehnacker remark indeed that the list of cities in Lazio listed by Pliny is alphabetized and that Rome is mentioned after last names beginning with V 5 .

However, it should be noted that if the enumeration of Pliny is strictly alphabetical for the first letter of the Poucet 2008, 2. 2 Liv., .6-7; Plut., Romulus, .22; D. H. .7, 2.50; Varr., LL, 5.. NH, .. Zenacker 985, -5. 5 NH, .: «In the interior, the colonies: Capua, so called from the word meaning countryside, Aquinum, Suessa, Venafrum, Sora, Teanum, nicknamed Sidicinum; Nola; Cities: Abellinum, Aricia, Alba Longa, Acerra, Allifa, Atina, Aletrina, Anagnia, Atella, Affile, Arpinum Auxima, Avella, Alfaterna (there are three, one Latin, one hernique a labicane) Boville , Calatiae, Casinum, Calenum, Capitulum Hernicum, the Cereatins, nicknamed Marians; Korans, descendants of the Trojan Dardanus; the Cubulterins the Castrimonienses the Cingulans the Fabienses on Mount Alba; the Foropopulienses, territory Falernian; the Frusinates the Ferentinates, the Freginates, Ies Fabraternes old, new Fabraternes, the Ficolenses, the Foroappiens, the Forebtans, the Gabiens, the lnteramnates, Succasins, also called Lirinates; the Ilionenses Laviniens, the Norbans, the Nomentins. Praeneste formerly called Stephane, Privernum, Setia, Signia, Suessula; the Telins, nicknamed the Tribulans Balinienses; the Trebans, the Tusculanse the Verulans, the Veliternes, the Ulubrenses, the Ulvernates, and finally Rome itself, whose mysterious rites, defend to utter the other name ». 6 Serv., Ad Aen., .27. 7 Solin., Polyhistor. 8 Ernout Meillet 979, 78. 9

Zenod., F. Gr. Hist. 82 F ap. Solin., Coll. 2.9. 0 Strab. 6.. 2. Paus. 8.5.7. 2 D. H. .72 quote Cephalon of Gergis, Démagos, Agathyllos, Xenagoras or Dionysius of Chalcis. Tradition that we find in Serv., Ad Aen., .27 «[...] this situation engendered a struggle in which Remus was killed; and the Romans took their name of Romus. If he was called Romulus instead of Romus, this is due to a sort of blarney, who loves the diminutive / […] quae res bellum creavit. Strab. 5..2: «Here the fable adds that the two children were son of Mars, and on the banks of the river where they were exposed a wolf was seen breastfeed them as she would have her young, and that someone called Faustule, one of many swineherds who were then graze their cattle along the river, collected them, made them eat with him, and called one Romulos and the other Romos (καλέσαι δὲ τὸν μὲν Ῥωμύλον τὸν δὲ Ῥῶμον)».

5

Verg., Aen., .292-29. 6 Serv., Ad Aen. or the third century B.C. with Alcimos [START_REF][END_REF] . And as M. Mirkovic wrote, «the story of the twins can be regarded as firmly established in Rome in the third century, when the statue of the twins was erected, 296 B.C. and when this statue group formed a reverse type on one of the earliest issues of silver coins to be minted in Rome, in 269 or 268 B.C. 9 ».

A set of contradictions in the sources, albeit very late, about the mythical origins of Rome in fact leads to strongly doubt the tradition of Romulus and Remus. J. Poucet recalls thus the anachronisms of Romulean gesture that refer mainly to the beginnings of the Republic 20 . F. Coarelli, G. Capdeville and D. Briquel also analyzed a kind of rewriting of the gesture of Servius Tullius from the one of Romulus 2 , where we find the active role of shepherds to oust the king in place, Tarquin the Elder in the case of Servius Tullius, Amulius to that of Romulus. But also that Romulus and Servius are generated both by a god (Mars for one, Vulcan for the other) or a maid fertilized by an igneous phallus, that both have divided the Roman nation (Romulus between tribes and curies, Servius between the classes and centuries) and the Roman space (thirty parts for Romulus, four for Servius). The mixing of the two stories was such that when Plutarch evokes the conception of Romulus by an igneous phallus, he places it in the court of a king called Tarchetius 22 , direct echo of Tarquinius. May be in fact Romulus was simply a nickname given to Servius Tullius at an indeterminate time? Tradition ascribes indeed Servius Tullius building the Servian wall, increasing the area of the City and moving pomerium. He could therefore be seen as a new founder of the city, that is to say a new Romus, so «Rom-ulus».

