Exact controllability to trajectories for semilinear heat equations with discontinuous diffusion coefficients

URL:http://www.emath.fr/cocv/ DOI: 10.1051/cocv:2002047

EXACT CONTROLLABILITY TO TRAJECTORIES FOR SEMILINEAR HEAT EQUATIONS WITH DISCONTINUOUS DIFFUSION COEFFICIENTS

Anna Doubova

, A. Osses

and J.-P. Puel

Abstract. The results of this paper concern exact controllability to the trajectories for a coupled system of semilinear heat equations. We have transmission conditions on the interface and Dirichlet boundary conditions at the external part of the boundary so that the system can be viewed as a single equation with discontinuous coefficients in the principal part. Exact controllability to the trajectories is proved when we consider distributed controls supported in the part of the domain where the diffusion coefficient is the smaller and if the nonlinear termf(y) grows slower than|y|log^3/2(1 +|y|) at infinity.

In the proof we use null controllability results for the associate linear system and global Carleman estimates with explicit bounds or combinations of several of these estimates. In order to treat the terms appearing on the interface, we have to construct specific weight functions depending on geometry.

Mathematics Subject Classification. 35B37.

Received October 23, 2001. Revised February 7, 2002.

1. Introduction and hypothesis

Let Ω ⊂ R^N, N ≥ 1 be a bounded connected open set with boundary Γ of class C². Let ω ⊂ Ω be a nonempty open subset and T >0. We will use the following notation: Q= Ω×(0, T), Σ = Γ×(0, T). For any p∈[1,+∞], we will denote by || · ||p the usual norm inL^p(Q).

There are two different situations that will be analyzed in this paper. More precisely, let Ω0 and Ω1 be a partition of Ω in two non empty open sets such that

Case 1: Ω0⊂⊂Ω, Ω1= Ω\Ω0 (see Fig. 1, left); (1) Case 2: Ω1⊂⊂Ω, Ω0= Ω\Ω1 (see Fig. 1, right). (2)

Keywords and phrases:Carleman inequalities, controllability, transmission problems.

1Departamento E.D.A.N., Universidad de Sevilla, Tarfia s/n, 41012 Sevilla, Spain and ´Ecole Polytechnique, 91128 Palaiseau Cedex, France; e-mail:dubova@numer.us.es, doubova@cmapx.polytechnique.fr

This work has been partially supported by D.G.E.S., Spain, Grants PB98–1134.

2Departamento de Ingener´ıa Matem´atica, Facultad de Ciencias de F´ısicas y Matem´aticas, Universidad de Chile, Casilla 170/3 - Correo 3, Santiago, Chile and Centro de Modelamiento Matem´atico, UMR 2071 CNRS-Uchile; e-mail:axosses@dim.uchile.cl This work has been partially supported by FONDECYT grants No. 1000955 and 7000955.

3Laboratoire de Math´ematiques Appliqu´ees, Universit´e de Versailles Saint-Quentin, 45 avenue des ´Etats Unis, 78035 Versailles Cedex, France and ´Ecole Polytechnique, 91128 Palaiseau Cedex, France; e-mail: jppuel@cmapx.polytechnique.fr

c EDP Sciences, SMAI 2002

Ω₀

Ω₁

S₋

S₊ Γ

n

n S_-

S₊

Ω₀ Ω₁

Γ n n

Figure 1. Two geometrical cases covered in this paper depending on Ω0⊂⊂Ω or Ω1⊂⊂Ω.

We denote by S= Ω0∩Ω1 the interface, which will be supposed of classC² and bynthe outward unit normal to Ω1 at the points of S and also the outward unit normal to Ω at the points of Γ. LetS⁺ (resp.S⁻) be the part ofS corresponding to the positive (resp. negative) direction of the normaln.

Remark 1.1. The two cases mentioned above are not exhaustive, we do not treat other possible geometrical situations in this paper.

