Denis Serre

Non-commutative standard polynomials applied to matrices

Keywords: MSC classification : 15A24, 15A27, 15A60 standard polynomial, random matrices

The Amitsur-Levitski Theorem tells us that the standard polynomial in 2n non-commuting indeterminates vanishes identically over the matrix algebra M n (K). For K = R or C and 2 ≤ r ≤ 2n -1, we investigate how big S r (A 1 , . . . , A r ) can be when A 1 , . . . , A r belong to the unit ball. We privilegiate the Frobenius norm, for which the case r = 2 was solved recently by several authors. Our main result is a closed formula for the expectation of the square norm. We also describe the image of the unit ball when r = 2 or 3 and n = 2.

The problem. First results

Let r ≥ 2 be an integer. The standard polynomial in r non-commuting indeterminates x 1 , . . . , x r is defined as usual by S r (x 1 , . . . , x r ) := ∑ {ε(σ)x σ(1) x σ(2) • • • x σ(r) : σ ∈ S r }, where S r is the symmetric group in r symbols and ε is the signature. Each monomial is a word in the letters x j , affected by a sign ±1. Despite its superficial similarity with the determinant of r × r matrices, S r is a completely different object: on the one hand, its arguments are non-commuting indeterminates, on the other hand, there are only r indeterminates instead of the r 2 entries of a matrix. We list here elementary properties of S r :

1. S r is alternating.

2. S r+1 (x 1 , . . . , x r+1 ) = ∑ i (-1) i+1 x i S r ( xi ), where xi := (x 1 , . . . , x i-1 , x i+1 , . . . , x r+1 ).

3. If r is even, and an x i commutes with all other x j 's, then S r (x 1 , . . . , x r ) = 0. Mind that this is false if r is odd.

The first polynomial x 1 x 2x 2 x 1 of the list is the commutator. When applied to the elements of an algebra A, it leads us to distinguish between commutative and non-commutative algebras. More generally, the polynomials S r measure somehow the degree of non-commutatitivity of a given algebra.

A classical theorem tells us that for a given matrix A ∈ M n (C), the commutator S 2 vanishes identically over the algebra A, A * (in other words, A is normal) if and only if A is unitarily diagonalizable. It is less known that S 2 vanishes identically over the algebra A, A * if and only if A is unitarily blockwise diagonalizable, where the diagonal blocks have at most size × ; see Exercise 324 in [START_REF] Serre | Matrices : Theory & applications[END_REF].

In addition, we have the theorem of Amitsur and Levitski [START_REF] Amitsur | Minimal identities for algebras[END_REF], of which an elegant proof has been given by Rosset [START_REF] Rosset | A new proof of the Amitsur-Levitski identity[END_REF].

Theorem 1.1 (Amitsur-Levitski.) Let K be a field (a commutative one, needless to say). The standard polynomial S 2n of degree 2n vanishes identically over M n (K). However the standard polynomials of degree less than 2n do not vanish identically.

In the sequel, we focus on the algebra M n (K) (K = R or C) of real or complex matrices. A norm over M n (K) is submultiplicative if it satisfies AB ≤ A B . The main examples are operator norms A := sup

x∈K n , x =0 |Ax| |x| ,
where | • | is a given norm over K n . One often says that • is induced by | • |. In particular, • 2 is the norm induced by the standard Euclidian/Hermitian norm. We are also interested in the Frobenius norm

A F := ∑ i, j |a i j | 2 ,
which is not induced, because

I n F = √ n > 1. Nevertheless, it is submultiplicative. A more exotic norm is the numerical radius h(A) := sup |x * Ax| x 2 2 : x ∈ C n \ {0} .
This is not a submultiplicative norm but it satisfies h(A k ) ≤ h(A) k . The numerical radius is therefore a super-stable norm.

The general question that we address is to find precise bounds of

S r (A 1 , . . . , A r ) ∏ r j=1 A j .
This can be cast into two sub-problems. On the one hand, we are interested in the best constant

C = C(r, n) satisfying S r (A 1 , . . . , A r ) ≤ C r ∏ j=1 A j , ∀A 1 , . . . , A r ∈ M n (K).
On the other hand, we may ask what is a typical ratio.

For the first task, we look for the smallest ball containing the image when each argument is taken out of the unit ball of M n (K). We may even ask for an accurate description of this image. For instance, we shall prove that for 2 × 2 matrices (i.e. n = 2) and the Frobenius norm, the image of the unit ball under S 2 (the commutator) is the ball of radius √ 2, while the image under S 3 is an ellipsoid. In order to tackle the second problem, we shall compute the closed form of

E( S r (A 1 , . . . , A r ) 2 F ) E(∏ r j=1 A j 2 F )
, when A 1 , . . . , A r are chosen independently and uniformly in the Frobenius unit ball. Strangely enough, our proof makes use of the Amitsur-Levitski Theorem.

When using a submultiplicative norm, the trivial bound

S r (A 1 , . . . , A r ) ≤ r! r ∏ j=1 A j , ∀A 1 , . . . , A r ∈ M n (K),
suggests to work with the normalized polynomial

T r := 1 r! S r , which now satisfies T r (A 1 , . . . , A r ) ≤ r ∏ j=1 A j , ∀A 1 , . . . , A r ∈ M n (K).
However this inequality ignores the cancellations that are likely to occur because of the signs ε(σ) in the definition of T r . For instance, the left-hand side vanishes whenever r ≥ 2n. For this reason, we are interested in the norm τ(r, n) of T r , defined by

τ(r, n) := sup T r (A 1 , . . . , A r ) ∏ r j=1 A j : A 1 , . . . , A r ∈ M n (K) \ {0 n } .
Alternatively,

τ(r, n) := sup{ T r (A 1 , . . . , A r ) : A 1 , . . . , A r ∈ M n (K), A 1 , . . . , A r ≤ 1} = C(r, n) r! .
Our definition above depends on the chosen norm on M n (K), and our notation should indicate this dependance. In this sense, the Frobenius norm and the operator norm • 2 yield the numbers τ F (r, n) and τ 2 (r, n), respectively. As said above, we always have τ(r, n) ≤ 1. However this bound is very poor as for instance τ(2n, n) = 0. Besides, one trivially has τ(1, n) = 1. The first non-trivial case comes when r = 2, where S 2 is the commutator. When n = 1, clearly τ(2, 1) = 0. But for n ≥ 2, the result depends on the norm we are considering. For instance, we know that

τ F (2, n) = √ 2 2 ,
the case n = 2 being due to Böttcher and Wenzel [START_REF] Böttcher | How big can the commutator be and how big is it typically?[END_REF], and the case n = 3 to László [START_REF] László | Proof of Böttcher and Wenzel's conjecture on commutator norms for 3-by-3 matrices[END_REF]. The equality for every n was conjectured in [START_REF] Böttcher | How big can the commutator be and how big is it typically?[END_REF] and proved by [START_REF] Vong | Proof of Böttcher and Wenzel's conjecture[END_REF], and independently by [START_REF] Lu | Proof of the normal scalar curvature conjecture[END_REF] for K = R and [START_REF] Audenaert | Variance bounds, with an application to norm bounds for commutators[END_REF] for K = C. See also [START_REF] Böttcher | The Frobenius norm and the commutator[END_REF]. The situation is significantly different with the standard operator norm, induced by the Hermitian norm. It is known (

[4], Example 5.2) that τ 2 (2, n) = 1, ∀n ≥ 2.
All the subtlety in the upcoming analysis comes from cancellations. We shall prove several inequalities that hold true for every submultiplicative norm. The simplest one, τ(r +s, n) ≤ τ(r, n)τ(s, n), is not sharp. But it suggests to extend our study to operators in infinite dimension. An interesting class in this respect is that of Hilbert-Schmidt operators, whose norm generalizes the Frobenius norm. We thus define the upper bound

τ F (r) := sup n≥1 τ F (r, n) = lim n→+∞ τ F (r, n).
Again, one has τ F (r + s) ≤ τ F (r)τ F (s). Since Theorem 1.1 is lost as n → +∞, we have τ F (r) > 0 for every r, and it becomes interesting to compute the rate of cancellation

ρ F := lim r→+∞ τ F (r) 1/r = inf r≥1 τ F (r) 1/r .
Because of Böttcher-Wenzel's inequality, we have τ F (2) = √ 2/2 and ρ F ≤ 2 -1/4 . One of our results from below is the improved bound

ρ F ≤ τ F (2k + 1) 1/2k , ∀k ≥ 1.
Theorem 2.1 Let n ≥ 2 be given. Then

(1) h([A, B]) ≤ 4h(A) h(B), ∀A, B ∈ M n (C).
The constant 4 is the smallest possible.

Proof If M ∈ M n (C), we define the real and imaginary parts of M by

ℜM = 1 2 (M + M * ) ∈ H n ℑM = 1 2i (M -M * ).
Then

h(M) = sup x sup ϕ ℜ(e iϕ x * Mx) x 2 2 = sup x sup ϕ x * ℜ(e iϕ M)x x 2 2 = sup ϕ sup x x * ℜ(e iϕ M)x x 2 2 = sup ϕ∈R/2πZ ℜ(e -iϕ M) 2 ,
where the supremum is actually a maximum, because ϕ → ℜ(e -iϕ M) 2 is continuous over the com-

pact set R/2πZ. Let A, B ∈ M n (C) be given. We write M = [A, B] and chose a ϕ with h(M) = ℜ(e -iϕ M) 2 . Let us denote X = ℜ(e -iϕ A), Y = ℑ(e -iϕ A), Z = ℜB and T = ℑB. From e -iϕ M = [X + iY, Z + iT ] = [X, Z] + [T,Y ] + i([X, T ] + [Y, Z])
and the fact that the commutator of Hermitian matrices is skew-Hermitian, we derive

ℜ(e -iϕ M) = i([X, T ] + [Y, Z]). We infer h([A, B]) = [X, T ] + [Y, Z] 2 ≤ [X, T ] 2 + [Y, Z] 2 ≤ 2( X 2 • T 2 + Y 2 • Z 2 ).

This proves immediately (1).