Finally, another argument supports the idea of a mythological built around the struggle between Vulcians and the king of Rome Tarquinius. Thus Varro discussing the etymology of Coelius hill says that Caelius Vibenna came to rescue Romulus against King Titus Tatius 2 , as Dionysius of Halicarnassus 2 . Varro demonstrating equivalence between Romulus and Servius Tullius, because François Tomb, Claudius 25 and Festus 26 attests that Caelius Vibenna was present in the battle against the King of Rome, Cnaieus Tarquinius, with Mastarna/ Servius Tullius. In the writings of Arnobius, the patronyms of Servius and Romulus are even merged into Servulus 27 . The murder of Remus by Romulus, if it contains obvious parallels with the Babylonian prototype of the Herculean myth where Gilgamesh kills his double Enkidu 28 , could therefore also be read as the struggle between Servius Tullius and Tarquinius, in other words between the new Romus and the bloodline could have given an Etruscan name to Rome, the Tarquinii. The myth of the fratricidal battle between Romulus and Remus would be therefore a later construction, pinning a very common myth in antiquity on a real historical context.

It is even possible that the episode of the struggle between Porsenna, Tarquinius Superbus and the Romans is only a repetition of the battle shown in the François Tomb: Tacitus can write, in opposition of the Vul- Fest. 86L: «Regarding the Vicus Tuscus, several authors write that he owes his name to those of the Etruscans who, when King Porsenna raised the siege, remained in Rome and received the place to dwell. Or it is so called because it was occupied by the brothers of Vulci, Caele and Vibenna, which, is said, came to Rome with Maxtarna to the king Tarquinius. For M. Varro, because [Etruscans] were taken from Caelius in this place». 27 Arnob. 6.7. 28 For B. Sergent, the real founder of Rome would be merely Cacus, obscured by rewrites (Sergent 999, 78).

gate, that Rome surrendered to Porsenna 29 , and Festus gave two etymologies of Tuscus vicus, one linked to Porsenna, the other to Vibenna brothers. There has also been noted the inconsistency that there could be for Porsena to come help Tarquinii while Tarquinius Superbus is supposed to find refuge at the end of his life among the tyrant Aristodemus of Cumae, an enemy of Porsenna 0 . This duplication seems to respond to the desire to expunge the negative sides of Tarquinius and Servius Tullius. We find also on the Francois Tomb both characters could explain the insertion of Lars Porsenna in the Roman story. Thus, a LARTH ULTHES kills LARIS PAPATHNAS VELZNACH, in other words Lars Papatius of Volsinii. And Pliny the Elder made of Porsenna a king of Volsinii, who, calling lightning, delivers the city of Volta monster that had depopulated the countryside . Volta seems to correspond to ULTHES of the François Tomb 2 . Lars Papatius was clearly on the side of Cnaieus Tarquinius while Lars Voltius was with Mastarna and Vibenna brothers. The fact that these two individuals have paenomen or titles very similar, or even identical, probably contributed to the duality of the character of Porsenna in tradition.

Similary, strangely, the two Tarquinii, the Elder and Superbus, are named both Lucius, and some have suggested a duplication of the same figure . The names of Servius Tullius seem eerily familiar of Tullus Hostilius. As explained Macrobius, Hostilius could mean the enemy , and François Tomb show that Servius Tullius was enemy of roman king. Tullus Hostillius is a warrior king, as Servius Tullius. And, finally, how not to link MARCE CAMITLNAS, getting ready to kill CNEVE TARCHVNIES RVMACH on the François Tomb, with the son of Ancus Marcius murdering Tarquinius the Elder according to tradition 5 ? What is the connection between this MARCE CAMITLNAS and Marcus Camillus, whose names seems to be identical 6 ? Livy and Plutarch call Camillus «second founder of Rome 7 », and Plutarch was surprised that Camillus, «which won so many brilliant successes at the head of armies, who was named five times dictator, who celebrated four triumphs, and was listed as the second founder of Rome, was not one time consul».

