Lorenzo Brasco
email: lorenzo.brasco@univ-amu.fr

AND Carlo Nitsch
email: c.nitsch@unina.it

Cristina Trombetti
email: cristina@unina.it

An inequality à la Szegő

Keywords: 2010 Mathematics Subject Classification. 35P30, 47A75, 34B15 Nonlinear eigenvalue problems, shape optimization

come

Since exact values of such quantities are known only for specific values of p and special domains Ω, it is important to give (sharp) estimates for these quantities in terms of (simple) geometric quantities such as measure, perimeter, diameter, relative isoperimetric constants and so on. In this direction, the reader could consult for instance [START_REF] Avinyo | Isoperimetric constants and some lower bounds for the eigenvalues of the p-Laplacian[END_REF][START_REF] Avinyo | Geometric inequalities of Cheeger type for the first positive eigenvalue of the n-dimensional free membrane problem[END_REF][START_REF] Brandolini | Sharp estimates for eigenfunctions of a Neumann problem[END_REF][START_REF] Brandolini | Optimal lower bounds for Neumann eigenvalues of linear and nonlinear Neumann problems[END_REF][START_REF] Ferone | A remark on optimal weighted Poincaré inequalities for convex domains[END_REF] and the references therein.

With this respect the most celebrated example is the Faber-Krahn inequality (see [START_REF] Henrot | Extremum problems for eigenvalues of elliptic operators[END_REF]Chapter 3] for example) which asserts that the following minimization problem (1.1) inf{λ p (Ω) : |Ω| ≤ c}, is (uniquely) solved by N -dimensional balls of measure c. By taking advantage of the homegeneity properties of the functional Ω → λ p (Ω), the previous can be summarized as

(1.2) λ p (Ω) ≥ λ p (B) |B| |Ω| p N
, where B is now any N -dimensional ball. Then λ p (Ω) can be bounded from below in a sharp way just in terms the measure of the set Ω. We point out that by using isoperimetric inequality or isodiametric inequality, from (1.2) we can infer similar lower bounds for λ p in terms of the perimeter or the diameter of Ω.

Observe that problem (1.1) becomes trivial, when we replace λ p (Ω) with µ p (Ω). Indeed, the latter is actually zero each time Ω is disconnected. It turns out that the natural counterpart for µ p is rather the maximization problem, i.e. (1.3) sup{µ p (Ω) : |Ω| ≥ c}.

Again, this is generally expected to be solved by N -dimensional balls of volume c. Unfortunately so far this problem has resisted all the attempts to be attacked with the unique exception of the case p = 2 (and partially of the limiting cases p = 1 and p = ∞, see [START_REF] Esposito | The longest shortest fence and sharp Poincaré-Sobolev inequalities[END_REF] and [START_REF] Esposito | The Neumann eigenvalue problem for the ∞-Laplacian[END_REF][START_REF] Rossi | On the first nontrivial eigenvalue of the ∞-Laplacian with Neumann boundary conditions[END_REF] respectively). The Szegő-Weinberger inequality [START_REF] Szegő | Inequalities for certain eigenvalues of a membrane of given area[END_REF][START_REF] Weinberger | An isoperimetric inequality for the N -dimensional free membrane problem[END_REF] states in fact that for p = 2 problem (1.4) is (uniquely) solved by N -dimensional balls of measure c. As before, the result can be rewritten in scaling invariant form as

(1.4) µ 2 (Ω) ≤ µ 2 (B) |B| |Ω| 2 N
, with equality holding if and only if Ω is an N -dimensional ball. We recall that the proof of (1.4) for p = 2 crucially exploits some pecularities of the Laplacian, like linearity and the knowledge of the explicit form of eigenfunctions on balls.

A couple of comments on the Szegő-Weinberger result are in order. First of all, inequality (1.4) says that µ 2 (Ω) can be estimated from above just in terms of the measure of Ω. But differently from the case λ p , now we can not directly infer similar upper bounds for µ 2 (Ω) in terms of perimeter or diameter. Then one may wonder whether such a kind of estimates hold true or not for every 1 < p < ∞, at least for some particular classes of sets.

Secondly, we notice that if B is any N -dimensional ball, using the fact that µ 2 (B) < λ 2 (B) (see [START_REF] Henrot | Extremum problems for eigenvalues of elliptic operators[END_REF] or Proposition 5.1 below), from (1.4) we can infer

(1.5) µ 2 (Ω) ≤ λ 2 (B) |B| |Ω| 2 N
, which can be see as a weak version of the Szegő-Weinberger inequality. Again, a natural question is whether inequality (1.5) can be extended to the case of p = 2 or not.

The last two questions are the starting point of our analysis. In this paper we prove indeed sharp upper bounds on µ p (Ω) in terms of diameter, as well as generalizations of (1.5) for p = 2, under the additional constraint that Ω is a convex set.

1.2.

A sharp upper bound. Then our main scope is to investigate the following shape optimization problem with convexity and diameter constraints

(1.6) sup{µ p (Ω) : Ω ⊂ R N convex, diam(Ω) ≥ 1}.
Of course, by homogeneity of the quantities involved the value 1 has no bearing and could be replaced by any constant c > 0. In Theorem 3.1 we show that the previous upper bound µ * is finite, then we compute it and show at the same time that this problem does not admit a solution. Notably, we show that for every admissible set Ω there holds

µ p (Ω) < µ * ,
and we exhibit a sequence {Ω n } ⊂ R N of convex sets suitably degenerating to a segment for which lim

n→∞ µ p (Ω n ) = µ * .
We refer to Section 3 for more details. As we will show, the previous result can be summarized by the following scaling invariant sharp inequality

(1.7) µ p (Ω) < λ p (B) diam(B) diam(Ω) p ,
where B is any N -dimensional ball and inequality sign is strict. The proof of (1.7) is based on a clever choice of a special test function which reminds idea exploited in [START_REF] Szegő | Inequalities for certain eigenvalues of a membrane of given area[END_REF][START_REF] Weinberger | An isoperimetric inequality for the N -dimensional free membrane problem[END_REF]. We point out that by joining (1.7) and the isodiametric inequality, we immediately get

µ p (Ω) < λ p (B) |B| |Ω| p N
, which generalizes (1.5) to p = 2 for convex sets, as announced above. By keeping in mind the way such an estimate was proved for p = 2, the previous can be seen as the trace of a potentially exhisting Szegő-Weinberger inequality for the p-Laplacian.

For ease of completeness and in order to neatly motivate some of the studies performed in this paper, it is useful to recall at this point that the minimization problem

(1.8) inf{µ p (Ω) : Ω ⊂ R N convex, diam(Ω) ≤ 1},
highlights the same features as problem (1.6). For example, here as well the infimum can be computed and is not attained. More interestingly, a minimizing sequence is again given by a family of convex sets collapsing on a segment. For p = 2 this is a celebrated result by Payne and Weinberger (see [START_REF] Payne | An optimal Poincaré inequality for convex domains[END_REF]), which has been recently generalized in [START_REF] Esposito | Best constants in Poincaré inequalities for convex domains[END_REF][START_REF] Valtorta | Sharp estimate on the first eigenvalue of the p-Laplacian[END_REF] to p = 2.

The result can be summarized by the sharp inequality

(1.9) µ p (Ω) > π p diam(Ω) p . 1.3. Generalized eigenvalues.
It is quite natural to wonder if similar conclusions can be drawn also in the case of the following generalized notion of eigenvalues

λ p,q (Ω) = min v∈W 1,p 0 (Ω) Ω |∇v| p dx : Ω |v| q dx = 1 , and
µ p,q (Ω) = min v∈W 1,p (Ω) Ω |∇v| p dx : Ω |v| q dx = 1 and Ω |v| q-2 v dx = 0 .
Quite interestingly, it turns out that for q > p one has the following picture:

• one can prove the analogue of (1.7);

• this estimate is not sharp;

• the maximization problem (1.6) now admits a solution;

• a lower bound like (1.9) is not possible (and the infimum in (1.8) is 0); On the contrary, for q < p all the previous statements have to be reverted. In particular, we have sup{µ p,q (Ω) :

Ω ⊂ R N convex, diam(Ω) ≥ 1} = +∞,
and it is rather the minimization problem for µ p,q which is now well-posed (see Section 4 for more details).

1.4. Plan of the paper. In Section 2 we prove some basic results concerning properties of µ p,q (Ω) and λ p,q (Ω). Section 3 is devoted to the investigation of problem (1.6). In Section 4 we consider the case 1 < p < q < p * , and we prove a lower bound for µ p,q (Ω) in terms of measure and diameter of Ω when p > q. In Section 5 we prove a nodal domain property which roughly speaking shows that for q > p eigenfunctions associated to µ p,q (Ω) can not have a closed nodal line. Finally, the last Section is devoted to investigate the limit cases q = p * and p = N .

during some visits of the first author to Napoli. The Departement of Mathematics of the University of Napoli and its facilities are kindly acknowledged.

Preliminaries

We fix two exponents p and q such that 1 < q, p < ∞ and

(2.1)

q < p * =      N p N -p , if 1 < p < N, +∞, if p ≥ N.
For every Ω ⊂ R N open bounded Lipschitz set, we use the standard Sobolev spaces

W 1,p (Ω) = u ∈ L p (Ω) : ∇u ∈ L p (Ω; R N ) ,
and W 1,p 0 (Ω), the latter being the closure of C ∞ 0 (Ω) with respect to the norm of W 1,p (Ω). We then define the two quantities

µ p,q (Ω) = inf v∈W 1,p (Ω)\{0}          Ω |∇v| p dx Ω |v| q dx p q : Ω |v| q-2 v dx = 0          , and
λ p,q (Ω) = inf v∈W 1,p 0 (Ω)\{0} Ω |∇v| p dx Ω |v| p dx p q
.

It is useful to recall that µ p,q (Ω) can be defined through the unconstrained minimization

µ p,q (Ω) = inf v∈ W 1,p (Ω) Ω |∇v| p dx min t∈R Ω |v -t| q dx p q
, where we set W 1,p (Ω) = {v ∈ W 1,p (Ω) : Ω |∇v| p dx > 0}. Also, we have that if λ is such that the equation

-∆ p u = λ u p-q L q (Ω) |u| q-2 u in Ω, u = 0, on ∂Ω,
admits a nontrivial solution in W 1,p 0 (Ω), then λ ≥ λ p,q (Ω). Remark 2.1. Observe that the quantity µ p,q (Ω) is always well-defined, we could have µ p,q (Ω) = 0 if Ω does not support a Poincaré inequality of the type

C min t∈R Ω |v -t| q dx p q ≤ Ω |∇v| p dx.
The same remark applies to the quantity λ p,q (Ω).

We start with a couple of preliminary results on the quantities µ p,q and λ p,q .

Lemma 2.2. Let 1 < p < ∞ and 1 < s < q < p * , then we have

λ p,q (Ω) ≤ |Ω| p s -p q λ p,s (Ω) and µ p,q (Ω) ≤ |Ω| p s -p q µ p,s (Ω).
Proof. The result is a plain consequence of Hölder inequality. Let us prove for example the second inequality: we pick u s ∈ W 1,p (Ω) a function minimizing the Rayleigh quotient which defines µ p,s (Ω). We then define t q as the minimizer of

t → Ω |u s -t| q dx,
thus we get

µ p,q (Ω) = min v∈ W 1,p (Ω) Ω |∇v| p dx min t∈R Ω |v -t| q dx p q ≤ Ω |∇u s | p dx Ω |u s -t q | q dx p q ≤ |Ω| p s -p q Ω |∇u s | p dx Ω |u s -t q | s dx p s ≤ |Ω| p s -p q Ω |∇u s | p dx min t∈R Ω |u s -t| s dx p s
which in turn gives µ p,q (Ω) ≤ |Ω| p s -p q µ p,s (Ω), as desired.