In both cases mentioned above, we will consider the following transmission problem for semilinear heat equation





















∂_ty−div(a0(x)∇y) +f(y) =v1_ω+g0 in Ω0×(0, T),

∂_ty−div(a1(x)∇y) +f(y) =v1_ω+g1 in Ω1×(0, T), y|S⁺×(0,T)=y|S⁻×(0,T),

a0(x)∂_ny|_S+×(0,T)=a1(x)∂_ny|_S−×(0,T),

y= 0, on Σ

y(x,0) =y0 in Ω.

(3)

Here f :R→ Ris a locally Lipschitz-continuous function, ∂_ny denotes the outward normal derivative to Ω1, y0∈L²(Ω) andv∈L^r(0, T;L^r(ω)),g_i∈L^r(0, T;L^r(Ω_i)),i= 0,1 withrsuch that



 1 r+N

2r <1 ifN ≥2,

r= 2 ifN = 1. (4)

Remark 1.2. We could in fact considerv∈L^p(0, T;L^q(ω)),g_i∈L^p(0, T;L^q(Ω_i)),i= 0,1 with 1/p+N/(2q)

<1 in order to haveL^∞ solutions, but in the sake of simplicity we takep=q=r.

Remark 1.3. Without loss of generality we can assume y0 ∈L^∞(Ω). Otherwise, takingv = 0 for t∈(0, δ), δ >0 and thanks to the regularizing effect of parabolic equations,y(δ)∈L^∞(Ω) for someδ >0 [21, 22].

In (3), y=y(x, t) is the state andv =v(x, t) is the control which acts on the system throughω since 1_ωis the characteristic function of the setω.

We will assume that the diffusion coefficient in (3) satisfies the following:

a_i∈C²(Ω_i) fori= 0,1,

a0|_S⁺6=a1|_S⁻. (5)

System (3) represents the coupling between two parabolic semilinear equations whose diffusion coefficient has a jump. At the interfaceS, we impose the continuity of the solutiony and also of the fluxes.

Let us set

a(x) =

a₀(x) if x∈Ω0,

a₁(x) if x∈Ω1. (6)

We also set

g(x) =

g0(x) if x∈Ω0,

g1(x) if x∈Ω1. (7)

Taking into account notations (6) and (7), problem (3) can be written in the divergence form (with discontinuous diffusion coefficients) as follows:







∂_ty−div(a(x)∇y) +f(y) =v1_ω+g in Q,

y= 0 on Σ,

y(x,0) =y0 in Ω.

(8)

We will require ato satisfy

a(x)≥α >0 a.e. in Ω (9)

and the following additional hypothesis:

a0|_S⁺ ≤a1|_S⁻. (10)

We assume that for eachη >0, there existsC_η >0 such that f(s)−f(s⁰)

s−s⁰

^2/3≤C_η+ηlog(1 +|s−s⁰|) ∀s, s⁰∈R. (11)

Let us also consider an “ideal” trajectoryy^∗, solution of the problem (without control)







∂_ty^∗−div(a(x)∇y^∗) +f(y^∗) =g in Q,

y^∗= 0 on Σ,

y(x,0)^∗=y₀^∗ in Ω

(12)

where y^∗₀ ∈ L²(Ω) and g ∈ L^r(0, T;L^r(Ω)), with r as in (4). We know that under conditions (9) and (11), problem (12) possesses exactly one local solution in time (cf.[21] and [22]). Moreover, we can say that there exists a time T^∗>0, such that forT < T^∗, the solutiony^∗ of (12) satisfiesy^∗∈C⁰([0, T];L²(Ω))∩L^∞(δ, T;L^∞(Ω)), for everyδ >0.

The main goal of this paper is to analyze the controllability properties of (8).

