CONTROLLABILITY OF A STOKES SYSTEM WITH A DIFFUSIVE BOUNDARY CONDITION

. We are interested by the controllability of a ﬂuid-structure interaction system where the ﬂuid is viscous and incompressible and where the structure is elastic and located on a part of the boundary of the ﬂuid domain. In this article, we simplify this system by considering a linearization and by replacing the wave/plate equation for the structure by a heat equation. We show that the corresponding system coupling the Stokes equations with a heat equation at its boundary is null-controllable. The proof is based on Carleman estimates and interpolation inequalities. One of the Carleman estimates corresponds to the case of Ventcel boundary conditions. This work can be seen as a ﬁrst step to handle the real system where the structure is modeled by the wave or the plate equation.


Introduction
Fluid-structure interaction systems are important systems for many applications such as aerodynamics, medicine (for instance the study of the motion of the blood in veins or in arteries), biology (animal locomotion in a fluid), civil engineering (design of bridges), naval architecture (design of boats and submarines), etc. Moreover, their mathematical studies can be very challenging due to several difficulties: in particular, the complexity of fluid equations such as the Navier-Stokes system, the strong coupling between the fluid system and the structure system and the free-boundary corresponding to the structure displacement.
In this article we consider a simplified fluid-structure interaction system. The corresponding system without simplification has been proposed in [50] as a model for the blood flow in a vessel. It writes as follows: we denote by I the torus (in order to consider periodic boundary conditions): Then the equations of motion are ∂ t w + (w · ∇)w − div T(w, π) = 0 t > 0, x ∈ Ω (t) , div w = 0 t > 0, x ∈ Ω (t) , w(t, x 1 , 1 + (t, x 1 )) = (∂ t )(t, x 1 )e 2 t > 0, x 1 ∈ I, H (w, π)(t, x 1 ) = (1 + |∂ x1 | 2 ) 1/2 [T(w, π)n] (t, x 1 , 1 + (t, x 1 )) · e 2 . (1.4) The two first lines of (1.2) correspond to the Navier-Stokes system for the fluid velocity w and the pressure π. The last line of (1.2) is a beam equation satisfied by the deformation . We have used the standard noslip boundary conditions (third and forth equations). To simplify, we assume that the viscosity of the fluid is constant and equal to 1. The vector fields n corresponds to the unit exterior normal to Ω (t) . This system has been studied by many authors: [16] (existence of weak solutions), [7,24,39,42] (existence of strong solutions), [52] (stabilization of strong solutions), [4] (stabilization of weak solutions around a stationary state). There are also some works in the case δ = 0, that is without damping on the beam equation: the existence of weak solutions is proved in [23] and in [46] (see also [15]). In [25], the existence of local strong solutions is obtained for a structure described by either a wave equation (α 1 = δ = 0 and α 2 > 0) or a beam equation with inertia of rotation (α 1 > 0, α 2 = δ = 0 and with an additional term −∂ ttss ). In [5,6], the authors show the existence and uniqueness of strong solutions in the case α 1 > 0, α 2 0 and δ = 0. Using similar techniques they also analyze the case of the wave equation (α 1 = δ = 0 and α 2 > 0) in [1] showing in particular that the semigroup of the linearized system is analytic. Let us mention also some results for more complex models: [37,38] (linear elastic Koiter shell), [47] (dynamic pressure boundary conditions), [48,49] (3D cylindrical domain with nonlinear elastic cylindrical Koiter shell), [58,59] (nonlinear elastic and thermoelastic plate equations), [41,43] (compressible fluids), etc.
The advantage of the damping in the beam equation is that the term −δ∂ t ∂ 2 x1 is a structural damping so that the corresponding beam equation becomes a parabolic equation (see, for instance, [18]). In this work, we consider a simplified model associated with (1.2). We neglect the deformation of the fluid domain due to the elastic déformation and we also linearized the Navier-Stokes system by only considering the Stokes system. Moreover, we replace the damped beam equation by a heat equation. By setting Ω = I × (0, 1) (see Fig. 1), we are thus considering the following system div w = 0 in (0, T ) × Ω, w = 0 on (0, T ) × Γ 0 , w = ζe 2 on (0, T ) × Γ 1 , ∂ t ζ − ∂ x1x1 ζ = −T(w, π)n · e 2 in (0, T ) × I, w(0, ·) = w 0 in Ω, ζ(t, 0) = ζ 0 in I. In the above system, ζ corresponds to the displacement velocity ∂ t in (1.2) and we do not consider anymore the displacement position . We have added a control f localized in the fluid domain, in an arbitrary small nonempty open set ω of Ω. Our goal is to show the null-controllability of this simplified system and to do this, as it is standard (see, for instance, [60], Thm. 11.2.1, p. 357), we prove an observability inequality on the adjoint system: We set and we define the space Remark 1.1. Using the particular geometry considered here, we can simplify the adjoint system. First on Γ 1 , n = e 2 and using (1.3), we deduce Moreover, using the incompressibility of the fluid and the boundary conditions, we deduce that Using this condition on the heat equation on the boundary and (1.8) yields In particular, in contrast with the standard Stokes system, the pressure is not determined up to a constant.