The following simple continuity result will be useful.

Lemma 2.3. Let 1 < p < ∞ and 1 < q < p * , then we have

lim s→q λ p,s (Ω) = λ p,q (Ω)
and lim s→q µ p,s (Ω) = µ p,q (Ω).

Proof. We just prove the second equality, the first one can be proved along the same lines. Let u q ∈ W 1,p (Ω) \ {0} be a minimizer for the variational problem defining µ p,q (Ω), i.e. such that

Ω |∇u q | p dx Ω |u q | q dx p q
= µ p,q (Ω), and

Ω |u q | q-2 u q dx = 0.
Observe that since u q = 0, the second condition above implies that u q is not constant. We then define t s to be the minimizer of

t → Ω |u q -t| s dx, t ∈ R, thus we obtain lim sup s→q µ p,s (Ω) ≤ lim sup s→q Ω |∇u q | p dx min t∈R Ω |u q -t| s dx p s = lim sup s→q Ω |∇u q | p dx Ω |u q -t s | s dx p s = Ω |∇u q | p dx Ω |u q | q dx p q = µ p,q (Ω).
where we used that 1 t s goes to 0 as s goes to q. In order to prove that lim inf

s→q µ p,s (Ω) ≥ µ p,q (Ω),
as well, we have to distinguish two cases. If s ր q, then we can simply apply Lemma 2.2, thus we obtain lim inf

sրq µ p,s (Ω) = lim inf sրq |Ω| p s -p q µ p,s (Ω) ≥ µ p,q (Ω).
On the contrary, for s ց q, we pick

u s ∈ W 1,p (Ω) such that Ω |∇u s | p dx = µ p,s (Ω), Ω |u s | s dx = 1, Ω |u s | s-2 u s = 0,
then in particular {u s } is a bounded sequence in W 1,p (Ω), thus there exists a subsequence (not relabeled) which strongly converges when s ց q in L q (Ω). If we call u this limit function, we have

Ω |u| p = lim sցq Ω |u s | s dx = 1 and Ω |u| q-2 u dx = lim sցq Ω |u s | s-2 u s dx = 0
and of course

µ p,q (Ω) ≤ Ω |∇u| p dx ≤ lim inf sցq Ω |∇u s | p dx = lim inf sցq µ p,s (Ω).
This concludes the proof.

We will also need the following very simple geometric result for convex sets. 1 The convex functions

t → Ω |u -t| s dx,
are converging locally uniformly to the function

t → Ω |u -t| q dx.
Moreover, for |s -q| sufficiently small, there exists m > 0 such that we have |ts| < m. Then by uniform convergence and uniqueness of the minimizer tq we can infer lims→q ts = tq = 0.

Lemma 2.4. Let Ω ⊂ R N be an open convex set, and let x 0 ∈ ∂Ω. Then

x -x 0 , ν Ω (x) ≥ 0, for H N -1 -a.e. x ∈ ∂Ω,
where ν Ω (x) denotes the outer unit normal to ∂Ω at the point x.

Proof. Since Ω is convex, given x ∈ ∂Ω we have that

Ω ⊂ {y ∈ R N : y -x, ν Ω (x) ≤ 0},
i.e. the hyperplane orthogonal to ν Ω (x) and passing from x is a supporting hyperplane for Ω. In particular, since x 0 ∈ Ω we get

x 0 -x, ν Ω (x) ≤ 0,
which concludes the proof.

A Szegő-Weinberger inequality for convex sets

The following is the main result of the paper. This shows that the nonlinear spectral optimization problem

sup{µ p,p (Ω) : Ω ⊂ R N convex, diam(Ω) ≥ 1},
does not admit a solution, but a maximizing sequence is given by a family of convex sets suitably degenerating to a segment. Of course, the value 1 for the diameter constraint plays no special role and could be replaced by any constant c > 0.

Theorem 3.1. Let Ω ⊂ R N be an open bounded convex set and 1 < p < ∞. Then we have

(3.1) µ p,p (Ω) < λ p,p (B) diam(B) diam(Ω) p ,
where B is any N -dimensional ball. Equality sign in (3.1) is never achieved but the inequality is sharp. More precisely, there exists a sequence {Ω k } k∈N ⊂ R N of convex sets such that:

• diam(Ω k ) = d > 0, for every k ∈ N;

• Ω k converges to a segment of length d in the Hausdorff topology;

• we have

(3.2) lim k→∞ µ p,p (Ω k ) = λ p,p (B d/2 ),
where

B d/2 is an N -dimensional ball having radius d/2.
Proof. We split the proof into two parts: at first we prove (3.1), then we construct the sequence {Ω k } k∈N verifying (3.2).

Proof of (3.1). First of all, we observe that inequality (3.1) is in scaling invariant form.

Then without loss of generality, we can confine ourselves to prove that where B is the ball centered at the origin such that diam(Ω) = diam(B). Let us take

µ p,p (Ω) < λ p,p (B),
u ∈ C 1,α (B) ∩ C ∞ (B \ {0}
) the first Dirichlet eigenfunction for the ball B, normalized by the condition u > 0. This solves

(3.3) -∆ p u = λ p,p (B) u p-1 and u = 0 on ∂B.
We then take two points x 0 , x 1 ∈ ∂Ω such that |x 0 -x 1 | = diam(Ω), and we define the two caps

Ω i = x : |x -x i | < diam(Ω) 2 ∩ Ω, i = 0, 1,
which are mutually disjoint (see Figure 1). We then take the function

ϕ(x) = u(x -x 0 ) • 1 Ω 0 (x) -c u(x -x 1 ) • 1 Ω 1 (x) ∈ W 1,p (Ω),
where c ∈ R is the constant given by

c = Ω 0 u(x -x 0 ) p-1 dx Ω 1 u(x -x 1 ) p-1 dx , so that Ω |ϕ| p-2 ϕ dx = 0.
By using this function ϕ in the Rayleigh quotient defining µ p,p (Ω), we get

µ p,p (Ω) < Ω 0 |∇u(x -x 0 )| p dx + c p Ω 1 |∇u(x -x 1 )| p dx Ω 0 |u(x -x 0 )| p dx + c p Ω 1 |u(x -x 1 )| q dx
, where the strict inequality holds since ϕ can not be an eigenfunction 2 . By performing an integration by parts in the integrals at the numerator, we obtain 3

Ω 0 |∇u(x -x 0 )| p dx = ∂Ω 0 |∇u(x -x 0 )| p-2 ∂u ∂ν Ω (x -x 0 ) u(x -x 0 ) dH N -1 (x) - Ω 0 ∆ p u(x -x 0 ) u(x -x 0 ) dx = ∂Ω 0 |∇u(x -x 0 )| p-2 ∂u ∂ν Ω (x -x 0 ) u(x -x 0 ) dH N -1 (x) + λ p,p (B) Ω 0 |u(x -x 0 )| p dx,
where we used the equation (3.3) solved by u. Observe that the first integral in the righthand side has a sign. Indeed u is a radially decreasing function, then (with a small abuse of notation) we have

∇u(x -x 0 ), ν Ω )(x) = u ′ (|x -x 0 |) x -x 0 |x -x 0 | , ν(x) ,
and the claim follows from Lemma 2.4. The same computations apply to the other terms appearing in the numerator, thus obtaining

µ p,p (Ω) < λ p,p (B) Ω 0 |u(x -x 0 )| q dx + c p Ω 1 |u(x -x 1 )| p dx Ω 0 |u(x -x 0 )| p dx + c p Ω 1 |u(x -x 1 )| q dx = λ p,p (B),
which concludes the proof of (3.1).

Optimality of (3.1). Let B be a ball of diameter d. We prove optimality of (3.1) by constructing a sequence of convex sets {Ω k } k∈N , all sharing the same diameter d, and such that

(3.4) λ p,p (B) ≤ lim inf k→∞ µ p,p (Ω k ) and lim sup k→∞ µ p,p (Ω k ) ≤ λ p,p (B).
2 Observe that if the Rayleigh quotient of ϕ achieves µp,q(Ω), then ϕ would solve

-∆pϕ = µp,q(Ω) |ϕ| q-2 ϕ, in Ω,
in a weak sense. Let us take y0 ∈ ∂Ω0 ∩ Ω, by picking a ball B̺(y0) with radius ̺ sufficiently small so that B̺(y0) ⊂ Ω \ Ω1, we would obtain that ϕ is a nonnegative solution of the equation in B̺(y0). Then by Harnack's inequality (see [21, Theorem 1.1]) one obtains 0 < max

B̺(y 0 ) ϕ ≤ C min B̺(y 0 ) ϕ = 0,
thus getting a contradiction. We point out that we are not using any unique continuation argument. 3 Observe that we have u(x -x0) = 0 on ∂Ω0 ∩ Ω. In view of (3.1) we only need to prove the liminf inequality in (3.4). For all s ∈ R and k ∈ N let us denote by

C - k (s) = (x 1 , x ′ ) ∈ R × R N -1 : (x 1 -s) -> 2 k |x ′ | and C + k (s) = (x 1 , x ′ ) ∈ R × R N -1 : (x 1 -s) + > 2 k |x ′ |
the left and right circular infinite cone in R N whose axis is the x 1 -axis, having vertex in (s, 0) ∈ R × R N -1 , and whose opening angle is α = 2 arctan 1 k . We set

(3.5) Ω k = C - k d 2 ∩ C + k - d 2 .
In dimension N = 2, Ω k is nothing but a rhombus of diagonals d and 1/k. In higher dimension Ω k is obtained by gluing together the basis of two right circular cones of height d/2 and radii 1/(2 k) (see Figure 2). We claim that for this family (3.4) holds true. We start observing that whenever u belongs to W 1,p (Ω k ) then the rescaled function v(x 1 , x ′ ) = u (x 1 , x ′ /k) belongs to W 1,p (Ω 1 ) and we have

Ω 1 |∂ x 1 v| 2 + k 2 |∇ x ′ v| 2 p 2 dx = k N -1 Ω k |∇u| p dx, Ω 1 |v| p dx = k N -1 Ω k |u| p dx,
and

Ω 1 |v| p-2 v dx = k N -1 Ω k |v| p-2 v dx = 0.
Thus we obtain

µ p,p (Ω k ) = min u∈W 1,p (Ω k )\{0}        Ω k |∇u| p dx Ω k |u| p dx : Ω k |u| p-2 u dx = 0        , = min v∈W 1,p (Ω)\{0}        Ω 1 |∂ x 1 v| 2 + k 2 |∇ x ′ v| 2 p 2 dx Ω 1 |v| p dx : Ω 1 |v| p-2 v dx = 0        =: γ k (Ω 1 ).
Now we denote by u k a function which minimizes the Rayleigh quotient defining µ p,p (Ω k ) and by v k (x 1 , x ′ ) = u k (x 1 , x ′ /k) the corresponding function which minimizes the functional defining γ k (Ω). Without loss of generality we can assume that v k L p (Ω) = 1. Since inequality (3.1) implies that

(3.6) Ω |∂ x 1 v k | 2 + k 2 |∇ x ′ v k | 2 p 2 dx ≤ C N,p,d , for all k ∈ N,
then there exists w ∈ W 1,p (Ω) \ {0} so that v k → w weakly in W 1,p (Ω) and strongly in L p (Ω). Moreover we also have 4 ∇ x ′ w ≡ 0, and