Definition 1.1. We say that (8) is exactly controllable to the trajectories if, for any trajectory y^∗ solution of (12) and for any initial condition y0∈ L²(Ω), for everyT < T^∗, there exists a controlv ∈L^r(0, T;L^r(ω)) such that (8) has a solutiony on (0, T) satisfying

y(x, T) =y^∗(x, T) in Ω. (13)

Definition 1.2. System (8) is said null controllable at time T if, for each y0 ∈ L²(Ω), there exists v ∈ L^r(0, T;L^r(ω)) such that the corresponding initial boundary problem (8) admits a solutiony∈C⁰([0, T];L²(Ω)) satisfying

y(x, T) = 0 in Ω. (14)

For linear problems, it is easy to see that the notions of null controllability and exact controllability to the trajectories are equivalent, but this is not true for nonlinear systems.

Definition 1.3. It will be said that (8) isapproximately controllablein L²(Ω) at timeT if, for anyy₀∈L²(Ω), any y_d ∈ L²(Ω) and anyε >0, there exists a controlv ∈L^r(0, T;L^r(ω)) such that the corresponding initial boundary problem (8) possesses a solutiony∈C⁰([0, T];L²(Ω)), with

ky(·, T)−y_dkL²(Ω)≤ε. (15) In the case in which the diffusion coefficients are sufficiently regular, the controllability of linear and semilinear parabolic systems has been analyzed in several recent papers. Among them, let us mention [1, 5, 11, 13, 15–17], and [8] concerning null controllability [9, 12, 13, 25] and [8] for approximate controllability [16] and [13] for exact controllability to the trajectories.

2. Main result

In order to state the main result of this work, we need the following geometrical conditions.

Condition 2.1 (corresponding to case (1)). We assume that there exists a vector field ζ : Ω1 7→ R^N, ζ ∈ C¹(Ω1), such that

ζ(x)·n(x)<0 ∀x∈Γ, (16)

ζ(x)·n(x)>0 ∀x∈S, (17)

ζ(x)6= 0 ∀x∈Ω1 (18)

and if we consider the characteristics associated to ζ



 dx(t)

dt =ζ(x(t)), t >0,

x(0) =x0, (19)

with x0∈Γ, we also assume that for some timeT1>0and for every x0∈Γ, there existst1(x0)< T1 such that the solution x(t) of(19)verifies

x(t)∈Ω1 for0< t < t₁(x₀) (20) and

x(t₁(x₀))∈S for x₀∈Γ. (21)

Remark 2.1. Condition 2.1 implies that Γ andS are isotopic, but it is not clear whether isotopy is sufficient to ensure this condition.

Remark 2.2. Notice that Condition 2.1 is fulfilled for usual domains, see for example the cases of Figure 2.

00 00 00 11 11 11

0000 0000 0000 0000 1111 1111 1111 1111

0000 0000 0000 1111 1111 1111

0000 0000 0000 1111 1111 1111 000 000

000 111 111 111

(a) (b) (c) (d) (e)

Figure 2. Condition 2.1 is fulfilled in situations (a, c, e) but not in (b) and (d). The boundary Γ is represented by a solid line and the interfaceSby a dashed line, the dashed region represents Ω0 and the black dot the location of the control zone.

00000 00000 00000 00000 11111 11111 11111 11111

(a)

00000 00000 00000 00000 11111 11111 11111 11111

(b)

00000 00000 00000 00000 11111 11111 11111 11111

(c)

00000 00000 00000 00000 11111 11111 11111 11111

(d)

00000 00000 00000 00000 11111 11111 11111 11111

(e)

000 000 000 111 111 111

000 000 000 111 111 111

000 000 000 111 111 111 000 000

111 111

Figure 3. Condition 2.2 is fulfilled in situations (a–d) but not in (e) with the same notations as in the previous figure.