Our main result states as follows: Theorem 1.2. Let γ > 1 and let ω be a nonempty open set of Ω. Then, there exists C 0 > 0 such that for any T ∈ (0, 1) and for any u 0 , η 0 ∈ H, the solution [u, η] of (1.6) satisfies In particular, for any w 0 , ζ 0 ∈ H and for any T > 0, there exists a control f ∈ L 2 ((0, T ) × ω) such that the solution [w, ζ] of (1.5) satisfies w(T, ·) = 0, ζ(T, ·) = 0. Remark 1.3. In comparison to the Stokes system with the Dirichlet boundary condition, one should expect that the above result remains true for γ = 1 (see [17]). However, our method is based on a spectral inequality and on a general result to pass from such a spectral inequality to an observability inequality. In this general result, stated in Theorem 1.8, we have the same restriction on γ. Note that the approach used here was already considered in [35,36] where the same restriction on γ also appears. Remark 1.4. As explained above, Theorem 1.2 can be seen as a first control result on a simplified model. We expect to extend some of the tools developed here to handle the control properties of the system (1.2) in future works. The controllability properties of fluid-structure interaction systems have been tackled mainly in the case where the structure is a rigid body (see, [10, 11, 19-21, 26, 40, 51, 54], etc.). In [45], the author shows an observability inequality for the adjoint of a linearized and simplified fluid-structure interaction system in the case of a compressible viscous fluid and of a damped beam. Note that the corresponding control problem involves two controls, one for the fluid and one for the structure. For the stabilization of fluid-structure interaction systems, one can quote some results: [4,52] (for the case of a damped beam), [2,3,55,57] (for the case of a rigid body).
Remark 1.5. The method proposed here to control a system involving the Stokes equations is quite different from the method used in a large part of the literature for the controllability of fluid systems. In general, the method is based on "global Carleman inequalities" (see, for instance [22,27]). Here, we follow another strategy as in [17,36]. Such a method is based on local Carleman inequalities for an "augmented" elliptic operator, from which one deduces a spectral inequality, in the spirit of [28,36]. We then use an adaptation of the original strategy of [35,36] in our context. This type of spectral inequality has already been used in the context of fluids in [14]. Controllability for coupled parabolic systems through the derivation of Carleman estimates using microlocal analysis near boundaries and interfaces has been widely studied, one can cite for instance [29,34] (see also [8,9,32,33] and the recent books [30,31] for elliptic counterparts).
Remark 1.6. In a large part of the works devoted to the controllability of fluid systems, a first step consists in removing the pressure by taking the rotational operator of the Stokes system (or another operator). As observed in [17], the unique continuation property does not hold for the augmented operator in the direction of the additional variable, due to the pressure. Here, due to the presence of the pressure in the structure equation, we choose, as in [22] to obtain a Carleman estimate of the fluid velocity by considering the pressure as a source term. Then, we obtain a Carleman estimate for the pressure by using that it is an harmonic function. However, due to the lack of boundary condition for the pressure, such an estimate contains boundary terms and one can see that we only need the high frequencies of the boundary value of the pressure. In order to estimate these high frequency terms, we derive a priori estimates of the system. Note that in this part, we use another important feature of our work which consists in considering two different weight functions to separate the characteristic manifolds of the concerned operators and perform proper elliptic estimates.
Remark 1.7. Let us point out that one can use Theorem 1.2 to handle nonlinear controllability issues by applying the general method proposed in [40]. For instance, one could replace in (1.5) the Stokes system by the Navier-Stokes system and by using a fixed-point argument, we would obtain the null-controllability of the corresponding system for small initial conditions An important ingredient in the proof of Theorem 1.2 is a general result showing that one can deduce the final-state observability (and thus the null-controllability) of a parabolic system from a spectral inequality. Similar results can be found in [35,36] but here, the main differences is that we assume that our operator is self-adjoint and that we do not assume the Weyl asymptotic formula that are needed in the previous references. The proof is similar to the proofs in [35,36], but for sake of completeness, we state below the result and prove it in the next section.
Let us consider a general Hilbert space H (not necessarily defined by (1.7)) and a nonnegative self-adjoint operator A : D(A) → H with compact resolvents. We denote by (λ j ) the nondecreasing sequence of eigenvalues and by (w j ) an orthonormal basis of H composed by eigenvectors of A: Aw j = λ j w j for j 1. We also consider a control operator B ∈ L(U, H). Theorem 1.8. Assume the above hypotheses. Assume moreover the existence of S 0 > 0, C > 0 and κ ∈ C ∞ 0 (0, S 0 ) such that that for any Λ > 0, and for any (a j ) j ∈ C N , Then for all γ > 1, there exists C 0 > 0 such that for any T ∈ (0, 1) and for any z 0 ∈ H, We recall that relation (1.11) implies the null-controllability of the system (1.12) The outline of the article is as follows: in Section 2, we prove Theorem 1.8 stated above. Then using this general result, we are reduced to show a spectral inequality that we state in Section 3 along with the functional framework. The spectral inequality is itself the consequence of an interpolation inequality that we obtain in Section 5. One of the main difficulties to obtain such an inequality comes from the fact that we need to estimate the pressure. Section 4 is devoted to such an estimate which is one of the main parts of this article. The proof of the spectral inequality and thus of Theorem 1.2 is obtained at the end of Section 5. In the appendix, we show an interpolation estimate for the Ventcel boundary condition that is mainly a consequence of a Carleman estimate obtained in [13]. Notation 1.9. In the whole paper, we use C as a generic positive constant that does not depend on the other terms of the inequality. The value of the constant C may change from one appearance to another. We also use the notation X Y if there exists a constant C > 0 such that we have the inequality X CY . In what follows, norms in the interior and at the boundary are denoted respectively by · and | · |.