Ω |w| p-2 w dx = 0.
Since w does not depend on the x ′ variable, we will write for simplicity w = w(x 1 ) with a slight abuse of notation. For all s ∈ [-d/2, d/2] we denote by Γ s the section of Ω which is 4 The bound (3.6) implies that for every given k0 ∈ N, we have

k p 0 Ω |∇ x ′ w| p dx ≤ Ω |∇x 1 w| 2 + k 2 0 |∇ x ′ w| 2 p 2 dx ≤ lim inf k→∞ Ω |∇x 1 v k | 2 + k 2 0 |∇ x ′ v k | 2 p 2 dx ≤ C,
which in turn gives ∇ x ′ w ≡ 0 by the arbitrariness of k0.

orthogonal to the x 1 -axis at x 1 = s and set g(s) = H N -1 (Γ s ). Then we get lim inf

k→∞ γ k (Ω) = lim inf k→∞ Ω 1 |∂ x 1 v k | 2 + k 2 |∇ x ′ v k | 2 p 2 dx Ω 1 |v k | p dx ≥ lim inf k→∞ Ω 1 |∂ x 1 v k | p dx Ω 1 |v k | p dx ≥ Ω 1 |w ′ | p dx Ω 1 |w| p dx = d/2 -d/2 |w ′ (s)| p g(s) ds d/2 -d/2
|w(s)| p g(s) ds

≥ min f ∈W 1,p (-d 2 , d 2 )\{0}          d/2 -d/2 |f ′ | p g ds d/2 -d/2 |f | p g ds : d/2 -d/2 |f | p-2 f g ds = 0          .
Let us denote by η the previous minimal value, then a minimizer f is a solution to the following boundary value problem s) it is easy to prove that f (0) = 0 and hence f solves

   -g |f ′ | p-2 f ′ ′ = η g |f | p-2 f, in (-d/2, d/2), f ′ (-d/2) = f ′ (d/2) = 0 Since g(s) = g(-
   -g |f ′ | p-2 f ′ ′ = η g |f | p-2 f, in (0, d/2), f (0) = f ′ (d/2) = 0
Finally, by reminding that g(s

) = ω N -1 (s -d/2) N -1 , if we set h(r) = f (d/2 -r) then    -r N -1 |h ′ | p-2 h ′ ′ = η r N -1 |h| p-2 h, in (0, d/2), h ′ (0) = h(d/2) = 0 which means that H(x) = h(

The case p = q

In this section we discuss variants and extensions of Theorem 3.1 for the quantity µ p,q when p = q.

4.1. The case p < q. Actually, with the very same proof of Theorem 3.1 we can prove the following upper bound (see Remark 4.2 below for a discussion on its sharpness). Theorem 4.1. Let Ω ⊂ R N be an open bounded convex set and 1 < p < q < p * . Then we have (4.1) µ p,q (Ω) < λ p,q (B) diam(B) diam(Ω)

p+ N p q -N
, where B is any N -dimensional ball.

Proof. We use the same notation as in the proof of Theorem 3.

1. With u ∈ C 1,α (B) ∩ C ∞ (B \ {0}
) we now indicate the function achieving λ p,q (B), normalized by the conditions (4.2) u L q (B) = 1 and u > 0.

Though we do not need this, we recall that such a function is unique for B and radially symmetric decreasing (see [START_REF] Erbe | Uniqueness Theorems for Positive Radial Solutions of Quasilinear Elliptic Equations in a Ball[END_REF]Main Theorem]). Moreover it solves (4.3) -∆ p u = λ p,q (B) u q-1 and u = 0 on ∂B.

As before, we consider the two caps Ω 0 and Ω 1 and take

ϕ(x) = u(x -x 0 ) • 1 Ω 0 (x) -c u(x -x 1 ) • 1 Ω 1 (x) ∈ W 1,p (Ω),
where c ∈ R is now given by

c = Ω 0 u(x -x 0 ) q-1 dx Ω 1 u(x -x 1 ) q-1 dx
.

By using this function ϕ in the Rayleigh quotient defining µ p,q (Ω) and proceeding as in Theorem 3.1 we now end up with µ p,q (Ω) < λ p,q (B)

Ω 0 |u(x -x 0 )| q dx + c p Ω 1 |u(x -x 1 )| q dx Ω 0 |u(x -x 0 )| q dx + c q Ω 1 |u(x -x 1 )| q dx p q
.

Observe that the term on the right-hand side is of the form

A + t p q B (A + t B) p q .
For p < q the previous expression is maximal for t = 1. By observing that this maximal value is given by (A + B) 1-p/q , we thus get (4.4) µ p,q (Ω) < λ p,q (B)

Ω 0 |u(x -x 0 )| q dx + Ω 1 |u(x -x 1 )| q dx 1-p q .
We have 1 -p/q > 0 and the sum of the two terms into square brackets is less than 1 by (4.2), thus we can finally infer (3.1).

Remark 4.2 (About sharpness). This time, the estimate (4.1) is not sharp. We keep the same notation as in the proof of Theorem 4.1 and still consider q > p. By adding and subtracting the term λ p,q (B) on the right-hand side of (4.4), recalling (4.2) and using the concavity of t → t 1-p/q , we get µ p,q (Ω) < λ p,q (B) + λ p,q (B)

Ω 0 ∪Ω 1 |u| q dx 1-p q - B |u| q dx 1-p q ≤ λ p,q (B) + q -p q λ p,q (B) Ω 0 ∪Ω 1 |u| q dx - B |u| q dx ≤ λ p,q (B) - q -p q λ p,q (B) B\T Ω |u| q dx,
Since u is radially decreasing, a simple rearrangement argument finally gives (4.5) µ p,q (Ω) < λ p,q (B) -q -p q λ p,q (B)

B\T Ω |u| q dx
where T Ω is the ball centered at the origin, such that |T Ω | = |Ω 0 ∪ Ω 1 |. Now observe that by using the quantitative isodiametric inequality (see [START_REF] Maggi | Quantitative stability in the isodiametric inequality via the isoperimetric inequality[END_REF]Theorem 1])

|B \ T Ω | = |B| -|Ω 0 ∪ Ω 1 | ≥ |Ω| + |Ω| C N A(Ω) 2 -|Ω 0 ∪ Ω 1 | = |Ω| -|Ω 0 ∪ Ω 1 | + |Ω| C N A(Ω) 2 ,
where C N > 0 is a dimensional constant and A(Ω) is the Fraenkel asymmetry of Ω, defined by

A(Ω) = inf 2 |Ω \ Ω # | |Ω # | : Ω # ball with |Ω # | = |Ω| .
The estimate (4.5) shows that there can not exist a convex set Ω such that µ p,q (Ω) = λ p,q (B), for q > p. Indeed, it this were true, one would obtain The first condition in (4.6) implies that Ω is a ball, in contrast with the fact that |Ω| > |Ω 0 ∪ Ω 1 | for a ball (see Figure 3).

From Theorem 3.1, we can also infer a couple of upper bounds on µ p,q , in terms of measure and diameter. µ p,q (Ω) < λ p,p (B) |Ω|

1-p q diam(B) diam(Ω) p ,
and also

(4.8) µ p,q (Ω) < λ p,q (B) |B| |Ω| p N + p q -1
, where B is any N -dimensional ball.

Proof. We first observe that (4.7) with p = q coincides with (3.1). Let p < q, by applying Lemma 2.2 we get µ p,q (Ω) ≤ |Ω| 1-p q µ p,p (Ω). It is then sufficient to apply Theorem 3.1 with q = p in order to estimate the right-hand side. This proves (4.7).

In order to prove (4.8), we can use the isodiametric inequality Remark 4.4. From estimate (4.7), we have that for p < q the quantity µ p,q (Ω) can not be bounded from below in terms on diam(Ω) only. In other words, for q > p we have

inf{µ p,q (Ω) : Ω ⊂ R N convex, diam(Ω) ≤ c} = 0.
Indeed, for any sequence of convex sets

{Ω k } k∈N ⊂ R N such that lim k→∞ |Ω k | = 0 and diam(Ω k ) = c,
we have that µ p,q (Ω k ) converges to 0, as k tends to ∞.

As in the case p = q, we can then ask whether the following shape optimization problem (4.10) sup{µ p,q (Ω) : Ω open and bounded convex set, diam(Ω) ≥ c}, admits a solution or not. We have the following existence result.

Theorem 4.5 (Existence of a maximizer). Let 1 < p < q < p * , for every c > 0 problem (4.10) admits a solution. In other words, there exists an open and bounded convex set K ⊂ R N such that µ p,q (Ω) diam(Ω)

p+ N p q -N ≤ µ p,q (K) diam(K) p+ N p q -N , for every Ω ⊂ R N open
µ p,q (Ω k ) ≥ N c 2 > 0, for every k ∈ N.
Since µ p,q scales like a length to a negative power, we can also assume that

diam(Ω k ) = c, for every k ∈ N.
Finally, thanks to (4.7) we can suppose that there exists a uniform constant δ > 0 such that (4.12)

|Ω k | ≥ δ, for every k ∈ N,
since otherwise we would have that µ p,q (Ω k ) goes to zero. Thanks to the bound on the diameters, we can assume that the whole sequence {Ω k } k∈N is contained in a common compact set D ⊂ R N . Thus the sequence is relatively compact for the complementary Hausdorff topology in D: more precisely, there exists an open set Ω ⊂ D such that Ω k (up to a subsequence) converges in the Hausdorff complementary distance to Ω (see [START_REF] Henrot | Shape variation and optimization] Une analyse géométrique[END_REF]Corollaire 2.2.24]). Moreover, Ω is still convex and its diameter equals c (see [START_REF] Henrot | Shape variation and optimization] Une analyse géométrique[END_REF]Section 2.2.3]). We also observe that the characteristic functions {1 Ω k } k∈N converges to 1 Ω strongly 5 in L 1 (D) and * -weakly in L ∞ (D).

Without loss of generality, we can assume that Ω contains the origin, since µ p,q (Ω) is not affected by translations. We are now going to prove that (4.13) lim sup

k→∞ µ p,q (Ω k ) ≤ µ p,q (Ω).
At this aim, let us take u ∈ W 1,p (Ω) a function attaining the infimum in the definition of µ p,q (Ω) > 0. Since Ω contains the origin, for every ε > 0 the set

Ω ε = (1 + ε) Ω is such that Ω ⋐ Ω ε ,
5 By convexity, a uniform bound on diam(Ω k ) implies a uniform bound on their perimeters and measures.

Then it is sufficient to use the compact embedding BV (D) ֒→ L 1 (D).

then by Hausdorff convergence for every ε > 0 there exists k ε ∈ N such that

Ω k ⊂ Ω ε , for every k ≥ k ε .
We also set

u ε (x) = u x 1 + ε , x ∈ Ω ε ,
then for every 0 < ε < 1 and every k ≥ k ε , we take t ε,k ∈ R such that

Ω k |u ε -t ε,k | q dx = min t∈R Ω k |u ε -t| q dx,
We claim that the sequence {t ε,k } k∈N in bounded uniformly in k and 0 < ε < 1, i.e. there exists C > 0 such that

(4.14) |t ε,k | ≤ C, for every 0 < ε < 1 and k ≥ k ε .
Indeed, observe that by convexity of the map τ → τ q and (4.12), we have

Ω k |u ε -t ε,k | q dx ≥ 1 2 q-1 |Ω k | |t ε,k | q - Ω k |u ε | q dx ≥ δ 2 q-1 |t ε,k | q - Ωε |u ε | q dx = δ 2 q-1 |t ε,k | q -(1 + ε) N Ω |u| q dx,
and on the other hand

Ω k |u ε -t ε,k | q dx ≤ Ω k |∇u ε | p dx µ p,q (Ω k ) ≤ 2 (1 + ε) N -p N c Ω |∇u| p dx,
where we used (4.11) and the very definition fo u ε . By keeping the two estimates together, we finally get (4.14).