Condition 2.2 (corresponding to case (2)). We assume that there exist two disjoint open sets O1,O2⊂⊂Ω1

(with always a unit outward normaln) and vectors fields ξⁱ: Ω17→R^N,ξⁱ∈C¹(Ω1),i= 1,2, such that ξⁱ(x)·n(x)>0 ∀x∈S,

ξⁱ(x)·n(x)>0 ∀x∈∂Oi, i= 1,2, ξⁱ(x)6= 0 ∀x∈Ω1\Oi

(22)

and for the characteristics associated to ξⁱ



 dxⁱ(t)

dt =−ξⁱ(xⁱ(t)), t >0,

xⁱ(0) =xⁱ₀, (23)

with xⁱ₀∈S, we assume also that for some timeT₂ⁱ>0, and for all xⁱ₀ ∈S, there exists tⁱ₂(xⁱ₀)< T₂ⁱ such that the solution xⁱ(t)of(23)verifies

xⁱ(t)∈Ω1\Oi for 0< t < tⁱ₂(xⁱ₀) and

xⁱ(tⁱ₂(xⁱ₀))∈∂Oi for xⁱ₀∈S,i= 1,2.

Remark 2.3. Notice that this hypothesis is essentially Condition 2.1 written for the case (2). It is also fulfilled in usual geometrical cases, see for example the cases in Figure 3.

The aim of this paper is to prove the following theorem:

Theorem 2.1. Assume that in problem (8) the coefficient a satisfies (5, 6, 9, 10), f is a locally Lipschitz- continuous function satisfying (11) or Condition 2.1 in case (1) or Condition 2.2 in case (2) are fulfilled. If ω∩Ωⁱ₀6=∅, for each connected component Ωⁱ₀ of Ω0, then for each case (1) or (2, 8) is exactly controllable to the trajectories.

The idea of the proof of Theorem 2.1 is the following. With a simple change of variables we reduce the problem of exact controllability to the trajectories for (8) to null controllability for a still nonlinear similar transmission problem. For this null controllability result we use approximate controllability to the zero state for an associated linear transmission problem with controls inL^r(0, T;L^r(ω)) forras in (4) and then we apply a fixed point method. For this we need explicit estimates on the cost of approximate controllability which is obtained from observability inequalities (see Props. 4.1 and 4.2). These estimates are deduced from global Carleman inequalities. In case (1), we use one single global Carleman inequality (see Th. 3.3) with a suitable weight function, whose construction is presented in Lemma 3.1. Case (2) is more complicated and we have to combine two different global Carleman inequalities (see Th. 3.4) with two appropriate weight functions whose construction are given in Lemma 3.2. The growth condition of the non linear term f is analyzed using the arguments of [13].

The idea of combining the controllability of a linearized system and a fixed point argument in the proof is rather general. It was introduced in [23] in the context of the boundary controllability of the semilinear wave equation. For other controllability results proved in a similar way, see for instance [9, 13, 15] and [8].

In the proofs we will suppose that Ω0 and Ω1 are connected sets and we assume the simpler hypothesis ω ∩Ω0 6= ∅. Otherwise the weight functions for Carleman inequalities are constructed analogously on each connected component of Ω0and Ω1.

The paper is organized as follows. In Section 3 we deduce global Carleman inequalities, that we use for proving the main result. Section 4 is devoted to obtain some observability estimates. In Section 5, we prove Theorem 2.1. Finally, in Section 6, we give an explicit construction of suitable weight functions, needed for the global Carleman inequalities.

2.2. Some consequences and extensions

1. Observe that, the controllability result holds if the control acts in the part of the domain where the diffusion coefficient is smaller. To our knowledge, this result is the first one in the literature related to exact controllability to the trajectories when the diffusion coefficients are discontinuous.

2. In the cases⁰= 0 and f(0) = 0, notice that assumption (11) can be simply read as follows:

|s|→lim+∞

f(s)

|s|log^3/2(1 +|s|) = 0. (24) The proof of Theorem 2.1 also gives the result of null controllability for (8) with a suitable hypothesis on g under the hypothesis and the same geometrical cases considered in Theorem 2.1 by taking condition (24) instead of (11).

3. Notice that approximate controllability for a linear transmission problem is always true and it is independent of the choice of the part of the domain where the control acts as a consequence of unique continuation property.

Nonlinear problem (8) with f growing as in (11) is still approximately controllable under the conditions of Theorem 2.1. This is due to the fact that approximate controllability in this case can be proved as a consequence of exact controllability to the trajectories. This idea is taken from [12], where approximate controllability for semilinear heat equations is obtained in such a way.