From a spectral inequality to the null-controllability
This section is devoted to the proof of Theorem 1.8. The proof follows closely the proof in [35,36].

Controllability of the first modes
We define We are interested here by the control problem for some τ > 0. We first prove an observability inequality associated with the above control problem, with an estimate of the cost of the control with respect to τ and Λ: Proof. We follow [35,36]: the proof is based on the use of a Gramian matrix G Λ that we conjugate with an adequate matrix. Using the Paley-Wiener theorem, we construct and estimate a control for (2.1) so that θ(τ ) = 0. Let us define the linear operator From (1.10), G Λ ∈ L(H Λ ) is symmetric, positive and invertible with We set Thus, h Λ ∈ Hol(C; U) and As in [56], we introduce Q Λ ∈ Hol(C; U) such that We deduce from the above relation and (2.8) that We define g Λ ∈ Hol(C; U) by From (2.3), (2.4) and (2.9), we have Since σ < 2, we can use a Paley-Wiener type theorem (see [35], Prop. A.3) and deduce the existence of In particular, from (2.11) and the Laplace method, for all t ∈ (0, τ ), Now, for any j such that λ j Λ, By a duality argument and (2.12), this implies that (2.2).

Proof of Theorem 1.8
We are now in a position to prove Theorem 1.8, adapting the method of [44]. The proof is based on Theorem 2.1 and on the exponential decay of the semigroup associated with A for high frequencies. This allows us to obtain an observability estimate between two times and then we apply this estimate between T /2 k+1 and T /2 k , k 0.

Functional framework and spectral inequality
In order to prove Theorem 1.2, we are going to apply Theorem 1.8. In this section, we first give the functional framework associated with (1.6). Then we write the spectral inequality that will be proven in the remaining part of the article.