Thus we can suppose that t ε,k converges (up to a subsequence) to t ε ∈ R as k goes to ∞, and t ε is in turn uniformly bounded. Then we get lim sup

k→∞ µ p,q (Ω k ) ≤ lim sup k→∞ Ω k |∇u ε | p dx Ω k |u ε -t ε,k | q dx p q ≤ Ω |∇u ε | p dx Ω |u ε -t ε | q dx p q
for every 0 < ε < 1, where we also used the * -weak convergence of the characteristic functions, recalled above. We now observe that

lim ε→0 Ω |u ε -t ε | q dx = Ω |u -t| q dx ≥ min t∈R Ω |u -t| q dx,
where t ∈ R is an accumulation point of the sequence {t ε } ε>0 , and also

lim ε→0 ∇u ε -∇u L p (Ω) = 0.
Thus it is now sufficient to take the limit as ε goes to 0 in order to get (4.13). This finally gives that Ω is a solution of (4.10). 4.2. The case p > q. In this case, we can show that an upper bound on µ p,q like that of (4.1) can not hold true and actually we have

sup{µ p,q (Ω) : Ω ⊂ R N convex, diam(Ω) ≥ c} = +∞. Indeed, for every sequence of open convex sets {Ω n } n∈N ⊂ R N such that diam(Ω n ) = c > 0 and lim n→∞ |Ω n | = 0,
we have lim

n→∞ µ p,q (Ω n ) = +∞.
Actually, this is a consequence of the following estimate.

Proposition 4.6. Let 1 < q < p and Ω ⊂ R N be an open and bounded convex set. Then we have

(4.15) π p diam(Ω) p |Ω| q p -1 ≤ µ p,q (Ω),
where the constant π p is given by

π p = 2 (p-1) 1 p 0 1 - t p p -1 -1 p dt.
Proof. Again by Lemma 2.2 with s = p > q, we get µ p,p (Ω) ≤ |Ω| 1-q p µ p,q (Ω).

By using the following lower bound on µ p,p (Ω) (see [9, Theorem 1.1])

µ p,p (Ω) ≥ π p diam(Ω) p ,
we can then conclude By combining (4.15) with the isodiametric inequality (4.9), we get the following counterpart of Theorem 4.1 for the case q < p (4.16)

µ p,q (Ω) ≥ π p |B| 1 q -1 p diam(B) p diam(B) diam(Ω) p+ N p q -N
, for every Ω ⊂ R N open and bounded convex set (as always B denotes any N -dimensional ball). Thus this time it is the minimum problem

inf{µ p,q (Ω) : Ω ⊂ R N convex, diam(Ω) ≤ c},
that actually makes sense. By proceeding as in the proof of Theorem 4.5, it is not difficult to see that the previous problem admits indeed a solution.

A nodal domain property

If u is a function achieving the infimum in the problem defining µ p,q (Ω), then by nodal domain we mean every connected component of the sets {x ∈ Ω : u(x) > 0} and {x ∈ Ω : u(x) < 0}.

As a consequence of Theorems 3.1 and 4.1, in the case p ≥ q we have the following result.

Proposition 5.1. Let Ω ⊂ R N be an open and bounded convex set and 1 < p ≤ q < p * . Then

(5.1) µ p,q (Ω) < λ p,q (Ω).

Moreover, every nodal domain of a function achieving µ p,q (Ω) has to intersect ∂Ω.

Proof. The proof of (5.1) immediately follows from (3.1) in connection with the Faber-Krahn inequality, i.e.

|B|

p q + p N -1 λ p,q (B) ≤ |Ω| p q + p N -1 λ p,q (Ω) and the isodiametric inequality.

To prove the second assertion, let us argue by contradiction. We take v achieving µ p,q (Ω) and we assume that the open set {x ∈ Ω : v > 0} has a connected component ω which is compactly contained in Ω. We can further suppose that v L q (Ω) = 1, then v ∈ W 1,p 0 (ω) and it solves -∆ p v = µ p,q (Ω) v q-1 , in ω, so that ω |∇v| p = µ p,q (Ω) ω |v| q dx ≤ µ p,q (Ω)

ω |v| q dx p q
, thanks to the fact that

1 = Ω |v| q dx ≥ ω |v| q dx,
and p/q ≤ 1. This implies that λ p,q (ω) ≤ µ p,q (Ω).

By using the strict monotonicity of λ p,q (Ω) with respect to set inclusion and (5.1), we then get λ p,q (Ω) < λ p,q (ω) ≤ µ p,q (Ω) < λ p,q (Ω), which gives the desired contradiction.

Remark 5.2. When p = q = 2, the previous argument to infer that first nontrivial Neumann eigenfunctions can not have a closed nodal line was originally due to Pleijel (see [START_REF] Pleijel | Remarks on Courant's nodal line theorem[END_REF]). For the Laplacian, inequality (5.1) was conjectured by Kornhauser and Stakgold (see [START_REF] Kornhauser | A variational theorem for 2u + u = 0 and its applications[END_REF]) and then proved by Szegő and Weinberger as a consequence of their celebrated inequality (1.4).

6. Limit cases 6.1. Sub-conformal case. We consider 1 < p < N and for an open bounded set Ω ⊂ R N we introduce the limit quantity

µ p,p * (Ω) = inf u∈W 1,p \{0}          Ω |∇u| p dx Ω |u| p * dx p p * : Ω |u| p * -2 u dx = 0          .
We also set

T N,p = min u∈W 1,p 0 (R N )\{0} R N |∇u| p dx R N |u| p * dx p p * ,
i.e. T N,p is the best constant in the Sobolev inequality for W 1,p 0 (R N ) (see for instance [START_REF] Talenti | Best constant in Sobolev inequality[END_REF]). We recall that the previous minimum is (uniquely) attained by functions of the form

x → c U x -x 0 λ , c ∈ R \ {0}, x 0 ∈ R N , λ > 0,
where U is the C ∞ decreasing function (6.1)

U (̺) = 1 1 + ̺ p/(p-1) N -p p , ̺ ≥ 0.
The following result is well-known. We provide a proof for the reader's convenience. Proof. At first we notice that by Lemma 2.2 λ p,p * (Ω) ≤ |Ω| p q -p p * λ p,q (Ω), while by the embedding W 1,p 0 (Ω) ⊂ W 1,p 0 (R N ) we get T N,p ≤ λ p,p * (Ω), thus we can conlude that T N,p ≤ lim inf q→p * λ p,q (Ω). To prove the limsup inequality, let r > 0 and x 0 ∈ Ω be such that the ball B r (x 0 ) is contained in Ω, which is always possible since Ω is open. Without loss of generality, we can suppose that r = 1 and x 0 = 0. By recalling the definition (6.1) of U , we define

u n (x) = U (n |x|) -U (n) + , n ∈ N,
which belongs to W 1,p 0 (Ω). We then observe that lim sup

q→p * λ p,q (Ω) ≤ lim sup q→p * R N |∇u n | p dx R N |u n | q dx p q = R N |∇u n | p dx R N |u n | p * dx p p * , n ∈ N.
By definition Proof. The equality (6.6) can be proved along the same lines as Lemma 2.3.

R N |∇u n | p dx = n p-N Bn(0) U ′ (
To prove the estimate (6.4) is sufficient to take the limit on both sides of (3.1) and use (6.6) and (6.2).

The simple estimate (6.4) can indeed be enhanced.

µ p,p * (Ω) ≤ inf x 0 ∈∂Ω γ Ω (x 0 ) 1-p p * T N,p ,
where for every x 0 ∈ ∂Ω the quantity γ Ω (x 0 ) is the blow-up measure of Ω at x 0 , i.e.

γ Ω (x 0 ) = lim r→+∞ |r (Ω -x 0 ) ∩ B 1 (0)| ω N .
Proof. We define

u n (x) = U (n 2 |x|) -U (n), if |x| < 1/n, 0,
otherwise, where U is still defined by (6.1). For every n ∈ N, the function u n is radially symmetric decreasing, supported in the ball B 1/n (0). For x 0 ∈ ∂Ω we define

Ω n = x ∈ Ω : |x -x 0 | < 1 n ,
and consider the functions

ϕ n (x) = u n (x -x 0 ) • 1 Ωn . For every n ∈ N there exists a unique t n ∈ R such that Ω |ϕ n -t n | p * -2 (ϕ n -t n ) dx = 0.
and we have t n = o(1) as n goes to ∞. Then we get (6.5)

µ p,p * (Ω) ≤ lim n→∞ Ω |∇ϕ n | p dx Ω |ϕ n -t n | p * dx p p * = lim n→∞ Ω |∇ϕ n | p dx Ω |ϕ n | p * dx p p *
, where we used that

lim n→∞ Ω |ϕ n -t n | p * dx Ω |ϕ n | p * dx = 1
With a change of variables, we get

Ω |∇ϕ n | p dx = n 2 (p-N ) n 2 (Ω-x 0 )∩Bn(0) |U ′ (|y|)| p dy,
and

Ω |ϕ n | p * dx = n -2 N n 2 (Ω-x 0 )∩Bn(0) |U (|y|) -U (n)| p * dy.
By using these in (6.5), we get

(6.6) µ p,p * (Ω) ≤ lim n→∞ n 2 (Ω-x 0 )∩Bn(0) |U ′ (|y|)| p dy n 2 (Ω-x 0 )∩Bn(0) |U (|y|) -U (n)| p * dy p p * .
We then observe that by convexity of Ω, the sets n (Ω-x 0 )∩B 1 (0) have uniformly bounded measures and perimters. Then we get that n (Ω -x 0 ) ∩ B 1 (0) converges (up to a subsequence) in the sense of characteristic functions to a set V. Here V ⊂ B 1 (0) has measure

|V| = γ Ω (x 0 ) ω N ,
and is a conical set in the following sense if x ∈ V then t x ∈ V for every t ∈ 0, 1 |x| .

This in turn implies that n 2 (Ω -x 0 ) ∩ B n (0) converges to the infinite cone centered at the origin and generated by V, i.e.

x ∈ R N \ {0} :

x |x| ∈ ∂V .

By using this information in (6.6), the radial symmetry of the functions involved gives the desired result

Observe that for a convex set we have γ(x 0 ) ≤ 1/2 for every x 0 ∈ ∂Ω. Moreover, if the convex set is C 1 , then γ(x 0 ) = 1/2 for every x 0 ∈ Ω. In this case, we have the following consequence. T N,p .

6.2. Conformal case. We consider the conformal case, i.e. we take p = N in (3.1) which gives (6.7)

µ N,q (Ω) < λ N,q (B) diam(B) diam(Ω) N 2 q
, where B is any N -dimensional ball. In what follows we set

α N = N (N ω N ) 1 N -1 .
We have the following preliminary result. Proposition 6.5. Let Ω ⊂ R N be an open and bounded convex set. There exists a dimensional constant β N such that we have

N N -1 β N e N -1 ≤ lim q→∞ q µ N,q (Ω) ≤ N N -1 α N e N -1
.

For β N we have the estimate α N 2 1/(N -1) ≤ β N ≤ α N .