4. We can also consider in (8) the more general case in which the diffusion coefficients are represented by a real symmetric uniformly elliptic matrixA,i.e. there exists a constantα >0 such that

A(x, ξ, ξ) = XN i,j=1

A_ijξ_iξ_j ≥α|ξ|² ∀ξ∈R^N, for a.e. x∈Ω (25)

and Ais regular in each Ω_i, i= 0,1. In this case, condition (10) has to be replaced by detA

An·n

S

≥0, (26)

where [ ]_S denotes the jump acrossS.

Until now, null controllability for semilinear parabolic systems (in the divergence form) has been analyzed when the diffusion coefficients are sufficiently regular. More precisely, when A = (A_ij), i, j = 1, . . . , N with A_ij∈C^1,2(Q) (see [15]).

2.3. Open problems related to Theorem 2.1

1. If ω ⊂Ω1 we do not know whether or not system (8) is exactly controllable to the trajectories. Is in this case null controllability also an open problem.

2. In [13], it is proved that even in the case of regular diffusion coefficients, for eachβ >2, there exist functions f =f(s) withf(0) = 0 and

|s|→∞lim

|f(s)|

|s|log^β(1 +|s|) =α withα >0, (27) such that the corresponding system (for the semilinear heat equation) is not null-controllable for anyT >0. In view of point 2 in Section 2.2, we see that, when f satisfies (27) with 3/2≤β≤2, null-controllability of (8) is an open question.

3. On the other hand, it is proved in [13], that also in the case of regular diffusion coefficients, for eachβ >2, there exist functions f satisfying (27) such that the corresponding system (for the semilinear heat equation) is not approximately controllable for allT >0. Then, approximate controllability for the transmission problem (8) with 3/2≤β≤2, is also an open question.

4. An abstract result due to Russell [20] shows that boundary exact controllability for the wave equation implies boundary exact null controllability for the heat equation with the same type of control and geometry. This result is proved in the case of smooth coefficients. If we consider this principle still true in the case of non smooth coefficients, the geometrical hypothesis that we consider here seems to be too restrictive in the case N = 1 (cf.[7] for the controllability of the corresponding wave equation) but not forN ≥2.

5. In [14], it is proved null controllability result for the one-dimensional linear heat equation like ρ(x)− (a(x)z_x)_x+m(x)z = 0 with only BV coefficients without any assumption on the control zone. However, the proof is definitely strictly one dimensional relying on the corresponding one for the wave equation and null controllability result is true if the potential mdepends only on space variable, but not on time. Then, even in the one-dimensional case it is not clear how to treat with this method a similar nonlinear problem.

3. Global Carleman inequalities

In this section we will deduce two global Carleman inequalities, that we need for the proof of Theorem 2.1.

For this purpose, we will introduce suitable weight functions. Let us first consider the situation of case (1) (see Fig. 1, left). The first weight function is given by the following result:

Lemma 3.1. Assume that we have the geometrical situation of case(1)(see Fig. 1). Assume that the function a defined in (5, 6) satisfies (9, 10) and that Condition 2.1 holds. If ω∩Ω0 6= ∅ then for every open set ω₀⊂⊂ω∩Ω0 there exists a functionβe∈C⁰(Ω),βe_i=βe|Ω_i ∈C²(Ω_i),i= 0,1,β >e 0 inΩ, such that

βe= 0 onΓ, (28)

∂_nβ <e 0 onΓ, (29)

βe= 1 onS, (30)

∂_nβe₀>0, ∂_nβe₁>0 onS, (31)

a₀∂_nβe₀=a₁∂_nβe₁ on S (32)

and

|∇βe|>0 inΩ\ω0. (33)

The proof of Lemma 3.1 will be given in Section 6.

Now, we consider the geometrical case (2) (see Fig. 1, right). For the second Carleman inequality, which we will use to treat the situation 2, we need two suitable weight functions.