The constant 4 is attained for the choice

A = 0 1 0 0 ⊕ I n-2 , B = 0 0 1 0 ⊕ I n-2 ,
for which

[A, B] = 1 0 0 -1 ⊕ I n-2 .
We have

h(A) = h(B) = 1 2 , h([A, B]) = 1.
This completes the proof.

Inequality ( 1) is of little interest; a similar proof yields the same optimal factor 4 if one replace the commutator by AB + BA: h(AB + BA) ≤ 4 h(A)h(B).

Consequently the numerical radius does not detect the possible cancellations, in contrast to the Frobenius norm.

When considering polynomials of higher degree, the same proof provides the inequality

h(S r (A 1 , . . . , A r )) ≤ 2 r-1 r! r ∏ j=1 h(A j ),
which seems to be very far from optimal.

General inequalities

We assume for the remainder that the norm is submultiplicative:

AB ≤ A • B .

Two formulae about S r

It is well-known that

S r (x 1 , . . . , x r ) = r ∑ i=1 (-1) i-1 x i S r-1 ( xi ),
where xi is obtained from x by deleting x i . The following formula generalizes the one above.

Proposition 3.1 Let r = r 1 + • • • + r be a partition into positive integers (k → r k need not to be monotonous). Denote by P(r 1 , . . . , r ) the set of ordered partitions of [[1, r]] = {1, . . . , r} into subsets I 1 , . . . , I such that |I k | = r k . If I = (I 1 , . . . , I ) ∈ P(r 1 , . . . , r ), and

I k = {i k,1 < • • • < i k,r k }, let us denote α(I)
the signature of the permutation ρ I defined by

ρ I ( j + r 1 + • • • + r k-1 ) = i k, j , 1 ≤ j ≤ r k , 1 ≤ k ≤ . Then (2) S r (x 1 , . . . , x r ) = ∑ I∈P(r 1 ,...,r ) α(I)S r 1 (X I 1 ) • • • S r (X I ),
where

X I k = (x k,1 , . . . , x k,r(k) ).
Proof Every σ ∈ S r can be factorized as

σ = (σ 1 × • • • × σ ) • ρ I
with I ∈ P(r 1 , . . . , r ) and σ k ∈ S r k . For this, take I 1 = σ({1, . . . , r 1 }), etc... This decomposition is unique; actually, one has

|S r | = r! = r 1 ! • • • r ! r r 1 r -r 1 r 2 • • • r r = |S r 1 × • • • × S r | × card P(r 1 , . . . , r ).
The signature of σ is obviously the product of all the signatures of σ 1 , . . . , σ , ρ I . Therefore the righthand side of (2) contains exactly once the monomial x σ(1) • • • x σ(r) , with the sign

α(I)ε(σ 1 ) • • • ε(σ ) = ε(σ).
Hence both sides are equal to each other. This proves the proposition.

Let 1 ≤ k < m ≤ be given, and denote ς the transposition (k, m). The ordered partition I ς is defined by

I ς k = I m , I ς m = I k
and I ς j = I j otherwise; mind that r k and r m have been flipped. We have α(I ς ) = (-1) r k r m α(I). Hence we derive from (2) the following identities.

Proposition 3.2 Let r = r 1 + • • • + r . Then, (3)
∑ I∈P(r 1 ,...,r ) α(I)S (S r 1 (X I 1 ), . . . , S r (X I )) = ! S r (X), if every r k is odd, 0,
otherwise.

The case r = 3 = 2 + 1 (hence = 2) of ( 3) is the Jacobi identity for the commutator.

Proof Let F(X) denote the left-hand side. The transposition i ↔ j induces an involution S over P(r 1 , . . . , r ) and we have α(SI) = -α(I). Therefore

F(x 1 , . . . , x i-1 , x j , x i+1 , . . . , x j-1 , x i , x j+1 , . . . , x r ) = -F(X).
Since F is a homogenenous polynomial, linear in each of the indeterminates, we deduce that F equals a multiple of S r . In order to determine the constant factor, we may focus on the monomial

X ↑ = x 1 x 2 • • • x r . This monomial occurs in the sum each time the interval [[1, r]] is split into consecutive intervals I ν(1) , . . . of respective lengths r ν(1)
, where ν ∈ S is arbitrary. It is accompanied by the sign α(I ν(1) , . . .)ε(ν). If all the r k are odd, then α(I ν(1) , . . .)ε(ν) = +1 for every ν and all the terms have a positive sign, whence the factor !.

Suppose on the contrary that some r k is even, say r 1 , and let us denote ς the transposition 1 ↔ 2. Then S is the disjoint union of A and ςA . If ν ∈ A , then α(I ςν(1) , . . .) = α(I ν(1) , . . .), while ε(ςν) = -ε(ν). Therefore half of the occurences of X ↑ have a positive sign, and half of them have a negative sign, whence the second formula and the proposition.

Submultiplicativity of τ(•, n)

The identity (2) allows us to derive a bound for τ(r, n) in terms of τ(s, n) and τ(rs, n). If A 1 , . . . , A r are in the unit ball, then each product in the sum above is bounded by

s!(r -s)!τ(s, n)τ(r -s, n) r ∏ j=1 A j .
Since there are r!/(s!(rs)!) terms in the sum, we immediately obtain

T r (A 1 , . . . , A r ) ≤ τ(s, n)τ(r -s, n) r ∏ j=1 A j ,
from which we derive the following estimate.

Proposition 3.3 If the norm is submultiplicative, then τ(r, n) ≤ τ(s, n)τ(r -s, n), ∀1 ≤ s < r.
In particular, τ F (r) ≤ τ F (s)τ F (rs). It is well known that for such a submultiplicative sequence, the sequence u r := τ F (r) 1/r converges to its lower bound. We call this limit the rate of cancellation and denote it by

ρ F = lim r→+∞ τ F (r) 1/r = inf r τ F (r) 1/r .
Because of τ(r, n) ≤ 1, we infer the next statement.

Corollary 3.1 As a function of its first argument r, τ(r, n) is non-increasing.

Improved submultiplicativity

The formula (2) has the drawback that it still involves the product of matrices, for which we have no gain in norm, since we cannot improve AB ≤ A B in general. To go further, we use (3), which involves operators T but no single matrix product. It can be recast as

∑

I∈P(r 1 ,...,r ) α(I)T (T r 1 (X I 1 ), . . . , T r (X I )) = cardP(r 1 , . . . , r ) • T r (X), if every r k is odd, 0, otherwise.
For instance, we have

8T 4 (A, B,C, D) = [A, T 3 (B,C, D)] + [B, T 3 (A, D,C)] +[C, T 3 (A, B, D)] + [D, T 3 (A,C, B)],
an identity which immediately gives

τ(4, n) ≤ τ(3, n)τ(2, n).
The latter inequality is tighter than τ(4, n) ≤ τ(2, n) 2 given by Proposition 3.3, since r → τ(r, n) is non-increasing.

More generally, we have

(5) 4rT 2r (A 1 , . . . , A 2r ) = 2r ∑ i=1 (-1) i+1 [A i , T 2r-1 ( Âi )],
with the classical notation Âi for the list A 1 , . . . , A i-1 , A i+1 , . . . , A r in which A i has been omitted. We deduce immediately

τ(2r, n) ≤ τ(2r -1, n)τ(2, n).
Using again submultiplicativity, this yields

τ(2r, n) ≤ τ(2, n)τ(3, n)τ(2r -4, n).
By induction, we infer

τ(4k, n) ≤ τ(2, n) k τ(3, n) k , τ(4k + 2, n) ≤ τ(2, n) k+1 τ(3, n) k ,
where submultiplicativity alone only grants

τ(2 , n) ≤ τ(2, n) .
More generally, (4) yields the inequality

(6) τ(r, n) ≤ τ( , n)τ(r 1 , n) • • • τ(r , n).
This improves the submultiplicativity in the following way: let us define a new sequence θ by shifting the first argument θ(s, n) := τ(s + 1, n).

Then ( 6) becomes

(7) θ(s, n) ≤ θ(s 0 , n)θ(s 1 , n) • • • θ(s , n),
for s = s 0 + • • • + s , whenever s 1 , . . . , s are even (s 0 may be odd). This is exactly submultiplicativity, up to the restriction on parity. In particular, the sequence µ(k, n) := θ(2k, n) is submultiplicative. Likewise, let us denote θ(s) = τ F (s + 1). Clearly, τ F (r) 1/r and θ(s) 1/s have the same limit, which must be equal to the infimum of the θ(2k) 1/2k . Combined, this delivers the following result.

Proposition 3.4 We have ρ F ≤ τ F (2k + 1) 1/2k , ∀k ∈ N.
The above bound improves, for odd arguments, the one we had before, namely τ(r) 1/r .

The Frobenius norm

The Frobenius norm, which it is the one studied in [START_REF] Böttcher | How big can the commutator be and how big is it typically?[END_REF], has several advantages for our study. First of all, it enjoys better bounds than either the operator norm, or the numerical radius: the limit n → +∞ seems non trivial. Next, it is a smooth, regular norm, and it is possible to use differential calculus when studying r-uplets which realize the norm of T r . At last, we have a duality principle, based on the inner product of two matrices

M F = sup{ℜTr (M * N) : N F ≤ 1}, whence [A, B] F = sup{ℜTr ([A, B]C) : C ∈ B F },
where B F denotes the unit ball for the Frobenius norm. Since 3Tr

([A, B]C) = Tr S 3 (A, B,C), we also have (say that K = R) τ F (2, n) = sup{Tr T 3 (A, B,C) : A, B,C ∈ B F }.
We warn the reader that this identity extends only to the even numbers r:

τ F (2s, n) = sup{Tr T 2s+1 (A 1 , . . . , A 2s+1 ) : A 1 , . . . , A 2s+1 ∈ B F }, while Tr T 2s (A 1 , . . . , A 2s ) ≡ 0.
The latter identity expresses the fact that T 2s (A 1 , . . . , A 2s ) can be written as a sum of commutators, an idea developed in Section 3.3. The vanishing of these traces is used in the proof by Rosset [START_REF] Rosset | A new proof of the Amitsur-Levitski identity[END_REF] of the Amitsur-Levitski Theorem.