Proof. To prove the upper bound, it is sufficient to multiply (6.7) by q and then use that lim q→∞ q λ N,q (B) = N N -1

α N e N -1 , which follows from 6 [17, Lemma 2.2]. To prove the lower bound, we can adapt the proof of [START_REF] Ren | On a two-dimensional elliptic problem with large exponent in nonlinearity[END_REF]Lemma 2.1]. First of all, we recall that in [START_REF] Cianchi | Moser-Trudinger inequalities without boundary conditions and isoperimetric problems[END_REF] it is shown that sup α N 2 1/(N -1) , since t = u Ω is always an admissible competitor. For every q ≥ N , let u q ∈ W 1,N (Ω) be a function achieving µ N,q (Ω), normalized by

Ω |∇u q | N dx = 1, q ≥ N.
We also set t q to the the minimizer of t → Ω exp α N |u q -t| N ′ dx, then we have min t∈R Ω |u q -t| q dx ≤ Ω |u q -t q | q dx = α

-q N ′ N Ω α N |u q -t q | N ′ q N ′ dx ≤ Γ q N ′ + 1 Ω exp α N |u q -t q | N ′ dx ≤ Γ q N ′ + 1 M N (Ω), which in turn implies 1 Γ q N ′ + 1 1 M N (Ω) ≤ 1 min t∈R Ω |u q -t| q dx = Ω |∇u q | N dx q N min t∈R Ω |u q -t| q dx , that is q   1 Γ q N ′ + 1 1 M N (Ω)   N q
≤ q µ N,q (Ω). By using Stirling formula, we can conclude. 6 The result in [START_REF] Ren | On a two-dimensional elliptic problem with large exponent in nonlinearity[END_REF] is for N = 2, but the very same argument can be easily adapted for a geneal N ≥ 2.

For completeness, we give the following technical result which shows that the maximization problem defining M N (Ω) is unchanged if we replace W 1,N (Ω) by any dense subset. Observe that we have Then there exists t ∈ R such that t n converges (up to a subsequence) to t. By Fatou Lemma we then get lim inf

M N (Ω) ≥ Ω exp β N |v n -t n | N ′ ≥ |Ω| + β N Ω |v n -t n | N ′ dx ≥ |Ω| + β N 2 N ′ -1 |Ω| |t n | N ′ -β N Ω |v n | N ′ dx,
n→∞ Ω exp β N |v n -t n | N ′ dx ≥ Ω exp β N |u -t| N ′ dx ≥ Ω exp β N |u -t u | N ′ dx ≥ M N (Ω) -ε,
thanks to the minimality of t u . Since the sequence {v n } n∈N is admissible for the problem defining M ′ N (Ω), we thus get M ′ N (Ω) ≥ M N -ε. which in turn gives the desired result, thanks to the arbitrariness of ε > 0.

1 . Introduction 1 . 1 .

 111 Overview. Given an open bounded connected Lipschitz set Ω ⊂ R N and an exponent 1 < p < ∞, we consider λ p and µ p the first nontrivial eigenvalues of the p-Laplace operator with Dirichlet and Neumann boundary conditions, respectively. We remind that these can be variationally characterized by λ p (Ω) = min v∈W 1,p 0 (Ω) Ω |∇v| p dx : Ω |v| p dx = 1 , and µ p (Ω) = min v∈W 1,p (Ω) Ω |∇v| p dx : Ω |v| p dx = 1 and Ω |v| p-2 v dx = 0 .

Figure 1 .

 1 Figure 1. The construction of the two caps Ω 0 and Ω 1 .

Figure 2 .

 2 Figure 2. The maximizing sequence Ω k of Theorem 3.1.

  |x|) is a Dirichlet eigenfunction of -∆ p of a N -dimensional ball of radius d/2, namely B. Hence η ≥ λ p,p (B) and we get lim inf k→∞ γ k (Ω) ≥ λ p,p (B), which concludes the proof.

  B\T Ω |u| q dx = 0, and since u > 0 in B, this would imply |B \ T Ω | = 0 and thus (4.6) A(Ω) = 0 and |Ω| = |Ω 0 ∪ Ω 1 |.

Figure 3 .

 3 Figure 3. The two caps Ω 0 and Ω 1 can not cover the whole ball.

Corollary 4 . 3 .

 43 Let Ω ⊂ R N be an open bounded convex set and 1 < p ≤ q < p * . Then we have (4.7)

Lemma 6 . 1 .

 61 Let Ω ⊂ R N be an open set with finite measure. Then (6.2) lim qրp * λ p,q (Ω) = T N,p .

Proposition 6 . 3 .

 63 Let Ω ⊂ R N be an open and bounded convex set. The following estimate holds true

Corollary 6 . 4 .

 64 Let Ω ⊂ R N be an open and bounded convex C 1 set. Then µ p,p * (Ω)

u∈W 1 ,

 1 N (Ω) Ω exp α N 2 1/(N -1) |u -u Ω | N ′ dx : Ω |∇u| N dx ≤ 1 < +∞,where u Ω denotes the mean of u over Ω. This implies that there exists a maximal constant β N such that the quantityM N (Ω) = sup W 1,N (Ω) min t∈R Ω exp β N |u -t| N ′ dx : Ω |∇u| N dx ≤ 1 ,stays finite and of course we haveβ N ≥

Lemma 6 . 6 .

 66 Let Ω ⊂ R N be an open and bounded convex set. If X(Ω) is a dense subset of W 1,N (Ω), thenM N (Ω) = sup X(Ω) min t∈R Ω exp β N |u -t| N ′ dx : Ω |∇u| N dx = 1 . Proof. Let us call M ′ N (Ω) = sup X(Ω) min t∈R Ω exp β N |u -t| N ′ dx : Ω |∇u| N dx = 1 , then of course we have M ′ N (Ω) ≤ M N (Ω).In order to prove the reverse inequality, for every ε > 0 let u ∈ W 1,N (Ω) be an admissible function such thatM N (Ω) -ε ≤ Ω exp β N |u -t u | N ′ dx,where t u attains the minimum ofΩ exp β N |u ε -t| N ′ dx.Let {u n } n∈N ⊂ X(Ω) be a sequence strongly converging to u, thus in particularlim n→∞ Ω |∇u n | N dx = Ω |∇u| N dx,and u n converges almost everywhere in Ω to u (up to a subsequence), then we define the new sequencev n = u n ∇u n L N (Ω) , n ∈ N. Let t n ∈ R N be such that Ω exp β N |v n -t n | N ′ dx = min t∈R Ω exp β N |v n -t| N ′ dx ≤ M N (Ω).

  which implies that the sequence of real numbers {t n } n∈N is bounded, sincelim n→∞ Ω |v n | N ′ dx = Ω |u| N ′ dx < +∞.

  and bounded convex set.Proof. By Theorem 3.1 we already know that this supremum is finite. Let us call N c this supremum and take a maximizing sequence of admissible sets {Ω} k∈N ⊂ R N , of course we

	can suppose that
	(4.11)

Acknowledgments. The first author has been partially supported by the Gaspard Monge Program for Optimization (PGMO), by EDF and the Jacques Hadamard Mathematical Foundation, through the research contract MACRO. Part of this work has been done


Jonathan Mercier-Ganady
email: jonathan.mercier-ganady@inria.fr

Fabien Lotte
email: fabien.lotte@inria.fr

Emilie Loup-Escande
email: emilie.escande@gmail.com

Maud Marchal
email: maud.marchal@irisa.fr

Anatole L Écuyer
email: anatole.lecuyer@inria.fr

The Mind-Mirror: See Your Brain in Action in Your Head Using EEG and Augmented Reality

Keywords: Augmented Reality, EEG, Brain Activity, Mirror, Visualization, BCI H.5.1 [Information Interfaces and Presentation]: Multimedia Information Systems-Artificial, augmented, and virtual realities, H.5.2 [Information Interfaces and Presentation]: User Interfaces-Input devices and strategies, I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism-Virtual reality

Imagine you are facing a mirror, seeing at the same time both your real body and a virtual display of your brain in activity and perfectly superimposed to your real image "inside your real skull". In this paper, we introduce a novel augmented reality paradigm called "Mind-Mirror" which enables the experience of seeing "through your own head", visualizing your brain "in action and in situ". Our approach relies on the use of a semi-transparent mirror positioned in front of a computer screen. A virtual brain is displayed on screen and automatically follows the head movements using an optical face-tracking system. The brain activity is extracted and processed in real-time with the help of an electroencephalography cap (EEG) worn by the user. A rear view is also proposed thanks to an additional webcam recording the rear of the user's head. The use of EEG classification techniques enables to test a Neurofeedback scenario in which the user can train and progressively learn how to control different mental states, such as "concentrated" versus "relaxed". The results of a user study comparing a standard visualization used in Neurofeedback to our approach showed that the Mind-Mirror could be successfully used and that the participants have particularly appreciated its innovation and originality. We believe that, in addition to applications in Neurofeedback and Brain-Computer Interfaces, the Mind-Mirror could also be used as a novel visualization tool for education, training or entertainment applications.

INTRODUCTION

Nowadays ElectroEncephaloGraphy (EEG) and Brain-Computer Interfaces (BCI) provide a unique access to brain activity in realtime. This enables new kinds of applications such as "Neurofeedback" which consists in progressively learning to control a brain pattern while continuously observing a real-time feedback about this pattern [START_REF] Demos | Getting Started with Neurofeedback[END_REF]. Neurofeedback is considered today in the medical field for treating different kinds of pathology, such as attention deficits [START_REF] Lubar | Evaluation of the effectiveness of EEG neurofeedback training for ADHD in a clinical setting as measured by changes in TOVA scores, behavioral ratings, and WISC-R performance[END_REF]. In most Neurofeedback applications, the feedback used is visual and consists in basic graphics such as 2D gauges increasing or decreasing over time. More elaborated visual feedback and 3D graphics can be used to display brain activity, but they are generally used in off-line visualization software [START_REF] Wolters | Influence of tissue conductivity anisotropy on EEG/MEG field and return current computation in a realistic head model: a simulation and visualization study using high-resolution finite element modeling[END_REF], for instance for medical diagnosis. In this paper we study the combination of Augmented Reality (AR), 3D Visualization, and ElectroEncephaloGraphy. We introduce a novel setup called "Mind-Mirror" which enables the visualization of our own brain activity "inside our own head" by superimposition (see Figure 1-left). The brain activity is extracted in realtime using an EEG acquisition machine and is displayed in a mirrorbased AR setup in front of the user's skull in semi-transparency. The Mind-Mirror could therefore be used for entertaining or educational activities, or for Neurofeedback applications, i.e., training to control our brain while being in a more natural, engaging, and immersive environment.

The remainder of this paper is organized as follows. Section 2 describes related work on augmented reality and brain activity visualization. Section 3 introduces the Mind-Mirror system and details its main components. Section 4 presents a pilot study comparing the Mind-Mirror to a classical 2D visualization used in current Neurofeedback applications, in the context of controlling brain patterns related to concentration versus relaxation mental activities. The paper ends with a general discussion and a conclusion.

RELATED WORK

Since the beginning of augmented reality the properties of halfsilvered screens have been used to superimpose various graphics on top of real objects. These objects could be buttons [START_REF] Knowlton | Computer displays optically superimposed on input devices[END_REF] or virtual enemies in a video game [START_REF] Piekarski | ARQuake: the outdoor augmented reality gaming system[END_REF]. In both cases, displayed objects would seem very realistic to the user since they are superimposed over real objects. These virtual objects are not directly dependent on the mental state of the user. A recent paper has shown that a high quality display of flames and smoke on a user's hand in AR could induce a heat sensation [START_REF] Weir | BurnAR: Feel the heat[END_REF]. Thus the realistic display of a virtual stimulation on a body part seems to have a strong impact on the sensations of users.