We have the following result:

Lemma 3.2. Assume that we have the geometrical situation of case (2) (see Fig. 1). Assume that the function a defined in (5, 6) satisfies (9, 10) and that there exist two open disjoint sets O1,O2⊂⊂Ω1 verifying Condi- tion 2.2. LetB_i andBe_i,i= 1,2 be balls such thatB₁⊂⊂Be₁⊂⊂ O1 andB₂⊂⊂Be₂⊂⊂ O2. Ifω∩Ω06=∅then for every open setω₀⊂⊂ω∩Ω0 there exist two functionsβe¹ andβe² such that

βe¹(x) =

( βe0(x) if x∈Ω0,

βe₁¹(x) if x∈Ω1, βe²(x) =

( βe0(x) if x∈Ω0,

βe₁²(x) if x∈Ω1, (34) with the following properties: βe0∈C²(Ω0),βe0>0 inΩ0,

βe0= 0on Γ, ∂_nβe0<0 onΓ, (35)

∂_nβe0>0 onS, βe0= 2 onS, (36)

|∇βe0|>0 in Ω0\ω0. (37) And fori= 1,2,βeⁱ₁∈C²(Ω1),βeⁱ₁>0 inΩ1,

βeⁱ₁=βe0= 2 onS, (38)

a₀∂_nβe₀=a₁∂_nβe₁ⁱ onS, i= 1,2, (39)

βe₁¹≥2βe²₁ inBe2, (40)

βe₁²≥2βe¹₁ inBe1, (41)

and

|∇βe₁ⁱ|>0 inΩ1\B_i, i= 1,2. (42) The proof of Lemma 3.2 will also be given in Section 6.

Remark 3.1. Notice that in geometrical case (2) (Ω1 ⊂⊂ Ω) it is impossible to have a function βewhich is constant on S and such that∇βe6= 0 in Ω1.

Let us consider the functions

β=βe+K, β =5 4max

Ω β, (43)

with K >0 such thatK≥5 max

Ω

β,e andβeis given by Lemma 3.1.

Let λbe a sufficiently large positive constant that only depends on Ω andω. It will be fixed later on. For t∈(0, T) and following [16] and [12], we introduce the following functions:

ϕ(x, t) = e^λβ(x)

t(T −t), η(x, t) = e^λβ−e^λβ

t(T −t) · (44)

Notice that

∇η =−λϕ∇β, ∇ϕ=λϕ∇β. (45)

Let us set

Z₀={q : q∈C²(Ω_i×[0, T]), i= 0,1, q|_S⁺_×(0,T)=q|_S⁻_×(0,T), a₀∂_nq|_S⁺_×(0,T)=a₁∂_nq|_S⁻_×(0,T), q= 0 on Σ}·

We have the following Carleman estimate:

Theorem 3.3. Assume thatω∩Ω06=∅,asatisfies (5, 6, 9) and(10)and Condition 2.1 in case (1) is fulfilled.

There existsλ1(Ω, ω, a)>0such that for eachλ > λ1there exists a positive constantCthat only depends onΩ, ω anda, ands1(λ)>0so that the following estimate holds

s³ ZZ

Q

e^−2sηt⁻³(T−t)⁻³|q|²dxdt+s ZZ

Q

e^−2sηt⁻¹(T−t)⁻¹|∇q|²dxdt

≤C s³ ZZ

ω×(0,T)e^−2sηt⁻³(T−t)⁻³|q|²dxdt +

ZZ

Q

e^−2sη|∂_tq+ div(a(x)∇q)|²dxdt

(46)

for all q ∈ Z₀ and s ≥ s₁. Moreover, s₁ is of the form s₁ = σ₁(Ω, ω, a, λ)(T²+T), where σ₁ is a positive constant that only depends on Ω,ω,aandλ.

Proof of the Theorem 3.3. In the sequel,C will stand for a generic positive constant only depending on Ω, ω and a, whose value can change from line to line. We will also use the usual convention of repeated indices.