Functional framework
We recall that H is defined by (1.7). We also define We denote by P 0 the orthogonal projection from L 2 (Ω) × L 2 0 (I) onto H. We now define the linear operator and for u, η ∈ D(A 0 ), we set Then one can check that (1.6) writes In the next proposition, we show in particular that A 0 is the infinitesimal generator of a semigroup so that z(t) = e tA0 z 0 for t 0.
Proposition 3.1. The operator A 0 defined by (3.1)-(3.2) has compact resolvents, and is self-adjoint negative on H. For all k 1, there exists a constant C such that Thus A 0 is symmetric and negative (by using the Poincaré inequalities).
In order to show that A 0 is self-adjoint it is sufficient to show that it is onto. Assume [f, g] ∈ H and let us solve the equation Multiplying the above equation by [v, ζ] ∈ V leads to the weak formulation Using the Poincaré inequalities, we see that we can apply the Riesz theorem and deduce the existence and uniqueness of [u, η] ∈ V solution of (3.5). Then if v ∈ C ∞ c (Ω) with div v = 0 and ζ = 0 in (3.5), we obtain that Using the De Rham theorem, we deduce the existence of p such that Using the elliptic regularity of the Stokes system, we deduce that (u, p) ∈ H 3/2 (Ω) × H 1/2 (Ω). Multiplying the first above equation by v, with [v, ζ] ∈ V, we deduce that for any ζ ∈ H 1 (I) ∩ L 2 0 (I), we deduce that η ∈ H 3/2 (I) and from (3.7) that (u, p) ∈ H 2 (Ω) × H 1 (Ω). Thus p |Γ1 ∈ H 1/2 (I), and from (3.8), we deduce that η ∈ H 2 (I). We conclude that [u, η] ∈ D(A 0 ) and satisfies (3.4). We also obtain relation (3.3) for k = 1 and with a similar argument we can also obtain relation (3.3) for any k 1.
The fact that A 0 has compact resolvents is coming from the compact embedding of H 2 into L 2 for bounded domains.
In particular, the eigenvalues λ j > 0 of −A 0 satisfy λ j → ∞ and there exists composed by eigenvectors of A 0 : (3.10) The above system can be written as and more precisely as Note that, as explained in Theorem 1.1, for all j 1, the pressure p j satisfies the relation and in particular, p j is uniquely determined. This differs from the standard Stokes system where the pressure is determined up to constant.

Spectral inequality
We are now in a position to state the spectral inequality for the operator A 0 defined in the previous section.
Theorem 3.2. Let ω 0 be a nonempty open subset of Ω and S 0 > 0. There exist C > 0 and κ ∈ C ∞ 0 (0, S 0 ) such that for any Λ > 0, and for any (a j ) j ∈ C N , ds. (3.13) In order to prove Theorem 3.2, we define for s ∈ (0, S 0 ) and x ∈ Ω, (3.14) and the domains From (3.12), we deduce that We thus consider χ ∈ C ∞ 0 ((0, S 0 )), satisfying 0 χ 1 and We work with the following localized solutions where We also have that As usual, we can use the three first equations to obtain the following equation for the pressure:

A global observability estimate on the pressure
In this section, we prove a global estimate on the pressure. We first introduce our weight and the corresponding conjugated operators. We then state our main result, that is Theorem 4.1. Then we show a first estimate on the pressure involving high frequency pressure terms at the boundary. Such terms are then estimated by showing some a priori estimates and this allows us to prove Theorem 4.1. and such that all its critical points belong to ω 0 :
We recall that we define (u, p) from (U, P ) by (3.18) (truncation in s) and that the source f i are defined by (3.20). We then define v := e τ ϕ u, q := e τ ϕ p, g i := e τ ϕ f i (i ∈ {1, . . . , 3}). (4.4) In order to take into account the dependence in s, we write and their tangential counterparts We keep our previous notation The equations satisfied by v and q can be written with the introduction of the following conjugated operators: Then we deduce from (3.19) and (3.22) the following conjugated system: The main result of this section is the following result.

A first estimate on the pressure
In order to prove Theorem 4.1, we exploit that q satisfies the third equation of (4.8), where D ϕ is defined by (4.6). Since we do not have any boundary condition, we need to split the boundary value of q into high and low frequencies. More precisely, for Q ∈ H 2 (Ω), we introduce the Fourier coefficients of the trace of Q : We then define the sets of low tangential frequencies and high tangential frequencies: In the above definition, the infimum of ∂ x2 ϕ is taken for x ∈ ∂Ω and s ∈ [0, S 0 ]. Due to (4.1) and (4.3), we have inf |∂ x2 ϕ| = λe −s 2 0 .