Mirror-based systems have already been used to see virtual objects mapped on users in so-called "magic mirror" augmented re-ality [START_REF] Maes | The ALIVE system: Full-body interaction with autonomous agents[END_REF]. For instance Maes et al. conceived the "Artificial Life Interactive Video Environment" (ALIVE) that allowed interaction between an autonomous virtual agent, e.g. a dog, and the real user. Users could perform gestures and see themselves in a magic mirror while the virtual agent would perform some actions depending on the gesture. In some cases, the mirror can be used to display virtual objects "inside" the user's real body. For instance Blum et al. used a half-silvered mirror to show various organs "in situ" [START_REF] Blum | mirracle: An augmented reality magic mirror system for anatomy education[END_REF]. However, the displayed organs were static and not dependent on the state of the real organs, since no device was used to capture their activity.

The visualization of brain activity has been extensively studied, often by using recorded data and viewing offline the most active areas of the brain [START_REF] Baillet | Academic software applications for electromagnetic brain mapping using MEG and EEG[END_REF]. These active areas can be represented through a flat 2D map of the brain (using changing colors to show high-and low-activity zones) or a topography of surface activity [START_REF] Mellinger | An MEG-based brain-computer interface[END_REF]. However the existing approaches do not currently incorporate a direct user interaction.

In a recent study on Neurofeedback Hwang et al. used a realtime visualization of brain activity to train the user to BCI control [START_REF] Hwang | Neurofeedback-based motor imagery training for brain-computer interface[END_REF]. Participants could visualize their cortical mu (8-12 Hz) rhythm activity in real-time, improving the BCI training and thus the performance when using the motor imagery paradigm. In this desktop setup participants were able to have a direct view of their brain activity using a simple 2D colored image of a virtual brain.

In the following section we introduce a novel approach relying on AR and 3D interaction for brain activity visualization in a magicmirror setup.

THE MIND-MIRROR SYSTEM 3.1 Concept

The Mind-Mirror is an augmented reality paradigm that enables users to see in a mirror both their real head and a virtual display of their brain in activity and perfectly superimposed to their real head. In other words, when using the Mind-Mirror, users can see "through their own head", visualizing their brain "in action and in situ".

Our approach relies on the use of a semi-transparent mirror positioned in front of a computer screen. A virtual brain is displayed on-screen and automatically follows the head movements using an optical face-tracking system. The brain activity is extracted and processed in real-time thanks to an EEG cap worn by the user. Since a mirror alone can only show the frontal parts of the brain, we have also added a rear-view. This view uses a webcam placed behind and above the head of the user.

System Description

The Mind-Mirror system is composed of several components which are displayed in Figure 2: • EEG recording and amplification: EEG acquisition and amplification is performed using a set of electrodes mounted on a cap (1) and an EEG amplifier (g.tec, g.USBamp) (4).

• Webcam: a webcam is used and positioned on a wall behind the user (2) in order to capture the image of the back of the head of the user and display it in a rear-view window on the Mind-Mirror.

• Face and head tracking: Face and head tracking are achieved using a Microsoft Kinect camera (3) and its development kit [START_REF]Kinect face tracking SDK[END_REF]. It is used to overprint the virtual brain at the position of the user's head in real-time. The Kinect development kit uses here depth and color sensing of the camera to track the human face using proprietary algorithms.

• Signal-processing: Signal processing is done on a computer (5) using the OpenViBE software platform [START_REF] Renard | OpenViBE: An open-source software platform to design, test, and use brain-computer interfaces in real and virtual environments[END_REF]. • Display: The display system consists in a computer screen [START_REF] Hwang | Neurofeedback-based motor imagery training for brain-computer interface[END_REF] that supports a half-silvered foil (7) applied onto a thin plastic plate. Due to the half-silvered foil only the screen parts that are bright can be seen. The rest of the screen surface appears opaque and reflective, as with a classical mirror. Using a real mirror enables users to see themselves directly, which would not be possible with a camera-based AR setup.

EEG data acquired with OpenViBE software [START_REF] Renard | OpenViBE: An open-source software platform to design, test, and use brain-computer interfaces in real and virtual environments[END_REF] and head movement data from the Kinect are retrieved and processed within a Unity3D-based application that simulates and displays the virtual brain. Virtual brain activity visualizations are displayed in real-time (framerates between 25 and 80 Hz depending on the visualization type). Face-tracking and head movements are detected and processed with a latency of less than 500 ms.

Visualization Tools

As mentioned previously, the Mind-Mirror can be used to display the brain in action and "in situ". Different visualization techniques can be used. We have explored the following ones for the Mind-Mirror.

Brain Topography

The brain topography visualization enables the display of the EEG signal power over the brain surface, as shown in Figure 3-left. This representation allows for a quick overview of the most active brain areas, i.e., the areas with the largest EEG signal power.

3D Volumetric Representation Using Inverse Solution

EEG signals correspond to brain activity at the surface of the scalp. Fortunately, it is possible to estimate the brain activity in the whole brain volume using algorithms known as inverse solutions [START_REF] Pascual-Marqui | Review of methods for solving the EEG inverse problem[END_REF].

The volumetric brain activity hence estimated can then be displayed with a 3D model of the brain using voxels. This visualization allows for a comprehensive 3D view of the brain activity, as shown in Figure 3-right.

Visualization Integrating Mental State Classification

One of the main goals of BCI or Neurofeedback training applications is to help users gain control of their own brain activity, in order to be able to modulate it and reach a specific mental state, e.g., a relaxed or a concentrated mental state. Therefore, in such applications, it is important for users to be aware of the mental state identified by the system, so that they can identify the relevant features of their brain activity and learn how to control them. Current brain activity visualization tools, e.g., the ones mentioned above, are not well suited for such purposes. Indeed, they do not show the mental state identified by the system, and display with equal focus the activity in the entire brain, making it difficult for the user to identify the relevant features in all the displayed information. In contrast, classical BCI feedback and Neurofeedback only show simple 2D graphics such as gauges [START_REF] Neuper | Brain-Computer Interfaces, chapter Neurofeedback Training for BCI Control[END_REF], only indicating the mental state identified, without giving information about the underlying brain activity despite this being recommended for a successful learning [START_REF] Lotte | Flaws in current human training protocols for spontaneous brain-computer interfaces: lessons learned from instructional design[END_REF].

Therefore, in this paper, we propose a new brain activity visualization technique that aims at using the best of both worlds. Indeed, our approach 1) gives information about the underlying brain activity, using surface topography, and highlights some specific brain areas that are relevant for the targeted mental state classification, and 2) displays simultaneously the mental state currently identified by the BCI, which altogether is expected to help users identifying the relevant features of their brain activity in order to control more efficiently the targeted brain activity. With this new representation, the whole virtual brain is colorized according to the mental state of the user, as identified by the BCI system. Currently, we have explored this approach with a BCI that estimates the concentration and relaxation level of the user (see Section 4). When the system detects that the user is relaxed, then the whole virtual brain is colored in blue, otherwise it is colored in red. Moreover, the more blue the brain is displayed, the more confident the BCI system is in the corresponding mental state estimation. This confidence is computed using the output of the linear regression used in the BCI system (see Section 4.1): the larger the absolute value of the linear regression output, the more confident the estimation [START_REF] George | Using scalp electrical biosignals to control an object by concentration and relaxation tasks: design and evaluation[END_REF]. Then, in order to help the user pay attention to the most relevant information and not overwhelm him, the most important brain areas for the estimation of the mental states are shown with a more intense color, as shown in Figure 4.

The most important brain areas were chosen using machine learning tools, as exposed in Section 4.1. In particular, the surface topography only displays the EEG signals in the frequency band that is the most relevant for classification (i.e., the most discriminative frequency band in the Filter Bank Common Spatial Pattern algorithm [START_REF] Ang | Filter bank common spatial pattern algorithm on BCI competition IV datasets 2a and 2b[END_REF]), and only the most relevant channels (i.e., the channels with the largest absolute weights in the Common Spatial Patterns filters for this frequency band) are displayed with more intense colors.

Surface topography is displayed on a 3D mesh of a brain using vertex coloration. In our current simple implementation the surface topography is computed as follows. The color of vertex j (V j ) is defined using Equation [START_REF] Ang | Filter bank common spatial pattern algorithm on BCI competition IV datasets 2a and 2b[END_REF]. A state color S is defined as red if the detected mental state is concentrated, blue otherwise. Electrode Relevance (R i ) represents the relevance of the electrode i for one mental state, computed during the initial calibration phase (see Section 4.3). P i represents the EEG band power at electrode i and D i, j the distance between electrode i and vertex j. The electrode positions used are those of the international 10-20 system. Note that other computations could be used such as Surface Laplacians [START_REF] Renard | OpenViBE: An open-source software platform to design, test, and use brain-computer interfaces in real and virtual environments[END_REF].

V j = S * n ∑ i=1 R i * P i D i, j (1)

Rear-View

The rear-view, which allows users to see what is happening on the rear parts of their brain, is displayed on top of our mirror (see Figure 5). This rear-view makes use of a standard computer webcam placed above and behind the user. Another virtual brain is used here and also superimposed over the image recorded by the webcam.

PILOT STUDY

In order to assess our Mind-Mirror system, we performed a pilot study in which the participants used it as feedback in a BCI based on the concentration and relaxation levels. More precisely, with this BCI, participants had to put themselves in a relaxed or in a concentrated state, such mental states being analyzed and identified by the BCI in real-time. These two mental states could then be assigned to two different commands (see, e.g., [START_REF] George | Using scalp electrical biosignals to control an object by concentration and relaxation tasks: design and evaluation[END_REF]), although this was not the case in this study. Here, we studied both the performance (in terms of successfully recognized mental state) and preference (using subjective questionnaires) of the participants using the Mind-Mirror to visualize their brain activity in real-time while using the BCI. We also compared the Mind-Mirror to a temporal gauge, a typical representation used within the BCI and neurofeedback community (see Figure 6). This indeed enabled us to study the impact of the more complex but also more immersive and informative brain activity representation that is the Mind-Mirror on participants' experience and BCI performance.

Experimental Apparatus

The experiment was conducted in a room without environmental noise or other source of distraction. The participants were comfortably seated in front of a computer screen.

EEG signals were acquired with a sampling frequency of 512Hz using a 16-channels g.USBamp acquisition system. EEG were recorded at channel locations T7, T8, F7, F8, Fp1, Fp2, C3, Cz, C4, O1, O2, F3, F4, P3, Pz and P4 according to the international 10-20 system. The ground electrode was placed on position FCz, and the reference electrode on the participant's left earlobe.

The concentration level of each participant was estimated using a subject-specific model obtained with machine learning [START_REF] George | Using scalp electrical biosignals to control an object by concentration and relaxation tasks: design and evaluation[END_REF]. More precisely, the Filter-Bank Common Spatial Pattern (FBCSP) algorithms [START_REF] Ang | Filter bank common spatial pattern algorithm on BCI competition IV datasets 2a and 2b[END_REF] was used to identify the most relevant EEG frequency bands and channels to discriminate the concentrated from the relaxed mental state [START_REF] George | Using scalp electrical biosignals to control an object by concentration and relaxation tasks: design and evaluation[END_REF]. Only frequency bands within the theta (4-7Hz) and alpha (8-14Hz) rhythms range were explored. Indeed, from a neurophysiological point of view, these rhythms are expected to be correlated with concentration [START_REF] Niedermeyer | Electroencephalography: Basic principles, clinical applications, and related fields[END_REF], and are less likely than higher frequency bands to be contaminated by muscle artifacts [START_REF] George | Using scalp electrical biosignals to control an object by concentration and relaxation tasks: design and evaluation[END_REF]. The FBCSP algorithm was optimized on training EEG signals collected during the initial calibration phase (see Section 4.3 for details on this phase). Then, FBCSP features were used to train a linear regression algorithm to estimate the participant's concentration level. Once the BCI is calibrated in this way, it can be used online. To do so, the FBCSP features were extracted from the EEG signals over the last 2s (using a sliding window scheme, with a 0.1s overlap between consecutive windows) and used as input for the linear regression, which output indicates the participant's concentration level. More precisely, a negative (respectively positive) output value means that the participant is in a relaxed (respectively concentrated) mental state.