Average estimate

Let us endow M n (R) with the usual probability measure, where the entries a i j are Gaussian independent variables:

dµ(A) = 1 V n e -A 2 da 11 • • • da nn , A 2 = Tr (A T A).
Here V n is a normalizing factor. For instance, we have

E( A 2 ) = n 2 ´x2 e -x 2 dx ´e-x 2 dx =: n 2 m 2 ,
where m 2 = 1 2 is the second moment of the Gaussian. We wish to calculate the expectation of S (A 1 , . . . , A ) 2 when A 1 , . . . , A are independent matrices. This amounts to calculating the average of S (A 1 , . . . , A ) 2 when A 1 , . . . , A all have unit norm.

Lemma 4.1 As a function of the size n of the matrices, the expression

n 2 E[ S (A 1 , . . . , A ) 2 ] is a polynomial. Proof Denoting A π = A π(1) • • • A π( ) , we have E[ S (A 1 , . . . , A ) 2 ] = E[Tr S (A 1 , . . . , A ) T S (A 1 , . . . , A )] = ∑ π∈S E[ε(π)Tr (A π ) T S (A 1 , . . . , A )] = ∑ π∈S E[ε(π)Tr (B id ) T S (B π -1 (1) , . . . , B π -1 ( ) )] = ! E[Tr (B id ) T S (B 1 , . . . , B )] = ! ∑ π∈S ε(π)E[Tr (B id ) T B π ].
Given π ∈ S , we have

Tr (A id ) T A π = a α -1 α • • • a 2 α 1 α 2 a 1 β 1 α 1 a π(1) β 1 β 2 • • • a π( ) β α ,
with Einstein's convention of summation over repeated indices. We stress that β 1 plays the role of an α 0 , and α plays the role of a β +1 . When taking the expectation of a given monomial, we obtain either (m 2 ) or 0, according to whether every entry shows up with a square, or not. A non-zero contribution happens when

(β k , β k+1 ) = (α π(k)-1 , α π(k) ),
or equivalently if

(8) α π(k)-1 = α π(k-1) = β k
for every k = 1, . . . , + 1. Hereabove we have to extend π by ( 9)

π(0) = 0, π( + 1) = + 1.
Let G π be the graph whose vertices are the indices 0 ≤ j ≤ for the α s, and the indices 1 ≤ k ≤ + 1 for the β s. Thus G π has 2( + 1) vertices. The edges correspond to every equality of the form either j = π(k -1) or j = π(k) -1. This includes the edge between the vertices j = 0 and k = 1, and the edge between the vertices j = and k = + 1. Notice that j and k may be connected by two edges, in case π(k) -1 = π(k -1) = j.

Given the permutation π, many among the monomials have zero expectation. The remaining ones have expectation m 2 ; they are parametrized by the maps

a α -1 α • • • a 2 α 1 α 2 a 1 β 1 α 1 a π(1) β1β2 • • • a π( ) β α
G π (α,β) -→ [[1, n]]
that are constant on each connected component. The number of such maps (α, β) is n N(π) , where N(π) is the number of connected components of G π . Therefore, we obtain

E[Tr (A id ) T A π ] = n N(π) m 2
and

E[ S (A 1 , . . . , A ) 2 ] = ! m 2 ∑ π∈S ε(π)n N(π) =: ! P (n)m 2 ,
for some P ∈ Z[X]. We infer ( 10)

E[ S (A 1 , . . . , A ) 2 ] = ! P (n) n 2 E[ A 1 2 • • • A 2 ].
We are now going to express the polynomial P in closed form. To begin with, we note that always N(π) ≥ 1, and therefore P (0) = 0. Actually, the quantity N(π) can further be restricted.

In addition,

(N(π) = + 1) ⇐⇒ (π = id).

Proof The edges of π always link an α-index j to a β-index k. In addition, every vertex has valence 2. Therefore a connected component is an even cycle in which the j's and the k's alternate. We may therefore construct a graph J π , whose vertices are the indices j ∈ [[0, ]] and there is an edge between j and j if { j, j } = {Sπk, πSk} for some k, where S is the shift k → kover [ [1, + 1]]. For completeness, we define S(0) = + 1, so that S is a permutation of [[0, + 1]]. This amounts to saying that J π is the graph associated with the permutation ρ := πSπ -1 S -1 in [[0, ]] (because of ( 9), + 1 is a fixed point of ρ). For instance, the examples displayed in Figure 1 yield ρ = (031) (left) or ρ = (021) (right); in both cases, the resulting J π consists of a three-node cycle and an isolated vertex. Every connected component

( j 1 , k 1 , j 2 , k 2 , . . .) of G π corresponds to a connected component ( j 1 , j 2 , . . .) of J π . Thus N(π) is nothing but the number of orbits of ρ.
Because ρ is a commutator, it is an even permutation of [[0, ]]. The number N even of its cycles of even length (these are odd permutations !) is thus even. Counting the elements modulo 2, we deduce that

N odd ≡ + 1. Hence N(π) = N even + N odd ≡ + 1.
If π is the identity, then ρ is the identity too and [START_REF] Vong | Proof of Böttcher and Wenzel's conjecture[END_REF], we deduce that π is the identity.

N(π) = + 1. Conversely, if N(π) = + 1, we have ρ = id, that is π(k -1) = π(k) -1 for every k ∈ [[1, + 1]]. With
Corollary 4.1 The monic polynomial P is odd if is even, and it is even if is odd. It has the form P (X) = X +1 + l.o.t, without constant term.

Finally, we invoke the theorem of Amitsur and Levistki. If 2n ≤ , we have P (n) = 0, thus Xn divides P (X). Because of the parity, we infer that X 2n 2 divides P . Hence

P (X) = Q (X)(X 2 -1)(X 2 -4) • • • (X 2 -r 2 ), r = 2 .
By the corollary, we know that Q is a monic polynomial of degree + 1 -2r. If is even, then Q is odd, of degree 1, hence equals X. If is odd, Q is even of degree 2, vanishes at 0 and is therefore equal to X 2 . Summarizing these thoughts, we obtain the desired relation.

Theorem 4.1 Let A j be i.i.d. Gaussians. Then,

E[ S (A 1 , . . . , A ) 2 ] = ! P (n) n 2 E[ A 1 2 • • • A 2 ] with P 2k (X) = X(X 2 -1 2 ) • • • (X 2 -k 2 ), P 2k+1 (X) = X 2 (X 2 -1 2 ) • • • (X 2 -k 2 ).

Asymptotic properties

We focus on the even case, = 2k which behaves a little nicer than the odd one, even if their asymptotics are quite similar.

Let us denote the ratio given by Theorem 4.1 by

ω(k; n) = (2k)! P 2k (n) n 4k .
First fix n and vary k from 1 to n -1. We have

ω(1; n) = 2 n 1 - 1 n 2 ,
a formula already known to Böttcher & Wenzel [START_REF] Böttcher | How big can the commutator be and how big is it typically?[END_REF]. Then

ω(k; n) ω(k -1; n) = 2k(2k -1) n 4 (n 2 -k 2 ). Denoting t = k n , this reads ω(k; n) -1; n) = 4t t - 1 2n (1 -t 2 ) < 4t 2 (1 -t 2 ) ≤ 1.
Therefore the sequence k → ω(k; n) is strictly decreasing. It has a critical point for

t n ∼ √ 2 2 (that is k n ∼ n √ 2
2 ). The decay is faster for small values of k, and also for k approaching n. We next investigate the behaviour of ω(k; n) as n → +∞, while k/n → t ∈ (0, 1). We start from

ω(k; n) = (2k)! (n + k)! n 4k (n -k -1)! .
The Stirling formula yields

ω(k; n) ∼ 2n -4k 2k e 2k n + k e n+k e n -k -1 n-k-1 πk n + k n -k -1 = 2 4k 2 (n + k)(n -k -1) e 4 n 4 k n + k n -k -1 n n -k -1 e πk n + k n -k -1 . Hence log ω(k; n) ∼ dk, where d = d(t) := log 4t 2 (1 -t 2 ) -4 + 1 t log 1 + t 1 -t .
A simple calculation gives that τ is increasing over (0, x) and decreasing over (x, 1), where x ≈ 0.95 ∈ (0, 1) is the unique root of (11

) 4t = log 1 + t 1 -t . Then max d(t) = d(x) = log 4x 2 (1 -x 2 ) < 0.
In conclusion, n → ω( tn ; n) decays exponentially fast, with a rate not larger than 4x 2 (1x 2 ), a number strictly less than one.

5 The law of distribution of the commutator (n = 2)

We continue our investigation for the Frobenius norm. We already know that

[A, M] F ≤ √ 2 A F M F ,
where the constant √ 2 is optimal [START_REF] Böttcher | How big can the commutator be and how big is it typically?[END_REF]. As noted by several authors, we may always restrict to the hyperplane H of zero-trace matrices, because on the one hand Therefore we have the commutative diagram

(A, M) ---→ √ 2 2 [A, M] L   L   (x, y) ---→ F(x × y)
where L is an isometry, and

F :   a b c   →   a c b   is an isometry in R 3 . Because the diagram's bottom line is onto B 3 , we already see that (A, M) ∈ B H × B H → √ 2 2 [A, M] ∈ B H is surjective, too.
Let us denote dν the pushforward of dA ⊗ dM, and dµ that of dx ⊗ dy. Because of the equivariance R(x × y) = (Rx) × (Ry) under every rotation R, we know that dµ is rotationally invariant. Therefore

ˆBH f (Z)dν(Z) := ¨BH ×B H f √ 2 2 [A, M] dA dM = ¨B3 ×B 3 f √ 2 2 [Lx, Ly] dx dy = ¨B3 ×B 3 f (LF(x × y)) dx dy = ˆB3 ( f • L • F)(z) dµ(z) = ˆB3 ( f • L)(Fz) dµ(z) = ˆB3 ( f • L)(z) dµ(z).
This shows that the distribution of (A, M) → √ 2 2 [A, M] over B H × B H is exactly the same as the distribution of (x, y) → x × y over the unit ball B 3 , up to the identification provided by the isometry L.