Estimates in the high frequency regime
We define We recall that (u, p) is defined by (3.18) and (f 1 , f 2 , f 3 ) is defined by (3.20). We define (u k , p k , f k Finally, with τ > 0 and ϕ 0 defined by (4.3), we set Note that h k i are the Fourier coefficients of the functions h i defined by (4.9). Since ϕ 0 only depends on s, and using (4.3), (4.4), we have Let us define the following conjugated operators: Then, for k ∈ Z * , (3.19) transforms into The relation (3.21) yields Proof. The proof is based on energy estimates on (4.19). We first obtain H 2 estimates on w k by using the high-frequency hypothesis to control the lower order terms. The estimate on π k follows then from (4.19 Now, since k ∈ HF τ , we have On the other hand, From the two previous relations, we deduce the existence of λ 3 = λ 3 (S 0 ) > 0 such that for λ λ 3 and for k ∈ HF τ k 2 − τ 2 |ϕ 0 | 2 1 2 (τ 2 + k 2 ).
Combining the above relation and (4.21) yields Now, we write (4.19) under the form where

From (4.22), we deduce
We multiply the first line (4.23) by −∂ 2 s w k 1 , the second line (4.23) by −∂ 2 s w k 2 , and the last line (4.23) by −∂ 2 s w k 2 . Integrating by parts, summing up and using (4.24) yield Next, we write (4.23) under the form where From (4.24) and (4.25), we deduce The first two lines of (4.26) can be written as and thus and using that k = 0, we deduce from the above estimate that and thus, with (4.27) and (4.22), On the other hand, differentiating the divergence equation of system (4.26) with respect to x 2 and using (4.22), (4.27) yield Then, combining the above relation with (4.32) and with a trace inequality, we deduce Using that k ∈ HF τ , we deduce the result.

Proof of Theorem 4.1
From Theorem 4.3 and from (4.18), for k ∈ HF τ and thus, with the Parseval formula, where we have used (4.17) and (4.9). Combining this estimate with (4.16) we finally obtain the sought result.

Proof of the spectral inequality
The proof of the spectral inequality, that is (3.13) is based on interpolation estimates. More precisely, it will be a consequence of Theorem 5.3 stated below. In order to show such a result, we first recall some interpolation inequalities available in the literature and then we combine them with the global pressure estimates, that is Theorem 4.1 to show Theorem 5.3. The last part of this section is devoted to the proof of the spectral inequality from the interpolation inequality.
First, we need the following notation for this section: Note that Z ⊂ Z and J 1 ⊂ J 1 .

Estimates on the velocity
The two components of the velocity satisfy different boundary conditions (see (3.16)). We start by an estimate on the first component U 1 that satisfies homogeneous Dirichlet boundary conditions. For the proof of this result, we refer to relation (1) in Section 3 of [36].
Theorem 5.1. There exist C 1 > 0 and µ 1 ∈ (0, 1) such that for all w ∈ H 2 (Z) such that w | J 0 ∪J 1 = 0, (5.4) Note that Theorem 5.1 is stated with an H 1 observation in [36], but we can transform it into an L 2 observation as in (5.4) by using a cut-off function and integrations by parts.
For the estimate of U 2 , we note that it satisfies a Ventcel boundary condition on J 1 and the Dirichlet boundary condition on J 0 . Hence, we use the following result, which is basically a consequence of a Carleman estimate obtained in [13]. However for the sake of completeness, we prove the next result in Appendix A.
Note that both Theorem 5.1 and Theorem 5.2 hold for µ 3 ∈ (0, 1) such that µ 3 µ 1 and µ 3 µ 2 (with a modification of the constants C 1 and C 2 ). Thus, with µ 3 = min(µ 1 , µ 2 ) and an adequate constant C 3 , we can apply Theorem 5.1 with w = U 1 and Theorem 5.2 with w = U 2 , where U satisfies (3.16) and we deduce We have used here the fact that on J 1 , We are going now to combine the above estimate with the estimates of the pressure terms obtained in Section 4.