Population

Twelve participants (aged from 21 to 30, mean=25, sd=2.8) took part in the experiment.

Experimental Plan

A calibration phase took place before the experiment. Participants were asked to concentrate for 60 seconds and then relax for 60 seconds without having any visual feedback. Participants were free to choose any cognitive activity for the concentration task. Several suggestions were made such as a "mental computation" task. The EEG data collected during this phase were used to calibrate the signal processing pipeline.

Two different representations were then compared: the Mind-Mirror and the temporal gauge. As the participants were concentrating or relaxing, the brain activity representations would show the current level of concentration or relaxation, respectively.

The experiment was divided into 12 trials. Each trial comprised a relaxation phase followed by a concentration phase, each of them lasting 25 seconds. The experiment lasted 10 minutes per representation and so 20 minutes for the whole experiment. The two brain activity representations were used by each participant. The half-silvered foil was manually applied on the monitor screen for the Mind-Mirror condition, and taken away for the gauge condition. Participants were divided into two groups. The first group started with the Mind-Mirror and the second group started with the temporal gauge.

Collected Data

For each trial and each participant, we recorded the BCI classifier output values resulting from the signal-processing pipeline. At the end of the experiment and for each representation, participants had to fill out a subjective Likert-scale questionnaire (1: strongly disagree to 7: strongly agree). Questions were: "Do you think that the representation is x", where x was one of the following criterion: (1) Comprehensible, (2) Motivating, (3) Simple, (4) Clear, (5) Innovative, (6) Original.

RESULTS

Classification performance results

We conducted statistical analysis in order to assess the BCI classification performance for each brain activity representation. To do so, we first compared the linear regression output between the relaxed state and the concentrated state for each of the two visualizations separately, in order to assess whether participants could control the BCI with them. A paired t-test showed significant differences in BCI output between the concentrated and relaxed states for the gauge visualization (t(11)=-4.82, p=0.0005) as well as for the Mind-Mirror visualization (t(11)=-2.73, p=0.02). We also compared the classification performances obtained with the two visualizations using a paired t-test. Such classification performances were assessed using the average difference between the linear regression output during the concentration and relaxation conditions, as in [START_REF] George | Using scalp electrical biosignals to control an object by concentration and relaxation tasks: design and evaluation[END_REF]. The larger the difference, the better the discrimination between the two mental states. We found no significant difference (p=0. [START_REF] Piekarski | ARQuake: the outdoor augmented reality gaming system[END_REF]) between the classification performance with the Mind-Mirror condition (average difference: 0.31 ± 0.39) and the gauge condition (average difference: 0.46 ± 0.33), although there might be a slight trend towards better performances with the gauge. The average values of the linear regression output for each mental state are provided in Table 1.

Questionnaire Results

We performed a Wilcoxon test on the visual representations for the different criteria. We found a significant effect for 4 criteria: Simplicity (z=-1.981, p=0.048), Clarity (z=-2.089, p=0.037), Innovation (z=-2.890, p=0.004) and Originality (z=-2.924, p=0.003). The participants found the Mind-Mirror less simple, less clear but more innovative and more original than the gauge representation. Table 2 summarizes the questionnaire results for the 6 criteria.

GENERAL DISCUSSION

The results obtained from the classification analysis and from the questionnaires enabled us to assess the pros and cons of the Mind-Mirror.

Concerning the pros, results from the classification analysis suggested that the participants were able to put themselves in a concentrated or in a relaxed state with the Mind-Mirror representation, hence showing that participants can successfully use the Mind-Mirror as a feedback for a neurofeedback/BCI application. Results also suggested that the Mind-Mirror, even if more complex than a classical gauge, does not seem to lead to substantially lower BCI performance, although there might be a trend towards slightly better performance with the gauge. Further studies with a larger population would be required to confirm this point. Results from the questionnaire suggested that participants found the Mind-Mirror to be more original and innovative than traditional BCI feedback (i.e., the gauge). In other words, this means that the Mind-Mirror gives a higher quality experience than a gauge, with a more engaging feedback environment. Moreover, as compared to the gauge, the Mind-Mirror provides more information, since it indicates in real-time which brain areas are active. The participants quoted the Mind-Mirror as "clear and easy to understand", "quite precise", "gives a good indication of the activated zones, without being too complex". The Mind-Mirror thus provides an explanatory feedback, informing about what is going on in the user's brain, whereas the gauge only provides a corrective feedback, only indicating whether the user is correctly relaxed or concentrated. These two feedback properties, namely, being engaging and explanatory, are recommended properties for the efficient learning of a skill such as here learning how to control brain activity [START_REF] Lotte | Flaws in current human training protocols for spontaneous brain-computer interfaces: lessons learned from instructional design[END_REF]. The evaluation reported in this paper is a short-term one, focused on feedback perception and usability. As such longer-term learning effects are unlikely to occur. In future works, it would be interesting to compare the classical gauge to the Mind-Mirror during multiple neurofeedback sessions, over multiple days. This would enable us to assess the users' learning curves (i.e., performance improvements over time), and confirm whether the Mind-Mirror -due to its properties -can make learning faster and better as theoretically expected [START_REF] Lotte | Flaws in current human training protocols for spontaneous brain-computer interfaces: lessons learned from instructional design[END_REF].

Concerning the Mind-Mirror cons, the user study highlighted that it was significantly more difficult to understand and to use than the classical gauge, at least for first-time users who qualified it as having a "slight lack of readability". This result is not unexpected since the Mind-mirror is a rich multidimensional feedback, thus a more complex one than the mono-dimensional gauge feedback. Users therefore need more time to get used to the Mind-Mirror feedback, to understand how it works and how to handle it. This may explain why the classification analysis revealed no increase in BCI performance with the Mind-Mirror. An alternative interpretation that could explain such performance is that the Mind-Mirror displays rich instantaneous information, but no temporal information, whereas the gauge displays an instantaneous information, more precisely the regression output, but also the previous outputs, hence showing a short history describing temporal variations. In the future it would be interesting to add this temporal information to the Mind-Mirror. This could be done, e.g., by displaying the last few surface topographies as well as the current one, with the older ones displayed in dimmer colors or steam-like appearance, and slowly fading away as time passes. It would also be interesting to perform a follow-up longer-term study, in order to assess users' performance and preference once they got used to the Mind-Mirror principle. Still regarding possible improvements to the Mind-Mirror design, some participants reported that they enjoyed the mirror principle with AR ("easier to control the movement without interfering with the task", "nice to look at and entertaining"), but that it made visually focusing on the brain or the user's reflection more difficult ("the virtual brain appears on the foreground whereas the reflection seems to be on the background"). In the future, it could be relevant to compare this mirror AR approach to a more classical AR design with a webcam filming the user and overlaying the brain, without the use of a mirror. This would prevent users from looking at themselves into the eyes as in a real mirror but could make the brain activity display easier to watch and to focus on.

As mentioned earlier, the Mind-Mirror has the potential to be used for multiple other applications, notably education, entertainment and neurofeedback training. Indeed, the Mind-Mirror could have educational purposes, as a support tool to teach brain biology and anatomy. For instance, users could select the brain area they are interested in by pointing to it using their own hands. Hand tracking could be done using the Kinect, and much meaningful information could be displayed about the selected brain area. This could be an engaging and effective way to learn, for instance, the different brain lobes names, their locations as well as how the brain works.

The Mind-Mirror could also be used for entertainment, as a motivating feedback in BCI-based video games for instance [START_REF] Lécuyer | Brain-computer interfaces, virtual reality, and videogames[END_REF]. It could also be used in serious games involving the brain, such as video games for brain fitness or for treating Attention Deficit Hyper Activity Disorders [START_REF] Lim | Effectiveness of a brain-computer interface based programme for the treatment of ADHD: A pilot study[END_REF].

Finally, and maybe more importantly, the Mind-Mirror could be a very powerful tool for Neurofeedback training, either for medical applications or for BCI control. Actually, it could be used to train users to learn how to modulate the brain activity in some areas of their brain. Indeed, as we have exposed before, the visualization tools offered by the Mind-Mirror are engaging, rich and explanatory which, from an instructional design point of view, is expected to lead to efficient skill learning, e.g., a BCI skill [START_REF] Lotte | Flaws in current human training protocols for spontaneous brain-computer interfaces: lessons learned from instructional design[END_REF]. The Mind-Mirror is thus expected to improve learning in BCI or neurofeedback applications in the long-term. We performed here a first short-term study with concentration and relaxation. In the future, it would be interesting to explore it with other BCI paradigms, e.g., with BCI based on imagination of movements [START_REF] Pfurtscheller | Motor imagery and direct braincomputer communication[END_REF].

CONCLUSION

In this paper we introduced the Mind-Mirror, a system which combines augmented reality and EEG in order to enable users to visualize their own brain operating inside their head. We proposed various brain activity visualization tools for this system, and identified multiple possible use-cases, including education, entertainment, neurofeedback and BCI. We also conducted a user study to compare feedback provided using the Mind-Mirror to a classical gauge feedback in a BCI experiment. Results suggested that participants found the Mind-Mirror more innovative and engaging than the gauge although more complex to use and understand. This complexity did not lead to a statistically significant drop in BCI performance. Overall, the Mind-Mirror seems to be a promising tool for a variety of applications, which would benefit from being validated by further studies.

Future work will concern more evaluations of the Mind-Mirror for BCI and neurofeedback applications. This implies formal experiments in other application contexts (e.g., game or education), as well as further studies to identify the best display parameters (e.g., color maps, contrasts, use of the mirror, etc.) for an optimal visualization and learning experience.

Figure 1 :

 1 Figure 1: The Mind-Mirror prototype. Left: A virtual brain superimposed to the real user's image (photomontage). Right: Our mirrorbased augmented reality setup.

Figure 2 :

 2 Figure 2: Mind-Mirror system overview. Components: (1) EEG cap, (2) webcam, (3) Microsoft Kinect camera, (4) g.tec g.USBamp EEG amplifier, (5) laptop PC, (6) computer screen, (7) half-silvered foil.

Figure 3 :

 3 Figure 3: Brain visualization techniques. Left: topography display of surface brain activity. Right: volumetric display of brain activity using Red and yellow colors correspond to the high-activity spots.

Figure 4 :

 4 Figure 4: Brain activity visualization integrating concentration/relaxation classification. The left image corresponds to a concentrated mental state (red color) while the right image shows a relaxed state (blue color). A brighter color represents a more intense electrical activity.

Figure 5 :

 5 Figure 5: Rear-view window: a virtual brain is superimposed with the back of the user's head. It follows the orientation of the real user's head and enables the perception of what happens at the back of the head (brain).

Figure 6 :

 6 Figure 6: The temporal gauge display. The red and blue bars respectively represent the evolution of the concentration and relaxation levels in time.

Table 1 :

 1 Classification performance results. Average values of the linear regression output and standard deviation are provided for each representation (Gauge; Mind-Mirror) and each state (Concentration; Relaxation), as well as the difference between the two states (Diff.)

			Gauge		Mind-Mirror
		Con. Relax. Diff.	Con.	Relax. Diff.
	Mean 0.21	-0.25	0.46 -0.004	-0.31	0.31
	SD	1.44	1.50	0.33	1.1	1.22	0.39

Table 2 :

 2 Subjective questionnaire results for the 6 criteria (1: worst to 7: best). Average values of mark and standard deviation were provided for each representation.