Proposition 5.2 The map (x, y) → x × y from B 3 × B 3 is onto B 3 . The pushforward dµ of dx ⊗ dy is radial, with density h(ρ)dρdω
where dω is the normalized area over S 2 , and

h(ρ) = 9ρ 2 1 -ρ 2 ρ + 2 arctan 1 + ρ 1 -ρ -π .
This instantly transfers to the commutator setting. We infer h(ρ) = 9 ρ ¨0<r,s<1,rs>ρ rs drds

r 2 s 2 -ρ 2 .
We integrate first with respect to s, which varies from ρ/r to 1. From d r 2 s 2ρ 2 = r 2 s ds Let us parametrize

r = ρ a 2 + 1 a 2 -1 , r 2 -ρ 2 = ρ 2a a 2 -1 , a > 1 + ρ 1 -ρ .
We obtain

h(ρ) = 9ρ 2 ˆ+∞ 1+ρ 1-ρ 4a 2 a 2 -1 da a 2 -1 - da a 2 + 1 = 9ρ 2 - 2a a 2 -1 -2 arctan a +∞ 1+ρ 1-ρ
, which is the required formula.

Let us compare the distribution dν with the uniform distribution dM = 3ρ 2 dρdω over B H . We have dν = g(ρ)dM, where

g(ρ) = 3 1 -ρ 2 ρ + 2 arctan 1 + ρ 1 -ρ -π .
Clearly, g(0) = +∞, g(1) = 0 and g (ρ) = -3ρ -2 1ρ 2 < 0. Therefore g is monotonous decreasing. Large commutators (with norms ≈ √ 2) are rare while small ones are likely. This phenomenon becomes stronger for larger matrix sizes n. On the one hand, the average ratio for [A, B] 2 F / A F B F is 2 n 1 -1 n 2 (see [START_REF] Böttcher | How big can the commutator be and how big is it typically?[END_REF] or Paragraph 4.1). On the other hand, when n ≥ 3, very few trace-less matrices of norms √ 2 can be written as [A, B] with A F = B F = 1; see Section 4 in [START_REF] Böttcher | The Frobenius norm and the commutator[END_REF], or [START_REF] Cheng | Commutators with maximal Frobenius norm[END_REF].

6 The law of distribution of S 3 over M 2 (R)

The situation changes significantly when passing from r = 2 (the commutator) to r = 3. On the one hand, the addition of tI n becomes harmful:

Figure 1 :

 1 Figure 1: The graph G π when = 3. The α-indices, from 0 to 3, are outer; the β-indices, from 1 to 4, are inner. Left: the cycle (123). Right: the transposition (12). In both cases, the graph has two connected components. For the identity, it should have four of them.

Proposition 4 . 1

 41 For every π ∈ S , we have 1 ≤ N(π) ≤ + 1 and N(π) ≡ + 1 mod 2.

2 2 5 . 1 2 [

 2512 [A + tI n , M] = [A, M] = [A, M + sI n ] for every s,t ∈ R, and on the other hand the projection A → A -1 n (Tr A)I n diminishes the Frobenius norm. We denote by B = B H the unit ball in H and ask two questions: • What is the range of the map (A, M) → √ [A, M] over B H × B H ? We already know that it is contained in B H . • What is the distribution law of [A, M] when A and M are chosen uniformly and independently in B H ? We solve these questions for the case n = 2. The following assertion is immediate Proposition The between H and the standard Euclidean space R 3 . Its inverse L : R 3 → H satisfies L(x × y) = √ 2 Lx, Ly] T .

Corollary 5 . 1 2 2 9 2 ˆ1 0 ˆ1 0 r 2 s 2 dr ds ˆπ 0 φ(rs sin θ) sin θ dθ = 9 2 ˆ1 0 ˆ1 0 r 2 s 2 dr ds ˆ1 - 1 φ(rs 1 -c 2 ) dc = 9 ˆ1 0 ˆ1 0 r 2 s 2 dr ds ˆ1 0 φ(rs 1 -

 51290211901 The image of B H × B H under the commutator is √ 2 B H . The pushforward dν of dA ⊗ dM under the map(A, M) → √ [A, M] is radial, with distribution determined by ˆBH φ( Z F ) dν(Z) = ˆ1 0 h(ρ)φ(ρ) dρ, ∀φ ∈ C([0, 1]),with h as in Proposition 5.2. Proof We only have to calculate the density h. Because it is radial, we only need to consider radial functions f (z) = φ(|z|); we have ˆ1 0 φ(ρ)h(ρ)dρ = φ(|x × y|)dxdy, where ffl denotes the normalized integral over B 3 . Using again the rotational invariance, we have ˆ1 0 φ(ρ)h(ρ)dρ = 3 ˆ1 0 r 2 dr φ(r| e 1 × y|) dy = 9 ˆ1 0 ˆ1 0 r 2 s 2 dr ds ˆS2 φ(rs| e 1 × e|)dω(e) = c 2 ) dc. Denoting ρ := rs √ 1c 2 , we have ρ ≤ rs, = rs ρ r 2 s 2ρ 2 |drdsdc|,

r 2 s 2 -ρ 2 ,ρ r 2 -

 22 ρ 2 dr r .

S 3 (aR 4 .

 34 A + tI n , B,C) = S 3 (A, B,C) + t[B,C].Therefore we may not restrict to zero-trace matrices; incidentally, the trace of S 3 (A, B,C) itself does not vanish in general.By direct inspection, one may verify the formula (12) S 3 (A, B,C) a 22 2b 11b 22 2c 11 -11 b 11 c 11 a 12 b 12 c 12 a 22 b 22 c 22 , Y = a 22 b 22 c 22 a 21 b 21 c 21 a 11 b 11 c 11 , a 11 2b 22b 11 2c 22c 11 . We interpret these formulae in terms of the mixed product in R 4 . If a, b, c ∈ R 4 , we denote a×b×c the vector defined by (a × b × c) • x = det(a, b, c, x), ∀x ∈

The commutator in terms of the numerical radius For the following result concerning the numerical radius h, we benefited of a fruitful disussion with Piotr Migdal.

Acknowledgement. I am grateful to the referee, who did a remarkable and detailed work. The report helped to improve significantly the presentation of the paper.

we have

Observe that the image of B 4 × B 4 × B 4 by the mixed product is B 4 itself. Therefore the image of B F × B F is given by the inequality

This is an ellipsoid centered at the origin, whose main semi-axis have lengths 1, 1, 3 and 3. We deduce the optimal inequality

The equality holds precisely when a, b, c form an orthogonal basis of the orthogonal complement of the vector

in other words when A, B and C are mutually orthogonal and trace-less. The factor 9 above must be compared with the average ratio given by Theorem 4.1,

We remark that this average ratio is larger than the inner radius of the ellipsoid. When A, B,C are drawn uniformly and independently from B F , the law of S 3 (A, B,C) is the pushforward of the law of a × b × c (a, b, c chosen uniformly and independently in B 4 ) under the linear map f → (W, X,Y, Z) declared previously. If R ∈ SO 4 , we have (Ra) × (Rb) × (Rb) = R(a × b × c), while the distribution of (Ra, Rb, Rc) is the same as that of (a, b, c). Therefore the law of a × b × c is radial.

Another consequence of ( 12) is a statement of the annihilating case.


Denis Serre

Alexis F Vasseur

Peter Lax

The relative entropy method for the stability of intermediate shock waves ; the rich case

 developed a new version of the relative entropy method, which is efficient in the study of the long-time stability of extreme shocks.

Introduction

We consider a strictly hyperbolic system of conservation laws in one space dimension

u t + f (u) x = 0, u(x, t) ∈ U,
where U is a convex open domain in R n and f : U → R n is a smooth vector field. By assumption, the Jacobian df (u) is diagonalizable with real eigenvalues λ 1 (u) < • • • < λ n (u) and eigenvectors r k (u). The fields u → (λ k , r k ) are smooth. An entropy-flux pair is made of functions η(u), q(u) for which every classical solution satisfies in addition ∂ t η(u) + ∂ x q(u) = 0.

We are interested only in entropies such that D 2 η is positive definite, which we call strongly convex entropies. Then we say that a bounded measurable solutions is admissible if it satisfies the entropy inequality ∂ t η(u) + ∂ x q(u) ≤ 0.

for every convex entropy. If a is a fixed state, the relative entropy η(u|a) and its relative flux q(u; a) are defined as usual by η(u|a) = η(u) -η(a) -dη a • (u -a), q(u; a) = q(u) -q(a) -dη a • (f (u) -f (a)).

We point out that η(u|a) > 0 unless u = a. The relative entropy-flux pair still satisfies ∂ t η(u|a) + ∂ x q(u; a) ≤ 0 for every admissible solution.

Let U denote a pure shock wave:

U (x, t) = u , if x < σt, u r , if x > σt.
Its data u ,r satisfy the Rankine-Hugoniot jump relation

(1) f (u r ) -f (u ) = σ(u r -u ).
We treat here the case of a Lax shock : there is an index 1 ≤ k ≤ n (we speak of a k-shock) such that the following inequalities hold true :

(2)

λ k-1 (u ) < σ < λ k+1 (u r ), λ k (u r ) < σ < λ k (u ).
P. Lax [START_REF] Lax | Hyperbolic systems of conservation laws[END_REF] proved that such shocks exist with u ,r ∼ u * if the k-th characteristic field is genuinely nonlinear at u * : (dλ k • r k )(u * ) = 0. The jump [u] := u r -u is then approximately colinear with r k (u * ).

The assumption (2), together with the Implicit Function Theorem, ensures that u r can be locally defined as a smooth function of the other parameters: (u , σ) → u r .

Leger & Vasseur showed in [START_REF] Leger | L 2 -stability estimates for shock solutions of scalar conservation laws using the relative entropy method[END_REF][START_REF] Leger | Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non-BV perturbations[END_REF] that the relative entropy method applied to a scalar (case n = 1) conservation law, with a convex (or concave) flux f , yield the strong property that

t → inf h∈R h -∞ η(u|u ) dx + +∞ h η(u|u r ) dx
is non-increasing for every shock U and every entropy solution u, whenever the initial data u(•, 0) belongs to U + L 2 (R). We interpret this property as saying that scalar shock waves are attractors up to a shift.