Patching the estimates together
Combining the previous estimates, we can now prove the following result.
Theorem 5.3. There exist C > 0 and µ ∈ (0, 1) such that for any Λ > 0 and for any (a j ) j ∈ C N , the function U defined by (3.14) satisfies Proof. We start with the estimate (4.10), where we recall that q is given by (3.18), (4.4) and h i , i = 1, 2, 3 by (4.9): Note that f 1 , f 2 (respectively f 3 ) are supported in In Z (respectively in J 1 ), χ(s) ≡ 1, and On the other hand, we deduce from (5.6) and a Young inequality that and combining this relation with (5.11), we deduce the existence of c 3 , c 4 > 0 such that for all τ τ 4 Now, we can use c P in (3.14) so that for all s ∈ [0, S 0 ], ω0 P (s, x) dx = 0 and using the Poincaré-Wirtinger inequality, we deduce that We deduce that for some constants c 5 , c 6 > 0, for all τ τ 4 , Optimizing this inequality with respect to τ τ 4 (see, for instance, [53], [13], Lem. 8.4) allows us to conclude the proof of Theorem 5.3.

From the interpolation inequality to the spectral inequality
Using Theorem 5.3, we are now in a position to prove Theorem 3.2. This inequality combined with Theorem 1.8 yields the main result of the article (Thm. 1.2).
Proof of Theorem 3.2. From (3.9) and (3.1), we deduce that Combining Theorem 5.3 with the previous relations, we deduce that Let us introduce a cut-off function ϑ ∈ C ∞ 0 (O) such that 0 ϑ 1 and ϑ = 1 in O 0 . By integration by parts, we obtain On the other hand, using the elliptic regularity of (3.3), the definition (3.14) of U and the orthogonality of the (5.14) Combining (5.12), (5.13) and (5.14), we deduce

Thus Theorem 3.2 follows.
Appendix A. Proof of Theorem 5.2

A.1 A Carleman estimate
The proof of Theorem 5.2 is mainly based on a Carleman estimate obtained in [13] that we recall here. We recall that Z, J 1 , J 0 are defined by (3.15) whereas Z and J 1 are defined by (5.2). In what follows, we consider z 0 ∈ J 1 , an open neighborhood V of z 0 in Z and a weight function ϕ ∈ C ∞ (V ). For any σ ∈ R, we define It is the principal symbol of the conjugated operator associated with −∆ z − σ 2 , that is, of the operator We assume the following hypotheses on ϕ: sub-ellipticity on V , that is the existence of τ 0 > 0 such that for any z ∈ V , ξ ∈ R 3 , |σ| 1 and τ τ 0 |σ|, and the two following conditions to handle Ventcel boundary conditions (see [13], conditions (23) and (24)): for ν 0 > 0 small enough. We recall that the Poisson bracket is defined by where we set here z 1 = s, z 2 = x 1 and z 3 = x 2 to simplify. Then we have the following result proved in [13]: Theorem A.1. Assume z 0 ∈ J 1 and V is an open neighborhood of z 0 in Z. Assume also that ϕ ∈ C ∞ (V ) satisfies the conditions (A.1) and (A.2) for ν 0 small enough. Then, there exist τ 0 > 0 and C > 0 such that for all |σ| 1, for all τ τ 0 |σ| and for all w ∈ C ∞ 0 (V ), .
First to precise the above statement, by w ∈ C ∞ 0 (V ) we mean that w is the restriction of a C ∞ function with compact support in V 0 where V 0 is an open set of R × I × R such that V 0 ∩ Z = V . Second, we use the above result in the case σ = 1, so that by taking τ 0 large enough, we obtain that for all τ τ 0 and for all w ∈ C ∞ 0 (V ),

A.2 Interpolation estimates for the Ventcel boundary condition
Using the Carleman inequality of the previous section, one can deduce, in a classical way, an interpolation inequality. First let us define the weight function that we are going to use.
Proof. We assume that λ 1 in all what follows. First, since ∇ z ψ = 0 in V , we deduce the first point of (A. From (A.4), there exist positive constants independent of λ such that C 1 |∇ z ψ| C 2 , ∇ 2 z ψ C 3 .
Proof of Theorem 5.2. By a compactness argument, one can deduce from Lemma A.3 an interpolation result on a neighborhood of J 1 . Then we combine this with classical interpolation estimates (see [36]) in the interior and at the boundary J 0 where Dirichlet boundary condition hold to conclude. A similar proof is done in [13] (see Lem. 8.3).