		Gauge	Mind-Mirror
		Mean	SD	Mean	SD
	Comprehensibility	5.67	1.37	5.17	0.94
	Motivation	4.75	1.05	4.67	1.37
	Simplicity	5.92	1.68	4.92	1.31
	Clarity	5.83	1.27	4.83	1.11
	Innovation	3.17	1.11	5.5	1.09
	Originality	3.25	1.14	5.5	0.8

ACKNOWLEDGEMENTS

We would like to thank Jozef Legény for his help as photographer and cameraman. This work was supported by the administrative region of Brittany and the French National Research Agency within the Homo Textilus project (grant ANR-11-SOIN-007).


Camille Jeunet
email: camille.jeunet@inria.fr

Alison Cellard

Sriram Subramanian

Martin Hachet

Bernard N'kaoua

Fabien Lotte

How Well Can We Learn With Standard BCI Training Approaches? A Pilot Study

. To determine to which extent such BCI training approaches (i.e., feedback and training tasks) are suitable to learn a skill, we used them in another context (without a BCI) to train 20 users to perform simple motor tasks. While such approaches enabled learning for most subjects, results also showed that 15% of them were unable to learn these simple motor tasks, which is close to the BCI illiteracy rate [1]. This further suggests that current BCI training approaches may be an important factor of illiteracy, thus deserving more attention.

Introduction

Brain-computer interfaces (BCIs) are communication systems allowing users to interact with the environment, using only their brain activity [START_REF] Wolpaw | BCI: Principles and Practice[END_REF]. BCIs, although very promising, remain barely used outside laboratories because they are not reliable enough [START_REF] Wolpaw | BCI: Principles and Practice[END_REF]. Two main reasons have been identified. The first one, extensively investigated, concerns brain signal processing, with current classification algorithms being still imperfect [START_REF] Allison | Could anyone use a BCI?[END_REF]. The second one concerns the users themselves. Indeed, many users seem unable to acquire good BCI skills (i.e. the capacity to generate specific and stable brain activity patterns): around 20% cannot control a BCI at all (the so-called "BCI illiteracy"), while most of the remaining 80% have relatively modest performances [START_REF] Allison | Could anyone use a BCI?[END_REF]. An appropriate training is needed to acquire these skills, especially for Mental Imagery-based BCI (MI-BCI). It has been suggested that currently used training and feedback protocols, which do not take into account recommendations from psychology to optimise human learning, might be partly responsible for BCI illiteracy and poor user performance [START_REF] Lotte | Flaws in current human training protocols for spontaneous BCI: lessons learned from instructional design[END_REF]. For instance, it has been shown that, for efficient learning, training protocols have to fit the user learning style and propose an increasing and adaptive difficulty [START_REF] Lotte | Flaws in current human training protocols for spontaneous BCI: lessons learned from instructional design[END_REF]. Yet standard BCI training protocols are the same for all users [START_REF] Lotte | Flaws in current human training protocols for spontaneous BCI: lessons learned from instructional design[END_REF]. While instructive, these studies only provide theoretical considerations about training approaches. It is therefore necessary to concretely assess whether training approaches used in BCI are appropriate to train a skill. Moreover, it is necessary to perform this evaluation independently of BCI, to rule out possible biases due to BCI complexity, non-stationarity and poor signal-to-noise ratio. Thus in this work, we propose to study these BCI training approaches without using a BCI: participants were asked to learn specific and simple motor tasks using the same feedback and training tasks used for MI-BCI. We then studied how well they could learn such motor tasks to assess the quality of the training approaches, independently of BCI use. We studied here two different approaches: 1) the training approach used in "standard" MI-BCI [START_REF] Pfurtscheller | Motor imagery and direct brain-computer communication[END_REF] and 2) a variant of it which provides some autonomy to the user. Indeed, with the "standard" approach, no autonomy is given to the user, who always has to perform the tasks required by the protocol. Yet, autonomy is known to increase motivation and learning efficiency in general [START_REF] Lotte | Flaws in current human training protocols for spontaneous BCI: lessons learned from instructional design[END_REF]. Interestingly enough, the study described in [START_REF] Neuper | Clinical application of an EEG-based BCI[END_REF] obtained promising results when providing more autonomy to a single BCI user.

Methods

Participants were asked to learn to perform two motor tasks: drawing triangles and circles with a pen on a graphic tablet (see Figure 1(b)), using standard MI-BCI training approaches [START_REF] Pfurtscheller | Motor imagery and direct brain-computer communication[END_REF]. Indeed, as with MI-BCI, in which users have to learn a suitable movement imagination strategy, the participants here had to learn the strategy which allows the computer to correctly recognise their drawing, e.g., they had to identify the suitable shape size, angles or speed of drawing.

Experimental protocol

Participants had to learn to draw circles and triangles that can be recognised by the computer during different runs, which were either standard (s) or self-paced (sp). S-runs were composed of 20 trials per task. As shown in Figure 1(a)), at the beginning of each trial a green cross was displayed. After 2s, an auditory cue (a beep) announced the beginning of the task. Then, after 3s, a red arrow was displayed, indicating which task the participant had to perform: continuously drawing circles or triangles upon appearance of a left or right arrow, respectively. After 4.25s, a blue feedback bar appeared and was updated continuously for 4s. Its direction indicated the shape recognized by the classifier (left: circle, right: triangle) and its length was proportional to the classifier output (i.e., the distance to the classifier separating hyperplane), as with MI-BCI. During sp-runs, no instructions were given: the participants were asked to do the motor tasks in an autonomous and free way. Half of the participants were asked to learn using a Standard (S) training approach: they did 4 seven-minute-long s-runs. The other half learned using a training approach with increased autonomy, denoted Partially Self-Paced (PSP) approach: the 1 st and 4 th runs were s-runs, while the 2 nd run was replaced by a 3.5 minute long sp-run followed by a shortened s-run (10 trials per task, 3.5 minutes), and the 3rd run was replaced by a shortened s-run followed by a 3.5 minute long sp-run. The training duration was the same in both conditions. We studied the impact of the condition, S vs. PSP, on the recognition accuracy of triangles and circles over runs (i.e., learning effects) and on subjective experience (using a questionnaire). 20 participants (10 per group) took part in our experiment.

Signal Processing

In order to discriminate triangle from circle pen gestures, we used a pattern recognition approach as in BCI. From the past 1s-long time window (in a sliding window scheme, 937.5ms overlap) of the 2D pen position (16Hz sampling rate), a histogram of angles was computed. More precisely, the angles between each consecutive segment of the time window were first computed.

Then the number of angles falling in the ranges 0-30 o , 30-75 o , 75-105 o , 105-150 o and 150-180 o were counted, and these 5 count values were used as input features for a Linear Discriminant Analysis (LDA) classifier. The (subject-independent) LDA classifier was trained on 60 trials from each gesture, from 2 persons (1 left-handed, 1 right-handed). The resulting classifier could discriminate triangles from circles with 73.8% classification accuracy (10-fold cross-validation on the training set), which is an accuracy equivalent to the average accuracy of a MI-BCI [START_REF] Blankertz | Neurophysiological predictor of SMR-based BCI performance[END_REF]. Classification accuracy was measured as the average number of 1s-long time windows correctly classified during the feedback period from each trial (see Figure 1(a)).

Analyses

In order to analyse the interaction between the "Condition" (2 modalities: S and PSP; independent measures) and the performance obtained at each "Run" (4 modalities: run1, run2, run3 and run4; repeated measures), we performed a 2-way ANOVA. Morevoer, we asked the participants to complete a Usability Questionnaire (UQ) which measured 4 dimensions: learnability/memorability (LM), efficiency/effectiveness (EE), safety (Saf.) and satisfaction (Sat.). Thus, we did a two-way ANOVA to analyse the interaction between the "Condition" and the "Evaluated Dimension" (4 modalities: LM, EE, saf. and sat.; repeated measures). Irrespectively of the condition, classification accuracy results (see Figures 2(a) and 2(b)) showed that 17 out of 20 participants managed to learn the task (best run accuracy ≥ 82.6%), with accuracies increasing with the number of runs on average, as can be observed in BCI. However, 3 participants did not manage to learn the tasks (best run accuracy < 70%). This is particularly interesting considering the simplicity of the motor tasks which ensured the users could technically perform them. Such training approaches thus seem suboptimal. Moreover, this rate of 15% of people who did not manage to learn is close to the BCI-illiteracy rate (20% [START_REF] Allison | Could anyone use a BCI?[END_REF]). Overall, this suggests that BCI illiteracy may not be due to the user only, but also substantially to the training protocol. Then, results showed neither a main effect of the Condition [F(1,18)=2.33; p=0.15] nor a Condition X Run interaction [F(3,45)=1.35;p=0.27] when considering all the participants. However, when the 3 illiterates are excluded, the 2-way ANOVA showed a main effect of the Condition [F(1,15)=7.48;p=0.01]: the PSP group seemed to perform better than the S group (meanPSP=91.00 ± 3.82, meanS=83.29 ± 6.95). However, random sampling of the participants led by chance to a PSP group with classification accuracies for the first run that are higher than that of the S group, which prevents us from drawing any relevant conclusion on comparative learning effects.

Results & Discussion

UQ results (see Figure 2(c)) showed no main effect of the Condition [F(1,18)=0.98; p=0.33]. However, they showed a trend towards a Condition X Evaluated Dimension interaction [F(3,54)=2.40; p=0.077], which is due to the better evaluation of LM, EE and Saf. in the PSP condition than in the S condition, which is not the case for the Sat. dimension. These results suggested that while the PSP approach is not more pleasant to learn with than the S approach, it is easier. Interestingly enough, 8 subjects reported in an open-question of the questionnaire that the feedback was very uninformative, which made learning the tasks difficult.

Conclusion

This study aimed to concretely assess how well one could learn a given skill with BCI training approaches. To do so, we proposed to study BCI training approaches without using BCI, i.e., we used feedback and training tasks from MI-BCI to train participants to draw triangles and circles (i.e., simple motor tasks) so these can be recognized by the computer. Half of the participants did so using a S training approach while the other half used a PSP one. In terms of learning effects, results unfortunately showed no relevant differences between conditions (S vs. PSP), due to initial performances that differed between conditions, by chance. However, irrespectively of the condition, 15% of the participants (3 out of 20) seemed unable to learn the motor tasks, despite their simplicity. This suggests that such training approaches are not optimal for learning and thus that they may be an important factor of BCI illiteracy. Concerning user experience, UQ showed a tendancy towards a better feeling of learnability/memorability, efficiency/effectiveness and safety with the PSP than with the S approach, while the satisfaction appeared similar for both. Overall, this study confirmed in practice the theoretical analyses from [START_REF] Lotte | Flaws in current human training protocols for spontaneous BCI: lessons learned from instructional design[END_REF] suggesting that current BCI training approaches were suboptimal and need to be changed. In the future, we will increase the number of participants, explore the PSP approach with actual BCIs, and propose new training approaches that consider the user's cognitive style and motivational states to improve both the learning experience and performance.

  (a) An 8 second-long trial (b) Experimental Setup

Figure 1 :

 1 Figure 1: (a) Outline of a trial from a standard run; (b) Experimental setup.

  (a) "Standard" Condition (b) "Partially Self-Paced" Condition (c) Questionnaire

Figure 2 :

 2 Figure 2: (a) (a) Performance (pen gesture classification accuracy) over runs in the S Condition; (b) Performance over runs in the PSP Condition; (c) Average UQ scores.