We considered in [START_REF] Serre | L 2 -type contraction for systems of conservation laws[END_REF] the relative entropy method applied to systems (n ≥ 2). We were interested in those shock waves that are attractors up to a shift. We found that, although the so-called Keyfitz-Kranzer system does admit such stable shocks, most systems resist to the method. A new tool was therefore needed, which was elaborated recently by Vasseur & M.-J. Kang [START_REF] Kang | Criteria on contractions for entropic discontinuities of systems of conservation laws[END_REF]. Their idea was, instead of integrating the relative entropy associated with the same η on both sides of the shock, to involve η on the left side of the shock, and aη on the right side, for some appropriate constant a > 0. Depending on whether a is small or large, they obtained that either 1-shocks or n-shocks (the so-called extreme shocks) are local attractors up to a shift. The method however, even with this improvement, does not work for intermediate shocks, as those encountered in elasticity or MHD. Our goal is therefore to adapt it to the case of a rich system (see [START_REF] Serre | Systems of conservation laws[END_REF] for this notion). We shall use two entropies η ± on both sides of the shock to measure the distance from u to v. We thus consider the quantity

E[u; h] := h -∞ η -(u|u ) dx + +∞ h η + (u|u r ) dx,
which incorporates a shift h. For intermediate shocks, η + and η -will not be linearly dependent modulo the affine functions. Our main result (Theorem 3.1) is that given a Lax shock of a rich system, strongly convex entropies η ± can be chosen so that the shock be a local attractor up to a shift.

The necessity of a shift tolerance

Let U be a Lax shock associated with a genuinely nonlinear field. Without loss of generality, we may assume that it is steady. If u 0 = U + φ is a compactly supported perturbation where φ is small, then the asymptotic behaviour as t → +∞ has been described by T.-P. Liu [START_REF] Liu | Pointwise convergence to N -waves for solutions of hyperbolic conservation laws[END_REF]. It consists of a superposition of so-called N -waves (which decay in L 2 -norm like t -1/4 ), of linear waves that just propagate at constant velocity, and of a shift of the shock from x = 0 to x = h. To determine h, we split the mass m = R φ dx into three parts X -+h[u]+X + , where X -(resp. X + ) belongs to the stable (resp. unstable) subspace of df (u ) (resp. df (u r )) ; thus h[u] = πm for some projection operator in R n . All these waves asymptotically separate from each other and therefore the limit of E[u(t); 0] equals either |h|η -(u r |u ) or |h|η + (u |u r ), depending on the sign of h.

If E[•; 0] was a Lyapunov function for the system, then this limit would not be greater than

E[U + φ; 0] = 0 -∞ η -(u + φ|u ) dx + +∞ 0 η + (u r + φ|u r ) dx ∼ R φ 2 dx.
This would imply an inequality

|u r -u l | 2 R πφ dx ≤ C • R φ 2 dx,
which is obviously false because L 2 (R) does not embed into L 1 (R). Therefore the uniform stability of shock waves cannot be obtained without involving a shift, as it was done by Leger & Vasseur in the scalar case. This explains why we consider

E min (t) := inf h E[u(t); h] = inf h h -∞ η -(u(x, t)|u ) dx + +∞ h η + (u(x, t)|u r ) dx .

Local analysis

If a shock U is stable, we expect, following [START_REF] Liu | Pointwise convergence to N -waves for solutions of hyperbolic conservation laws[END_REF], that for every admissible solution u, close enough to U , there is a shock curve x = X(t) with X (t) → σ and u ± := u(X(t) ± 0, t) → u ,r as t → +∞. For such a solution, we expect that E min (t) is achieved by the shift h = X(t), for large enough t.

The function

ψ(h) := h -∞ η -(u|u ) dx + +∞ h η + (u|u r ) dx
has left and right derivatives at h = X(t), with

ψ (X(t) -0) = η -(u -|u ) -η + (u -|u r ), ψ (X(t) + 0) = η -(u + |u ) -η + (u + |u r ).
When t is large enough, we obtain ψ (X(t) -0) ∼ -η + (u |u r ) < 0 and ψ (X(t) + 0) ∼ η -(u r |u ) > 0, which ensures that ψ has a local minimum at h = X(t).

If instead u(•, t) is continuous at h where ψ( h) is minimum, then we must have

(3) η -(ū|u ) = η + (ū|u r ), ū := u( h).
Let us assume that the minimum E min (t) is achieved at some h(t), a piecewise smooth function of time. Then t → E min (t) is a continuous function, piecewise C 1 , and its decay is equivalent to the fact that the derivative be non-positive. We compute

d dt E min = d dt h(t) -∞ η -(u|u ) dx + +∞ h(t) η + (u|u r ) dx = ḣ(η -(u -|u ) -η + (u + |u r )) + h(t) -∞ ∂ t η -(u|u ) dx + +∞ h(t) ∂ t η + (u|u r ) dx ≤ ḣ(η -(u -|u ) -η + (u + |u r )) - h(t) -∞ ∂ x q -(u; u ) dx - +∞ h(t) ∂ x q + (u; u r ) dx = q + (u + ; u r ) -ḣη + (u + |u r ) -q -(u -; u ) + ḣη -(u -|u ),
where we have used the entropy inequalities for η ± . If u is continuous at h(t), we have

d dt E min ≤ D c (ū; u ,r ) := q + (ū; u r ) -q -(ū; u ),
where ū = u(h, t) is constrained by [START_REF] Leger | L 2 -stability estimates for shock solutions of scalar conservation laws using the relative entropy method[END_REF]. If instead u is discontinuous at h(t), we have

d dt E min ≤ D s (u ± ; u ,r ) := q + (u + ; u r ) -ḣη + (u + |u r ) -q -(u -; u ) + ḣη -(u -|u ),
where (u ± ; ḣ) are constrained by the Rankine-Hugoniot relation and the Lax entropy inequality ( 4)

f (u + ) -f (u -) = ḣ(u + -u -), q ± (u + ) -q ± (u -) ≤ ḣ(η ± (u + ) -η ± (u -)).
We ask therefore whether, given a shock U , there is a pair of strongly convex entropies η ± such that D c (ū; u ,r ) ≤ 0 for all ū satisfying (3), and D s (u ± ; u ,r ) ≤ 0 for every (u -, u + ) satisfying [START_REF] Leger | Relative entropy and the stability of shocks and contact discontinuities for systems of conservation laws with non-BV perturbations[END_REF]. If this holds true, we may say that the shock is a global attractor. If we only have D s (u ± ; u ,r ) ≤ 0 for every (u -, u + ) close to (u , u r ) and satisfying (4), then we say that the shock is a local attractor. Notice that we exclude the "continuous" case in the latter definition, because a continuous u(x) cannot be uniformly close to the shock U , even up to a shift.

Local attractors

Because η(b|a) ∼ 1 2 D 2 η a (b -a, b -a), q(b; a) ∼ 1 2 D 2 η a (df a (b -a), b -a),
we see on the one hand that D s behaves approximately quadratically around the pair (u , u r ) :

2D s (u ± ; u ,r ) ∼ D 2 η +r ((df r -σ)δu + , δu + ) -D 2 η -((df -σ)δu -, δu -),
where δu + := u + -u r and δu -:= u --u . We have written D 2 η +r for D 2 (η + ) r = D 2 η + | u=ur . On the other hand, differentiating the Rankine-Hugoniot relation (remember that u + is a smooth function of u -and σ), we have

(df r -σ)du r -(df -σ)du = (dσ)(u r -u ),
from which we infer (df r -σ)δu + -(df -σ)δu -∼ (u r -u ) for = ḣ -σ. In order that U be a local attractor, it is therefore necessary that the quadratic form

(v -, v + ) Q -→ D 2 η +r ((df r -σ)v + , v + ) -D 2 η -((df -σ)v -, v -)
be negative semi-definite over the linear space defined by

(df r -σ)v + -(df -σ)v -∈ R(u r -u ).
Conversely, if this quadratic form is negative definite, then there is a neighbourhood of (u , u r ) in which D s has a constant negative sign, and then the shock is a local attractor. Let us recall that df a is D 2 η a -symmetric: the product

S := D 2 η a df a
is a symmetric matrix. In other words, the eigen-basis of df a is orthogonal with respect to D 2 η a , that is (i = j) =⇒ (D 2 η(r i , r j ) = 0).

We deduce that, if the matrix df a -σ is non-singular, then

M a := (df a -σ) -T D 2 η a ∈ Sym n (R).
We point out that, because of D 2 η a is positive definite, the signature (n -, n + ) of M a is made of the number n ± of positive/negative eigenvalues of df a -σ.

Our quadratic form can be re-written in the form

Q(v, s) := (v + s[u]) T M +r (v + s[u]) -v T M -v, where (df -σ)v -=: v and (df r -σ)v + =: v + s[u].
The negative definiteness of Q is equivalent to that of the matrix

Q := M +r -M - M +r [u] [u] T M +r [u] T M +r [u] .

The choice of entropies η ±

A general system of conservation laws does not always admit a non-trivial entropy (that is, a non-affine one). If n = 1, every function is an entropy, thus every convex function is a convex entropy. If n = 2, the system admits a lot of entropy-flux pairs, among which there is an infinite dimensional cone of convex entropies (Serre [START_REF] Serre | Systems of conservation laws[END_REF], Chapter 12). More generally, we may declare that physically relevant systems do admit at least one strongly convex entropy. However, if n ≥ 3, this convex entropy η is unique up to the addition of an affine function and the multiplication by a positive constant. The former modification does not change the relative entropy.

In the general case, we have therefore only the choice η ± = a ± η where a ± are positive constants and only the ratio a + /a -matters. This is the approach made by Kang & Vasseur [START_REF] Kang | Criteria on contractions for entropic discontinuities of systems of conservation laws[END_REF]. In the rich case (see [START_REF] Serre | Systems of conservation laws[END_REF]), we have a more general choice, which we exploit in Section 3.

One entropy only doesn't work

It turns out that in the scalar case with a convex flux f , we may choose a + /a -= 1, that is η + = η -. It is therefore of some importance to recognize whether this is also possible for systems. The following example shows that the answer is negative.