Let us assumeq∈Z0and s >0. We set

f =∂_tq+ div(a(x)∇q)

and

ψ= e^−sηq. (47)

Notice that

ψ(0) =ψ(T) = 0. (48)

We have the following equality:

e^−sη(∂_t(e^sηψ) + div(a(x)∇(e^sηψ)) = e^−sηf. (49) Using (45), we can write (49) in the form

M1ψ+M2ψ= e^−sηf+sλϕdiv(a(x)∇β)ψ−sλ²ϕa(x)|∇β|²ψ, (50) where

M1ψ= div(a(x)∇ψ) +s²λ²ϕ²|∇β|²a(x)ψ+s∂_tηψ (51) and

M₂ψ=∂_tψ−2sλϕa(x)∇β∇ψ−2sλ²ϕa(x)|∇β|²ψ. (52) Let set

f_s= e^−sηf+sλϕdiv(a(x)∇β)ψ−sλ²ϕa(x)|∇β|²ψ. (53) From (50), we obtain

kM₁ψk²2+kM₂ψk²2+ 2(M₁ψ , M₂ψ) =kf_sk²2, (54) where (·,·) denotes the scalar product inL²(Q). Let us compute the scalar product in the left hand side of (54).

We can write

(M1ψ , M2ψ) =I11⁰+I12⁰ +I13⁰+I21⁰ +I22⁰+I23⁰+I31⁰+I32⁰+I33⁰. (55) In (55), all the integrals denote the respective scalar products for the terms of M₁ψ andM₂ψ. For simplicity, in the sequel, we will write ainstead ofa(x). We have

I₁₁0 = ZZ

Q

div(a∇ψ)∂_tψdxdt=− ZZ

Q

a∇ψ ∂_t(∇ψ) dxdt

+ Z _T

0

Z

S

a1∇ψ·n ∂_tψdxdt− Z _T

0

Z

S

a0∇ψ·n ∂_tψdxdt=−1 2

ZZ

Q

a∂_t(|∇ψ|²) dxdt= 0.

(56)

Here we have used (48) which says that ψ(0) =ψ(T) = 0.

I₁₂0 =−2sλ X1 l=0

Z _T

0

Z

Ω_l

ϕdiv(a_l∇ψ)a_l∇β∇ψdxdt= 2sλ X1

l=0

Z _T

0

Z

Ω_l

a_l∂_xiψ∂_xi(ϕa_l∂_xjβ∂_xjψ) dxdt

−2sλ Z _T

0

Z

S⁻

ϕa²₁(∇β1· ∇ψ)(∇ψ·n) dσdt+ 2sλ Z _T

0

Z

S⁺

ϕa²₀(∇β0· ∇ψ)(∇ψ·n) dσdt

−2sλ Z _T

0

Z

Γ

ϕa²₁(∇β· ∇ψ)(∇ψ·n) dσdt.

(57)

Let us consider the first term of (57). Also for simplicity, we will make the computation only for the integrals in Ω1. We set

I₁₂¹0 = 2sλ Z _T

0

Z

Ω₁

a₁∂_xiψ ∂_xi(ϕa₁∂_xjβ∂_xjψ) dxdt.

We will use (45) and

∇β= (∇β·n)n+∇τβ, (58)

∇ψ= (∇ψ·n)n+∇τψ, (59)

where ∇τβ and ∇τψ denote the tangential gradients. Thanks to the choice of β we know from (30) and (43) that β is a constant onS, then∇τβ= 0 onS and we can write

I₁₂¹0 = 2sλ² Z _T

0

Z

Ω₁ϕ(a1∂_xiψ∂_xiβ)(a1∂_xjβ∂_xjψ) dxdt+ 2sλ Z _T

0

Z

Ω₁ϕa1∂_xiψ ∂_xi(a1∂_xjβ)∂_xjψdxdt +sλ

Z _T

0

Z

Ω₁

ϕa1(a1∂_xjβ)∂_xj(|∇ψ|²) dxdt.