Although it is very difficult to unravel the tangled lines of myth and history, we believe that the François Tomb provides a credible witness, at a time when Vulci was still not under Roman rule -there is general agreement today for dating François Tomb of the fourth century B.C., possibly in the second half of this century [START_REF][END_REF] -and that it can help to illuminate the versions provided by tradition. It seems the whole tradition of the kings of Rome, and may be the early days of Republic, in fact refers to one or more episodes described by the frescoes in the François Tomb, that of the struggle between Vulci and Etruscan cities of a possible coalition involving Rome, Volsinii, and two others cities that could be Sovana and Falerii. What was the date of that or those wars?

We often link the AVLE VIPINAS of François Tomb with a bucchero found in the Portonaccio sanctuary of Veii, dated to the first half of the sixth century B.C., and marked AVILE VIPIIENNAS [START_REF]assignment of a late chronology of events and historical characters doesn't deny early occupation of Rome attested by archaeology[END_REF] . Another inscription has been linked with Aulus Vibenna, the Rodin kylix marked AVLES V[I]PINAS / NAPLAN and dated to around 60 B.C. 0 . The origin of this cup is unknown, but it seems to come from Vulci . The chronological discrepancy between the inscription, apparently meaning «cup of Aulus Vibenna», and the supposed date of Aulus Vibenna according to tradition was a problem. J. Such dating obviously calls into question the whole chronology rooted from the annalistic tradition. It would nevertheless be consistent with a dating advanced sometimes for the Servian wall [START_REF][END_REF] . These fragments of walls scattered in Rome may not have evidentiary value and dating is discussed between the sixth and fourth century B.C. In both cases, this is unfortunately based on data from tradition. In 92, T. Franck tried to argue his late dating with the use of anathyrosis on the tuff blocks [START_REF]assignment of a late chronology of events and historical characters doesn't deny early occupation of Rome attested by archaeology[END_REF] . We must admit that the dating of this type of structure is problematic. However, the recent excavation of the walls of Vulci, similar to those of Rome, has allowed to date the construction in the second half of the fourth century B.C. 50 . So, a dating from the fourth century for the Servian wall would be strongly considered. Similarly, M. Humm saw in the reforms attributed to Servius Tullius anachronisms in fact referring to the fourth century B.C., as the military lifted (dilectus) and the introduction of tributum, but especially the timocratic organization 5 . We could cite Timaeus via Pliny attributed to Servius Tullius introducing the aes signatum 52 , monetary system that we consider today to have appeared earlier at the end of the fourth century B.C. 5 .