If U is a local attractor when we choose only one entropy (denoted η), then the matrix

M r -M M r [u] [u] T M r [u] T M r [u]
is non-positive. We show below that this does not happen in the simple example of the p-system

v t + w x = 0, w t + p(v) x = 0, u = v w ,
where p > 0 and p does not vanish. The natural entropy is η(u) = 1 2 w 2 + e(v) with e a primitive of p. We have

D 2 η = c 2 0 0 1 , c := p and df (u) = 0 1 c 2 0 .
This yields

M = 1 c 2 -σ 2 σ c 2 1 σ c 2 0 0 1 = 1 c 2 -σ 2 σc 2 c 2 c 2 σ .
We obtain therefore

M r -M = σ(c 2 -c 2 r ) (c 2 r -σ 2 )(c 2 -σ 2 ) σ 2 σ σ 1 = κY Y T , Y := σ 1 .
Because M r -M is only rank-one, we see that Q cannot be negative definite. We show below that it is not even negative semi-definite. On another hand, the Rankine-Hugoniot conditions are

[w] = σ[v], [p(v)] = σ[w],
hence

σ 2 = [p(v)]/[v], [u] = [v] 1 σ .
When U is a 2-shock, that is σ > 0, the Lax shock inequalities give c r < σ < c ; we thus have κ < 0. If U is a 1-shock, then -c r < σ < -c and we obtain again κ < 0. In both cases, the block M r -M is negative semi-definite. We also have

[u] T M r [u] = σ[v] 2 c 2 r -σ 2 (3c 2 r + σ 2 ) < 0.
The Sherman-Morrison criterion therefore says that Q is negative semi-definite if and only if

M r -M - 1 [u] T M r [u] M r [u][u] T M r ≤ 0 2 , or equivalently [u] T M r [u](M r -M ) ≥ M r [u][u] T M r .
Both sides are rank-one non-negative matrices ; this inequality has the form ZZ T ≤ βY Y T with β > 0 and Z = M r [u]. It implies that Z is colinear to Y ; in other words, we must have

0 = 1 -σ Z = [v] 1 -σ M r 1 σ = σ[v].
This is absurd because σ = 0 and [v] = 0.

In conclusion, a shock U is not a local attractor for the p-system if we only choose η + = η - equal to the mechanical energy.

In particular, we may specify the Hessian matrix of an entropy at one given point ū, provided that it is diagonal in the basis {r 1 (ū), . . . , r n (ū)}. Choosing a positive definite Hessian, the entropy is strongly convex in some neighbourhood of ū.

Let U be a k-shock. The stable subspace S r of df (u r ) -σ is spanned by r 1 (u r ), . . . , r k (u r ). Likewise, the unstable subspace U r is spanned by r k+1 (u r ), . . . , r n (u r ). They have dimensions k and n -k, respectively. The matrix df (u ) -σ has also a stable and an unstable subspaces S and U , but then dim S = k -1 and dim U = n -k + 1.

Because we may choose arbitrarily the Hessian of η + at u r , among the positive diagonal matrices in the eigenbasis B r = {r 1 (u r ), . . . , r n (u r )}, we see that the quadratic form q + associated with M +r can be any difference q Sr q U r , where q U r and q Sr are diagonal in B r as well, and are positive semi-definite, with ker q U r = U r and ker q Sr = S r respectively. Likewise the quadratic form q -associated with M -has the form q S q U where each form is diagonal in the eigenbasis B , et caetera.

By choosing q S large enough and then q U small enough we see that the form Q can be made negative definite, provided that the restriction of

Q r := M +r M +r [u] [u] T M +r [u] T M +r [u] to S × R is negative definite. Because Q r = X T M r X, X = I n [u] ,
this amounts to requiring that M r is negative definite over S + R[u], and that the last sum is a direct sum: [u] ∈ S . Finally, choosing q U r large enough and then q Sr small enough, we see that this M r can be made negative definite over S ⊕ [u] if and only if (S ⊕ [u]) ∩ U r = {0}.

Because the dimensions sum up to n, we obtain Theorem 3.1 For a rich system, the following properties are equivalent to each other.

• There exist two entropies η ± , strongly convex about u ,r respectively, such that the matrix Q is negative definite (and then the shock U is a local attractor up to a shift for the corresponding E).

• One has

R n = S ⊕ [u] ⊕ U r .
The latter property is nothing but the Liu-Majda's condition det(r 1 (u ), . . . , r k-1 (u ), [u], r k+1 (u r ), . . . , r n (u r )) = 0, under which the evolution of the shock wave (a free boundary-value problem) is locally wellposed (again, see [START_REF] Majda | The stability of multidimensional shock fronts[END_REF][START_REF] Majda | The existence of multidimensional shock fronts[END_REF]). It can be viewed as a Lopatinskii condition for the shock stability.

Acknowledgements. We are grateful to the referee for her/his careful reading and useful comments.

towards the University of Texas, Austin and to its Department of Mathematics for the kind hospitality and the nice working atmosphere. The second author was partially supported by NSF grant DMS-1209420 while completing this work.

Extreme shocks (after Kang & Vasseur)

We now allow that η -= aη + for some positive constant a. We denote η = η + , a convex entropy usually provided by some physical principle.

An "extreme" shock is either a 1-shock or an n-shock. Up to a switch x → -x, we may suppose that U is a 1-shock:

In the construction above, df (u )-σ has positive eigenvalues, while df (u r )-σ has n-1 positive and one negative eigenvalues. Therefore M -is positive definite while M +r is of signature (n -1, 1).

Because M -is positive definite, Q is negative definite provided a is large enough, and

Remarking that the restriction of M r to the hyperplane spanned by r 2 (u r ), . . . , r n (u r ) is positive definite, we see that ( 5) is slightly stronger than the Liu-Majda's criterion

We recall that the latter is the condition under which the local-in-time stability of the shock wave U holds true: the free boundary value problem associated with the evolution of the shock under initial disturbances is locally well-posed in spaces of smooth functions. : See [START_REF] Majda | The stability of multidimensional shock fronts[END_REF][START_REF] Majda | The existence of multidimensional shock fronts[END_REF].

If the shock strength is small, the direction [u] is approximately that of r 1 (u r ). Then

because of the Lax shock inequality. Then the criterion ( 5) is satisfied trivially.

Kang & Vasseur actually proved that when 1 < k < n (non extreme case), a small shock cannot be an attractor if we deal with a pair η ± = (aη, η). This can be checked by following the arguments developed in the next paragraph.

Intermediate shocks in rich systems

The class of rich systems generalizes that of 2 × 2 systems ; it was introduced in [START_REF] Serre | Richness and the classification of quasilinear hyperbolic systems[END_REF]. A rich system has a lot of entropies. Specifically, if ū ∈ U and R j (ū) is the integral curve of the j-th characteristic field passing through ū, then for every set of prescribed functions φ j : R j (ū) → R vanishing at ū, there exists one and only one entropy coinciding with φ j over R j for all j = 1, . . . , n.

Comments

• The result stated in the theorem suggests that every Lax shock satisfying Liu-Majda's condition is actually asymptotically stable ; a similar result is known for viscous profiles, but only for extreme shocks, see [START_REF] Benzoni-Gavage | Alternate Evans functions and viscous shock waves[END_REF].

• When we consider an extreme shock, say a 1-shock, Theorem 3.1 actually gives a slightly better result than that of Kang & Vasseur : if we had to choose η -= aη + with a large enough, the attractivity would require the stronger condition

which is not needed here. Notice that our result is a genuine improvement only for large amplitude shocks: if [u] is small, then its direction is approximately that of r 1 , and ( 6) is automatically satisfied.

• Kang & Vasseur's restricted choice is nevertheless relevant if the system is the limit of some dissipation process, because then we cannot prove the entropy inequality for every convex entropy, but only for one of them ; see [START_REF] Serre | The structure of dissipative viscous system of conservation laws[END_REF].

• When the shock strength tends to zero, the matrices df (u r, ) -σ become singular. A good question is whether we can choose η ± independently of the shock triplet (u , u r ; σ) in the neighbourhood of (ū, ū; λ k (ū)) when the k-th field is genuine nonlinear. The answer is positive in the scalar case (convex flux), where we may just choose η + = η -, a strongly convex function.


UMPA Denis Serre

Gradient estimate in terms of a Hilbert-like distance, for minimal surfaces and Chaplygin gas

Keywords: Chaplygin gas, Riemann problem, Keldysh-type degeneracy, Hilbert distance. MSC classification: 76N10, 35J93, 52A10

We consider a quasilinear elliptic boundary value problem with homogenenous Dirichlet condition. The data is a convex planar domain. The gradient estimate is needed to ensure the uniform ellipticity, before applying regularity theory. We establish this estimate in terms of a distance which is equivalent to the Hilbert metric.

This fills the proof of existence and uniqueness of a solution to this BVP, when the domain is only convex but not strictly, for instance if it is a polygon.

Introduction

Let Ω be a connected planar open domain. We are interested in the following elliptic boundaryvalue problem: div ∇w

1 + |∇w| 2 + 2 w 1 + |∇w| 2 = 0 in Ω, (1)
w > 0 in Ω, [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF] w = 0 on ∂Ω.

Because of ellipticity we anticipate that the solution is classical, hence (1) amounts to

(4) (1 + |∇w| 2 )∆w -D 2 w : ∇w ⊗ ∇w + 2 w (1 + |∇w| 2 ) = 0.
The principal part in [START_REF] Anderson | Complete minimal varieties in hyperbolic space[END_REF] is the operator of minimal surfaces. There are at least three interpretations of the PDE (1) :

• Equation ( 1) is the Euler-Lagrange equation for the functional

A[w] = 1 + |∇w| 2 w 2 dx.
The graph x 3 = w(x 1 , x 2 ) is therefore a complete minimal surface in the half-space x 3 > 0 endowed with the Poincaré Riemannian metric

ds 2 = dx 2 1 + dx 2 2 + dx 2 3 x 2 3
.

The curve ∂Ω is the asymptote of the surface at the "infinity" x 3 = 0. We notice that the total area of the graph is infinite; see [START_REF] Serre | Multi-dimensional shock interaction for a Chaplygin gas[END_REF], especially a remark after Theorem 5.1.

General minimal surfaces associated with the metric ds 2 , with prescribed asymptote at infinity have been studied by Anderson [START_REF] Anderson | Complete minimal varieties in hyperbolic space[END_REF] in terms of currents. When ∂Ω is not a convex curve, a minimal surface is not a graph over Ω.

• The hypersurface of revolution in R 4 defined by

x 2 3 + x 2 4 = w(x 1 , x 2 )
satisfies the equation

κ 3 = 1 2 (κ 1 + κ 2 ),
in terms of its principal curvatures; see [START_REF] Serre | Multi-dimensional shock interaction for a Chaplygin gas[END_REF], especially a remark after Theorem 5.1.

• Our original motivation in [START_REF] Serre | Multi-dimensional shock interaction for a Chaplygin gas[END_REF] was the resolution of a 2-dimensional Riemann problem for a gas that obeys the equation of state of Chaplygin.

The function φ := 1 2 w 2 is the potential of a self-similar irrotational isentropic flow. The independent variable is a self-similar coordinate x = y t , where y, t are the space and time variables, respectively. The cone defined by y ∈ tΩ is the domain where the flow is pseudo-subsonic. The velocity and the density are given by

u(y, t) = y t + (∇ x φ) y t , ρ(y, t) = a 2φ + |∇φ| 2
for some constant a > 0. The domain Ω can be determined from the Riemann data, by solving explicitely the flow in its pseudo-supersonic regime.

The data of the BVP is nothing but the domain Ω. The function φ defined in the third item above helps us to guess for which domains the BVP is likely to be well-posed. The equation satisfied by φ is

(5) (2φ + |∇φ| 2 )∆φ -D 2 φ : ∇φ ⊗ ∇φ + 4φ + |∇φ| 2 = 0.
The positivity of φ is the condition that this second-order equation is elliptic. Of course the ellipticity degenerates at the boundary because of (3). However this degeneracy is of a rather strong form, that called Keldysh type; formally, if φ could be extended such that φ < 0 away from Ω (hyperbolic type), then the characteristic curves would be tangent to ∂Ω. This is the symptom that if φ ∈ C 2 (Ω), the normal derivative can be calculated in terms of the curvature of ∂Ω (again, see [START_REF] Serre | Multi-dimensional shock interaction for a Chaplygin gas[END_REF], Paragraph 5.4) :

(6)

κ ∂φ ∂ν = -1.
Because φ is positive in Ω and vanishes at the boundary, we must have

∂φ ∂ν ≤ 0,
which yields the necessary condition that κ > 0. In other words, Ω needs to be convex, in a strong sense. We proved in [START_REF] Serre | Multi-dimensional shock interaction for a Chaplygin gas[END_REF] that this necessary condition is also sufficient:

Theorem 1.1
Let Ω be a bounded convex planar domain, whose boundary ∂Ω is a piecewise-C 2 curve, with uniformly strictly positive curvature.

Then there exists one and only one strictly positive solution φ ∈ Lip(Ω) ∩ C ∞ (Ω) of (5) satisfying the boundary condition φ = 0.

Of course, w = √ 2φ is the unique solution of our BVP in this situation. We remark that, although w is smooth in Ω, it is not globally Lipschitz, as it experiences a square root singularity at the boundary.

Theorem 1.1 raises the question whether a solution exists when Ω is a convex domain but the curvature vanishes in some part of the boundary ; for instance, we are interested in the case where the domain is polygonal. Then (6) suggests that even φ will not be globally Lipschitz over Ω. This observation is meaningful because the Lipschitz estimate of φ was a crucial step in the proof of Theorem 1.1. We recall now the procedure that we followed in [START_REF] Serre | Multi-dimensional shock interaction for a Chaplygin gas[END_REF] :

Step #1. Construct a sequence of approximate solutions w m . There are several possible choices, but they need to be consistent with the estimates described below.

Step #2. Equation (1) satisfies the maximum principle. One may compare w to exact solutions w p,r = 2φ p,r where

φ p,r = 1 2 (r 2 -|x -p| 2 ).
We obtain upper/lower bounds w ± , given by

φ -= inf{φ p,r | D(p; r) ⊂ Ω}, φ + = sup{φ p,r | Ω ⊂ D(p; r)},
where D(p; r) is the disk of radius r, with center at p.

The solution is therefore expected to satisfy

(7) φ -≤ φ ≤ φ + .
The lower bound ensures that w > 0 in Ω, while the upper bound provides an L ∞ estimate. Of course, the approximate solutions must be constructed so as to fill the same inequality, or an approximate form of it.

Step #3. Because the curvature is everywhere strictly positive (this is the assumption that we intend to drop below), we have φ ± ≡ 0 on the boundary. Therefore (7) ensures that w is continuous at the boundary, with w = 0 along ∂Ω.

Actually, (7) implies the bounds

0 ≥ ∂φ ∂ν ≥ ∂φ + ∂ν ≥ - 1 κ ,
where κ is a lower bound of the κ along ∂Ω. This can be recast as

(8) |∇φ| ≤ 1 κ along ∂Ω.
Step #4. It turns out that if α ∈ (2, 3), then αφ + |∇φ| 2 must reach its maximum on the boundary. This fact and (8) provide the needed a priori Lipschitz estimate.

Step #5. The Lipschitz estimate provides the relative compactness of w m in C(Ω). It also tells us that ( 1) is a uniformly elliptic equation. Therefore the regularity theory provides interior estimates for D 2 w m and higher derivatives (see [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF]). With Ascoli-Arzela Theorem, this ensures the relative compactness of ∇w m in C(K) for every compact K ⊂ Ω. We may therefore pass to the limit in a subsequence : the limit w satisfies the PDE. We obtain in the limit φ -≤ φ ≤ φ + , which guaranties (2,3).

When the curvature vanishes somewhere, two arguments in the strategy above fall down. On the one hand, it may happen that φ + does not vanish on the boundary. When Ω is a triangle, one finds that φ + coincide with φ P ;R where D(P ; R) is the circumscribed disk ; then φ + vanishes only at the vertices. On the other hand, even if κ vanishes only at isolated points and therefore φ + ≡ 0 on the boundary, its normal derivative must be infinite at these exceptional points. Therefore one cannot conclude in the Step #4 above.

The purpose of this article is to provide new a priori estimates for both w and ∇w, which are valid for arbitrary bounded convex domains. The L ∞ estimates involve one-dimensional barrier functions ; this part is far from original. The technique employed to estimate the gradient is more interesting and, up to our knowledge, rather new. We use the scale invariance of (1) and compare w to itself, after translation and dilation. We obtain a Lipschitz estimate of the form

| log w(y) -log w(x)| ≤ d(x, y),
where d is a distance on Ω, which is locally equivalent to the Euclidian distance. For the metric d, ∂Ω is a horizon at infinity ; this ressembles the situation when Ω is equipped with the Hilbert distance, although our metric does not coincide with the latter.

With these estimates in hands, we may proceed as in Step #5 above. Passing to the limit from uniformly convex domains to arbitrary ones, we obtain the more general conclusion : Theorem 1.2 Let Ω be a bounded planar convex domain. Then the boundary-value problem (1,2,3) admits one and only one solution w ∈ C(Ω) ∩ C ∞ (Ω).

Plan of the paper: We construct barrier functions in Section 2. Even if they are new with respect to [START_REF] Serre | Multi-dimensional shock interaction for a Chaplygin gas[END_REF], they are hardly surprising. We introduce our (new ?) distance on convex domains in Section 3, and compare it with the Euclidian distance and the Hilbert metric. Our most original idea is the Lipschitz estimate, presented in Section 4. We use both estimates to prove Theorem 1.2 in Section 5.

2 Barrier functions, L ∞ -estimate Let y := x • ξ be a one-dimensional variable, where ξ is a unit vector. The function x → W (y) solves (1) if and only if Proof Let v(t) be the solution of the Cauchy problem

(9) 1 2 W W + W 2 + 1 = 0.
v = 1 -9v 4/3 , v(0) = 0.
The function v is increasing from 0 to the maximum

1 3 √ 3 reached at y * = 1 3 √ 3 0 dv √ 1 -9v 4/3 = 1 3 √ 3 1 0 du √ 1 -u 4/3 .
The function v can be extended over (0, 2y * ) by v(2y * -y) = v(y), as a solution of

v 2 + 9v 4/3 = 1.
Then z := v 1/3 satisfies z 2 + 1 = 1 9 z -4 , or

z 4 z 2 + z 4 = 1 9 .
Differentiating once, we obtain 1 2 zz + z 2 + 1 = 0.

There remains to define

W I (y) = b -a 2y * z 2y * y -a b -a .
A more careful analysis provides a positive solution such that W (a) = W (b) = 0, and W (a + b -y) = W (y). We don't need it to establish the following upper bound: Corollary 2.1 Let Ω be a bounded planar convex domain. Then there exists a super-solution w + ∈ C(Ω) of (1), positive in Ω and vanishing along ∂Ω. It satisfies

w + ≤ diam Ω 2y * √ 3 .

Proof

Given a direction ξ ∈ S 1 , let us define

a(ξ) = inf x∈Ω x • ξ, b(ξ) = sup x∈Ω x • ξ.
Let us denote I(ξ) = (a(ξ), b(ξ)). Lemma 2.1 provides a solution of (1) of the form

w ξ (x) = W I(ξ) (x • ξ). Then w + (x) = inf ξ w ξ (x)
is a continuous positive super-solution of (1). The bound

w + (x) ≤ diamΩ 2y * z min ξ x • ξ -a(ξ) b(ξ) -a(ξ)
shows that w + (x) → 0 as x → x ∈ ∂Ω (consider the inward unit normal at x).

Of course, if Ω is a polygon, a super-solution vanishing at the boundary can be defined as the minimum of finitely many one-dimensional solutions of (1).

A distance over a bounded convex domain

As above, Ω is a non-void, bounded convex open domain. The fact that Ω is 2-dimensional is not essential here.