There is also the thorny issue of the dating of the cippus of the Lapis Niger, usually fixed to the middle of the sixth century B.C. As it's well known, the sector of the Lapis Niger in the Roman Forum was excavated in 898 and 899 by G. Boni 5 . He discovered a black marble paving covering a sort of altar in the shape of U, a cippus with an inscription in Greek characters and a tapered base (Fig. 2A). The dating of the U-shaped altar and inscribed cippus were the subject of heated debate between the currents called «hypercritic» and «fideist» in the early twentieth century 55 -and still today. Stratigraphically, it is not possible from the data published by G. Boni, to precisely date the altar U-shapped and inscribed cippus (Fig. ). We just know that the top of the altar and the cippus were included in that Boni described as a «votive deposit», containing animal bones but also fragments of weapons, ceramics, bronze statuettes and objects , fibulae and fragments of aes rude. Moreover, at the altar plinth level was found a bronze statuette identified as Vertumnus. Ceramics contained fragments of black bucchero datable between the seventh and fifth centuries, Etruscan vase in black figure, now generally dated from the fifth century, probably presigillata of the second or first century and finally a gray ceramic with green varnish. It can be ceramica invetriata, which may not be earlier than the first century 56 . The graphy of an inscription on this pottery and the colour of the varnish describe more returns to the High Empire. A ceramic described by L. Savignoni as «olla a doppio manico a nastro, portante in rilievo, sì nel diritto che nel rovescio, una faccia umana accennata in una maniera assai schematica e primitiva, che ricorda certi vasi di Troia» 57 (Fig. 2C) is in fact an anthropomorphic vase, as drawing given by D. Vaglieri shows 58 . This vase must be dated of the first half of the first century A. D. 59 Another object described by Boni as a «borchie a disco concoidale forato» (Fig. 2B) actually corresponds clearly to a mirror box of a type that appears in Greece until the end of the fifth century and is produced in Etruria until the late fourth century 60 . Boni also gives a brief description and photographs of discovered fibulae. It contains simple bow fibulae, datable to the eighth or seventh century, fibulae a navicella (obviously without decoration), dated to the seventh century, and fibulae with buttons datable from the seventh or sixth century. As already fell L. Savignoni in 900, the supposed «votive deposit» was made up of objects belonging to an extensive chronological period 6 , from the seventh or sixth century B.C. to the first or second century A.D. As, according to the stratigraphic sequence identified by Boni, this material was found in the layer including the top of the altar and inscribed cippus, and it is not credible to considering that a monument of the Forum is abandoned for more than seven centuries, this layer can be only reported embankment. It lets just say that earlier monuments Lapis Niger are also predate the first or second century A.D., the realization of the Lapis Niger pavement meanwhile be between the first and the late second century when Festus mentions. It seems logical to think that the earlier monuments of the Lapis Niger were destroyed in the great fire of Rome in 65, Tacitus indicating that the regia and the temple of Vesta were in flames 62 , explaining that the artefacts of the supposed «votive deposit» were mixed with ash.

To return to previous monuments of the Lapis Niger, only a comparative method can therefore help advance a date, even though we know from recent excavations, they are after the ninth-eighth centuries B.C. In 899, inscribed cippus was dated by Gamurrini of the sixth century by comparison with the inscription of Formello 6 . This dating was not accepted by all epigraphists, some attributing the inscription of the Lapis Niger between the sixth and the fifth centuries 6 . But today we know that if the artefacts found in the tumulus of Formello date for most of the late seventh or early sixth, more recent objects, from the late fourth or early third century, were subsequently filed in the grave 65 . Now, one of the two alphabets of the Formello vase has a nu wich graphy refers to neo-Etruscan 66 , that is to say the period of the most recent objects. The write in «boustrophedon» is meanwhile not synonymous with a high dating since found shapes on inscriptions of the fourth century 67 . If kappa largely disappears from the southern Etruscan inscriptions in the fourth century, it still in use on the margins of the area at that time, while the koppa remain in use in Latin. The paleography elements thus refer instead to the fourth century to the cippus of Lapis Niger (Fig. ). A parallel between the inscription of the cippus of the Lapis Niger and the altar of Tibur or the dedication to Castor and Pollux of Lavinium is often made. At Tibur, for now, the oldest remains are dated to the fourth century 68 . And for the dedication of Lavinium, it was dated of the sixth century compared with the cippus of Lapis Niger. with those of Lavinium, of similar shape and profile, where the oldest date back to that time. The chronology established in Lavinium is however questionable. For the altar XIII, considered the oldest, excavations have unearthed a little above the base an Attic vase fragment of black figure datable of 570-550 B. C. 69 But as in the case of the Lapis Niger, this element was in an embankment and lets just say that the layer covering the altar is later than mid-sixth century. Similarly, in the area of the altar VII-VIII-IX, the excavators discovered mixed objects dating differently, leaving puzzled R. Turcan on the chronological evolution pattern of thirteen altars 70 .