Given two points p, q ∈ Ω, Ω -p contains a ball centered at the origin and is therefore absorbing. Thus there exists some λ > 0 such that Ω -q ⊂ λ • (Ω -p). If µ > λ, then also Ω -q ⊂ µ • (Ω -p), by convexity. Likewise, the infimum m(p, q) of all such numbers satisfies the same inclusion, by continuity. Hence the set of these numbers is of the form [m(p, q), +∞). Considering the volumes, we have

|Ω| = |Ω -q| ≤ m(p, q) 2 |Ω -p| = m(p, q) 2 |Ω -q|, which implies (10) m(p, q) ≥ 1.
The equality in (10) stands only if

Ω -q = Ω -p, that is if p = q. If r ∈ Ω is a third point, then Ω -r ⊂ m(q, r) • (Ω -q) ⊂ m(q, r)m(p, q) • (Ω -p)
and therefore m(p, r) ≤ m(q, r)m(p, q).

All this shows that the logarithm of m is a non-negative function over Ω × Ω, which vanishes only along the diagonal and satisfies the triangle inequality. In other words, the function

d Ω (p, q) = log m(p, q) + log m(q, p) is a distance over Ω.
Our distance is finer than the Euclidian one:

Proposition 3.1 Denote M the diameter of Ω. Then |q -p| ≤ M 2 d Ω (p, q).

Proof

Let J = (α, β) be the segment in Ω passing through p and q, with α, β ∈ ∂Ω. We may assume that α, p, q, β are ordered in that way along the line, and we use the same order along

J below. If Ω -q ⊂ λ • (Ω -p), then J -q ⊂ λ • (J -p), which means α -q ≥ λ(α -p), that is q -p ≤ (q -α) 1 - 1 λ .
We deduce

|q -p| ≤ M 1 - 1 m(p, q) ≤ M log m(p, q).
The opposite inequality is non-uniform at the boundary:

Proposition 3.2
The following inequality holds true:

d Ω (p, q) ≤ log 1 + |q -p| dist(p, ∂Ω) + log 1 + |q -p| dist(q, ∂Ω) ≤ |q -p| 1 dist(p, ∂Ω) + 1 dist(q, ∂Ω) ,
where dist(•, ∂Ω) denotes the Euclidian distance to the boundary.

Proof

One has

Ω -q = p -q + Ω -p ⊂ D(0; |q -p|) + Ω -p ⊂ |q -p| dist(p, ∂Ω) + 1 • (Ω -p), whence m(p, q) ≤ |q -p| dist(p, ∂Ω) + 1.
We conclude with the inequality log(1 + t) ≤ t.

Using the fact that the inclusion between two disks D(P ; r) ⊂ D(Q; s) is equivalent to s -r ≥ |P -Q|, we calculate easily the distance associated with the unit disk D :

d D (p, q) = 2 log 1 -p • q + |q -p| 2 -|p| 2 |q| 2 sin 2 θ -log(1 -|p| 2 ) -log(1 -|q| 2 ),
where θ denotes the angle between the vectors p and q. When |q| → 1, the first term above is equivalent to log(2 -2|p| cos θ). If p is kept fixed, we thus have

d D (p, q) ∼ log 1 1 -|q| → +∞.
Therefore the boundary is the infinite horizon of D. In other words, (D, d D ) is a complete metric space. For a general domain, we have Proposition 3.3 The following inequality holds true:

d Ω (p, q) ≥ | log dist(q, ∂Ω) -log dist(p, ∂Ω)|.

In particular, (Ω, d Ω ) is a complete metric space.

Proof Denote r = dist(q, ∂Ω) and s = dist(p, ∂Ω). We have

D(0; r) ⊂ Ω -q ≤ m(p, q) • (Ω -p), whence D 0; r m(p, q)
⊂ Ω -p.

We deduce m(p, q) ≥ r s , or log m(p, q) ≥ (log r -log s) + .

Evaluating this inequality at q, we obtain w(q) ≤ w(q) = m(p, q)w(p), that is (log w(q) -log w(p)) + ≤ log m(p, q). This yields our Lipschitz estimate:

Proposition 4.1 Let Ω be a bounded convex planar domain, and w be a solution of the BVP [START_REF] Anderson | Complete minimal varieties in hyperbolic space[END_REF][START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF][START_REF] Serre | Multi-dimensional shock interaction for a Chaplygin gas[END_REF] in Ω. Then we have

| log w(q) -log w(p)| ≤ d Ω (p, q).
Using Proposition 3. We begin by constructing a one-parameter family of bounded convex domains Ω ( > 0), with three properties:

1. The boundary ∂Ω is a C 1 -curve, piecewise C 2 . Its curvature is bounded below by .

2. The family → Ω is decreasing for the inclusion.

More precisely

Ω ⊂ Ω + D(0; h( )), lim →0+ h( ) = 0.
To begin with, we define U as the intersection of all disks D(p; + 1 ) such that Ω ⊂ D(p; 1 ).

Each of this disk is convex, with a curvature less than , and so is their intersection. The boundary has at most denumerably many vertices. Each vertex, being at a distance ≥ of Ω, can be smoothed out by placing a bitangent arc of circle of radius 2 . This is done in a unique way and yields a domain Ω ⊂ U which satisfies the requirements above.

A converging approximation

According to [START_REF] Serre | Multi-dimensional shock interaction for a Chaplygin gas[END_REF], Theorem 5.1, the BVP (1,2,3) admits a unique solution w ∈ C(Ω) ∩ C ∞ (Ω).

If < η then w η is a super-solution to the BVP in Ω and thus w ≤ w η by the maximum principle. The map → w is therefore non-decreasing.

A positive lower bound is the one established in [START_REF] Serre | Multi-dimensional shock interaction for a Chaplygin gas[END_REF]. It uses the solutions w p,r of (1) defined in Step #2 above. If D(p; r) ⊂ Ω, w p,r is a sub-solution for all these BVPs and we infer w ≥ w p,r in D(p; r). We deduce

w ≥ w -:= sup{w p,r | D(p; r) ⊂ Ω}.
If K is a compact subset of Ω, this lower bound is uniformly positive over K.

By monotonicity, the limit w(x) = lim We may therefore apply Theorem 13.6 in [START_REF] Gilbarg | Elliptic Partial Differential Equations of Second Order[END_REF] to derive uniform estimates of D 2 w in C α (Ω ) whenever Ω Ω .

Concluding. Applying repeatedly Theorem 6.17, we actually obtain uniform estimates of higher derivatives in every sub-domain ω Ω. This ensures that w ∈ C ∞ (Ω). With (11), we have in particular w ∈ C(Ω). Finally, w → w in C ∞ (Ω) and we may pass to the limit in and infer that w solves (1). This proves the existence part of Theorem 1.2. The uniqueness is just a consequence of the maximum principle, as in [START_REF] Serre | Multi-dimensional shock interaction for a Chaplygin gas[END_REF].

Lemma 2 . 1

 21 Let I = (a, b) be a non-void interval. There exists a positive solution W I of (9) over I, with the property that W I (a) = W I (b) = 0.

→0+w|p| 2 -

 2 (x) exists and satifies w -≤ w. Therefore w > 0 in Ω. On the other hand, passing to the limit in the bound w ≤ w + obtained in Section 2, yields the upper bound w ≤ w + , implying (11) limd(x;∂Ω)→0 w(x) = 0.To each w , we apply the L ∞ estimate (Section 2) and the Lipschitz estimate (Corollary 4.1) above. This shows that the operatorL = (1 + |∇w | 2 )∆ -∇w ⊗ ∇w : D 2is uniformly (in and x) elliptic in Ω , provided Ω Ω.C 2,α -Estimates. Let us fix a sub-domain Ω Ω. Our Lipschitz estimate tells us that there exists a finite number such that |∇ (x)| ≤ for every < η and x ∈ Ω . In this region, w is therefore a solution of the quasilinear uniformly elliptic equation (1 + |f (∇w )| 2 )∆w -D 2 w : f (∇w ) ⊗ f (∇w ) |p| p, if |p| ≥ .

( 1 +

 1 |∇w | 2 )∆w -D 2 w : ∇w ⊗ ∇w + 2 w (1 + |∇w | 2 ) = 0

  When the boundary has a positive curvature, we know that w 2 is Lipschitz up to the boundary, with non-zero normal derivative. The estimate above is thus improved into

	2, we deduce
	Corollary 4.1 We have the gradient estimate
	|∇w(p)| ≤	2w(p) dist(p; ∂Ω)	.
	|∇w(p)| = O	1 dist(p; ∂Ω)	.
	For general convex domain, the accuracy of Corollary 4.1 is unclear. The barrier function w +
	constructed in Section 2 satisfies a slightly better bound
	|∇w + (p)| = O	1 (dist(p; ∂Ω)) 2/3 .
	5 Proof of Theorem 1.2
	5.1 Outer approximation of Ω by strictly convex domains

Acknowledgement. I am indebted to the referee for his/her careful reading and pointing out the correct reference for the regularity estimates.

Propositions 3.2 and 3.3 show that when p is kept fixed and q tends to ∂Ω, then d Ω (p, q) blows up like log 1 dist(q, ∂Ω) .

When Ω = (a, b) is one-dimensional, our distance coincides with that defined by Hilbert:

For a convex domain Ω of arbitrary dimension, the Hilbert distance d H (p, q) between two points p, q ∈ Ω is defined as d J (p, q) where J = Ω ∪ Aff(pq) is the segment obtained by intersecting Ω with the line passing through p and q. The coincidence of d Ω with d H holds true only in one space dimension. In higher dimension, it fails, as shown by our formula above for a disk. Instead, we have

Proof

Let L be an affine subspace, and define ω := L∩Ω. If p, q ∈ ω and if Ω-q ⊂ λ•(Ω-p), then also ω -q ⊂ λ • (ω -p), because L -q = L -p is a vector space. We deduce m ω (p, q) ≤ m Ω (p, q), hence d ω (p, q) ≤ d Ω (p, q)

Choose L the line passing through p an q and remark that d L (p, q) is nothing but d H (p, q).

Lipschitz estimate

Suppose w is a solution of (1,2,3) in a bounded convex planar domain Ω.

Observe that the equation ( 1) is translation invariant as well as dilation invariant: if r ∈ R 2 and µ > 0, then

is also a solution. Given p, q ∈ Ω, we thus define w(x) := λw p + x -q λ , with λ = m(p, q), which is a positive solution of (1) in the domain q + λ(Ω -p).

Because the latter contains Ω, w is a super-solution of (1,2,3) and we infer w ≤ w.