It is therefore impossible to establish a chronological comparison. The U-shaped altars discovered in 97 in Sant' Omobono are traditionally awarded, following the 97-975 surveys 7 , in the fourth century, but the dedication of the circular support is dated by an inscription around 26 B.C. 72 F. di Mario meanwhile offers to date the U-shaped altar of Ardea of the fourth-third centuries because of the tuff use 7 . Today, it is indeed generally accepted that the construction of the altar of the Lapis Niger date from the second half of the fourth century B. C. 7 The Lapis Niger is mentioned under this name by Festus, who also says that this place is considered as the tomb of Romulus, sometimes as that of Faustulus or to Hostilius 75 The traditional reading is «in columna ahenea» form the manuscripts P1, but the manuscripts P2 rell. gives «in columna aenea». The traditional reading is not possible because the latin locution in is following by ablative. The ablative of aena is aena also, but the ablative of AEneas is AEnea. So P2 rell. manuscripts must be following . 8 Plut. Romulus: «Ἐν δὲ τοῖς ἄλλοις λαφύροις καὶ χαλκοῦν ἐκόμισε τέθριππον ἐκ Καμερίας· τοῦτο δ' ἀνέστησεν ἐν τῷ ἱερῷ τοῦ Ἡφαίστου, ποιησάμενος ἑαυτὸν ὑπὸ Νίκης στεφανούμενον.» his achievements.» [START_REF] Halshs-02275960 | ἀνέθηκε τῷ Ἡφαίστῳ καὶ παρ´ αὐτῷ τὴν ἰδίαν ἀνέστησεν εἰκόνα ἐπιγράψας Ἑλληνικοῖς γράμμασι τὰς ἑαυτοῦ πράξεις[END_REF] Plutarch evokes finally a version of the death of Romulus killed by senators in the temple of Vulcan [START_REF][END_REF] . All texts are consistent with the discoveries of the Lapis Niger site and suggest that there was confusion between different mythical founders of Rome.

Finally, the name of the supposed deceased of the François Tomb, Vel Saties, is strongly similar to the legendary king Titus Tatius, supposed to have shared power with Romulus. We indicated above Varro's version that Caelius Vibenna would rescue Romulus against Titus Tatius. It's pretty obvious that Tatius replaced Tarchetius mentioned by Plutarch, in other words Tarquinius. The Romulean gesture also resonates at the frescoes in the François Tomb. According Dionysius of Halicarnassus, Romulus crushes initially Caenina soldiers 8 , a name that seems to be the compression of Caelius Vibenna, while the frescoes in the François Tomb present Caelius Vibenna prisoner. Then, in the Romulean gesture, the Sabinian reach, thanks to the betrayal of Tarpeia, to seize Rome, as Vibenna brothers and Mastarna will clearly do.

In fact, we believe that Vel Saties, Mastarna, Rasce, Titus Tatius, Romulus, Servius Tullius, are only one individual. We developed the reasons that encouraged us to this conclusion for Servius Tullius and Romulus. The equivalence between Mastarna and Servius Tullius has long been established from the testimony of Claudius speech in Lyon. Regarding Vel Saties and Mastarna and Rasce, we believe that this identity can be deduced because of the lack of praenomen of the last two characters. It would therefore be more titles than names. Long time ago, it was thus suggested that MACSTRNA might be a title similar to that of magister 85 . RASCE evokes archaic latin RECEI on the cippus of the Lapis Niger 86 , which identifies the classic latin rex 87 . It is generally accepted that the Francois Tomb is the tomb of the Saties family and that Vel Saties was its sponsor 88 . Vel Saties could therefore have been at first a kind of supreme magistrate to Vulci 89 and then «king» of Rome after the victory, as suggested by literary sources.

Our analysis leads us to think that the whole tradition of the kings of Rome is a late construction built from a real episode in which a mythological construction was developed. Apparently, different versions have existed in this episode. Faced with these different versions, the first historians of Rome have attempted to give a consistency by providing a chronological narrative with dates which had maybe another meaning. François Tomb does not tell a different story in fact, because it already placed in parallel the exploits of Vel Saties with those of Achilles. This concept of mythologizing was not peculiar to Etruscans, since Greeks clearly did the same as some could analyze for sculptures of the Parthenon or the Pergamon Altar 90 . We believe that the frescoes in the François Tomb depict the exploits of Vel Saties, probably a leading figure in Vulci, in his conquest of several Etruscan cities, including Rome, during the fourth century B.C. Echoed of these wars are visibly located in Dionysius of Halicarnassus, according to who Servius Tullius helped King Tarquinius to «completely subjugate the Etruscans 9 ». This character of Vel Saties seems to have been the basis for the construction of several legendary kings of Rome afterwards. We are conscious to deliver a version profoundly changing the vision of the early days of Rome, with a very late chronology for historical figures, and that it should be discussed. We hope that this analysis and the conclusions will be viewed calmly and without a priori. The

  In quo extinctus est Remus, et a Romi nomine Romani appellati. Ut autem pro Romo Romulus diceretur, blandimenti genere factum est, quod gaudet diminutione.» D. H. .72 quote Hellanicos Lesbos and Callias. Perhaps due to the possible etymology of Rome from ruma, which Festus and Plutarch (Romulus) gives the sense of udder.

  Heurgon therefore believed that it was counterfeit of artist wanting to assign his cup to a famous name2 . J. Poucet proposes for the Rodin cup a namesake of Aulus Vibenna, who was living in the fourth century B.C. . We could however overturn the idea of J. Poucet, especially as dating and origin assumed from the Rodin cup is consistent with the François Tomb. Besides, a mirror discovered in the Bolsena region, that is to say Volsinii, represent the two brothers, named by the inscriptions AVLE VIPINAS and CAILE VIPINAS surrounding Cacus, mythological character eminently linked to the Roman origins5 . And this mirror is dated from the third century B.C. by J. Poucet 6 , but the fourth century B.C. advanced by D. Briquel is more plausible 7 . Thus, war or wars reproduced on the François Tomb and its protagonists could be contemporaries of the fourth century B.C.

	29	Tac., Hist., .72..
	0	Ogilvie 965, 255.
		NH, 2.0.
	2	Bonfante 978, 0.
		Heurgon 99, 228.
		Macrob., Saturn., .6.
	5	Liv. .0; D. H. .72-7.
	6	Holloway 99, 6; Ridley 20, .
	7	Liv. 7.; Plut., Camillus.
	8	Cristofani 967.

9 TLE, sec. ed., 5. 0 TLE, sec. ed., 92. Lubtchansky 20, . an

  We find the story of the capture of Caelius Vibenna figured on the Tomb Francois in a fragment of Aulus Gellius preserved by Solinus (Polyhistor): «In relation to Aulus Gellius, Cacus thrown into chains by Tarchon, King of the Tuscans, who had sent to him by the King Marsyas, and giving for companion the Phrygian Megale, escaped, returned to the places he lived first, and then, with a large force, seized the banks of the Volturno and Campania». Obviously Caelius Vibenna was mythologized in Cacus.

	2	Heurgon 966, 55-528.
		Poucet 2000,97.
		CIE 085.
	5
	6	Poucet 2000, 95.
	7	Briquel 997, 69.
	8	On the tradition attributing this wall to Servius Tullius, cf. Platner 929, 50-55.
	9	Franck 92, 7.
	50	Moretti Sgubini 2006, 9; Moretti Sgubini, 2008, 76-77.
	5	Humm 2005, 255, 28-288.
	52	NH, 2.: «Servius rex primus signavit aes; antea rudi usos Romae Timaeus tradit».
	5	Sutherland 97, 7; Schmitt Prieur 200, 8.
	5	Boni 899; Boni 900; Boni 9, 5-7.
	55	Poretta 2005.

A.D., but

can also refer to late antiquity or middle age

  ..

	On the altar U-shaped of the Lapis Niger, it was also found confirmation of dating from the sixth century
	56	Guerini Mancini 2007, 20-2.
	57	Savignoni 900, .
	58	Vaglieri 90, 22.
	59	Benedetti 2007; Guerini Mancini 2007, 20, 229; Schindler-Kaudelka 202, 9-6.
	60	Lightfoot De Puma 20, 88-89; Anderson 98.
	6	Savignoni 900, 5.
	62	Tac., Ann., 5, 6.
	6	Gamurrini 899, 62.
	6	Thompson 92, 5; Grenier 92, 7.
	65	Michetti Van Kampen 20.
	66	Haynes 2000, 66.
	67	Emiliozzi 2008.
	68	Rous 200, 6.

  . According Porphyrion and Commentator Cruquianus, Varro wrote that the tomb of Romulus was behind the Rostra76 . Commentator Cruquianus and the Pseudo-Acron also mentions two lions above the tomb of Romulus 77 . For Dionysius, there was a single stone lion on the Forum over the grave of Faustulus. He also write that «kings» (which should match Romulus and Numa) buried Hostilius on the Forum «and erected a pillar with an inscription.» 78 In an other hand, it was traditionally understood that Cicero spoke of a bronze column placed behind the Rostra on which was engraved the foedus cassianum79 . Some manuscripts, however, give a different lesson that it should understand by «the column of AEneas» 80 , echoing Plutarch speaking Romulus dedicated a bronze chariot in the temple of Vulcan and «his own statue crowned by Victory» 8 while Dionysius wrote that Romulus settled near the bronze chariot dedicated to Vulcan «his own statue with an inscription in Greek characters giving the list of Hoc sic dicitur, quasi Romulus sepultus sit, non ad caelum raptus aut descerptus, nam Varro post Rostra fuisse sepultum Romulum dicit»; Commentator Cruquianus: «Varro pro rostris sepulcrum Romuli dixit; ubi etiam in huius rei memoriam duos leones erectos fuisse constat». 77 Pseudo-Acron: «plerique aiunt in Rostris Romulum sepultum esse et in memoriam huius rei leones duos ibi fuisse, sicut hodieque in sepulcris videmus, atque inde esse ut pro rostris mortui laudarentur». 78 D. H. 3.1.2.: «Οὗτος ὁ ἀνὴρ πολλοὺς συνδιενέγκας Ῥωμύλῳ πολέμους καὶ μεγάλα ἔργα ἀποδειξάμενος ἐν ταῖς πρὸς Σαβίνους μάχαις, ἀποθνήσκει καταλιπὼν παιδίον μονογενὲς καὶ θάπτεται πρὸς τῶν βασιλέων ἐν τῷ κρατίστῳ τῆς ἀγορᾶς τόπῳ στήλης ἐπιγραφῇ τὴν ἀρετὴν μαρτυρούσης ἀξιωθείς. Ἐκ δὲ τοῦ μονογενοῦς παιδὸς εἰς ἄνδρας ἀφικομένου καὶ γάμον ἐπιφανῆ λαβόντος υἱὸς γίνεται Τύλλος Ὁστίλιος ἀνὴρ δραστήριος, ὃς ἀπεδείχθη βασιλεὺς ψήφῳ τε πολιτικῇ διενεχθείσῃ περὶ αὐτοῦ κατὰ νόμους».

	69	Dury 98,.
	70	Turcan 98, 55-56.
	7	Smith 205, 7.
	72	CIL VI 0895.
	7	Di Mario 2007, 8-85.
	7	Smith 205, 76.
	75	Fest. 2: «Niger lapis in Comitio locum funestum significat, ut ali, Romuli morti destinatum sed non
	usu ob in [venisse ut ibi sepeliretur, sed Fau]stulum nutri[cium eius, ut ali, dicunt Hos]tilium avum Tu[lli
	Hostili]».
	76 Porphyrion: «79 Cic. Pro Balbo 5-5: «Cum Latinis omnibus foedus esse ictum Sp. Cassio Postumo Cominio consu-
	libus quis ignorat ? Quod quidem nuper in columna AEnea meminimus post rostra incisum et perscriptum
	fuisse».
	80

Travertine crepidoma

Name of inscription

Language

Writting direction Graphy of inscription Fig. : Comparison between inscription of Lapis Niger cippus and other archaic italian inscriptions