(60)

Integrating now by parts in the third term of (60) we obtain I₁₂¹0 = 2sλ²

Z _T

0

Z

Ω₁

ϕ(a1∂_xiψ∂_xiβ)(a1∂_xjβ∂_xjψ) dxdt+ 2sλ Z _T

0

Z

Ω₁

ϕa1∂_xiψ ∂_xi(a1∂_xjβ)∂_xjψdxdt

−sλ² Z _T

0

Z

Ω₁

ϕ|a₁∇β|²|∇ψ|²dxdt−sλ Z _T

0

Z

Ω₁

ϕa₁∂_xj(a₁∂_xjβ)|∇ψ|²dxdt

−sλ Z _T

0

Z

Ω₁

ϕ(∂_xja₁)(a₁∂_xjβ)|∇ψ|²dxdt+sλ Z _T

0

Z

S⁻

ϕa²₁(∂_nβ₁)|∇ψ|²dσdt +sλ

Z _T

0

Z

Γ

ϕa²₁(∂_nβ)|∇ψ|²dσdt.

(61)

For the integrals in Ω0 it is sufficient to take into account thatnis the outward unit normal to Ω1 and replace in (61), nby−n,S⁻ byS⁺ and Ω1by Ω0.

Consequently, from (57) and (61), using again (59) we deduce

I12⁰ =−sλ² ZZ

Q

ϕ|a∇β|²|∇ψ|²dxdt+ 2sλ² ZZ

Q

ϕ(a∇ψ· ∇β)²dxdt

−sλ Z _T

0

Z

S⁻

ϕ(∂_nβ1)|a1∂_nψ|²dσdt+sλ Z _T

0

Z

S⁺

ϕ(∂_nβ0)|a0∂_nψ|²dσdt +sλ

Z _T

0

Z

S⁻

ϕa1(a1∂_nβ1)|∇τψ|²dσdt−sλ Z _T

0

Z

S⁺

ϕa0(a0∂_nβ0)|∇τψ|²dσdt

−sλ Z _T

0

Z

Γ

ϕa²₁(∂_nβ)|∂_nψ|²dσdt+X₁

(62)

with

X₁ = 2sλ ZZ

Q

ϕa∂_xiψ ∂_xi(a∂_xjβ)∂_xjψdxdt−sλ ZZ

Q

ϕa∂_xj(a∂_xjβ)|∇ψ|²dxdt

−sλ ZZ

Q

ϕ(∂_xja)(a∂_xjβ)|∇ψ|²dxdt.

(63)

Finally, we get

I12⁰ =−sλ² ZZ

Q

ϕ|a∇β|²|∇ψ|²dxdt+ 2sλ² ZZ

Q

ϕ(a∇ψ· ∇β)²dxdt +sλ

Z _T

0

Z

S

ϕ|a∂_nψ|²[∂_nβ]_S dσdt−sλ Z _T

0

Z

S

ϕ|∇τψ|²(a∂_nβ)[a]_Sdσdt

−sλ Z _T

0

Z

Γ

ϕ|a∂_nψ|²(∂_nβ) dσdt+X1,

(64)

where X1 is given by (63), and [·]_S denote the jump on S. Notice that in (64), due to the choice of βe the boundary integrals are nonnegative and this is essential. In fact, from (9) and (10) we have

[a]_S =a₀−a₁≤0 on S. (65)

On the other hand, from (10, 31, 32) and (43), we deduce that

[∂_nβ]_S=∂_nβ0−∂_nβ1≥0 on S, (66)

since n is the outward unit normal to Ω1. Moreover, thanks to (29) and (43), we have∂_nβ ≤0 on Γ. This justifies the above statement.

Let us compute the scalar product of the first term ofM1ψand the third one ofM2ψ.

I13⁰ =−2sλ² X1

l=0

Z _T

0

Z

Ω_ldiv(a_l∇ψ)ϕa_l|∇β|²ψdxdt= 2sλ² ZZ

Q

a∇ψ∇(ϕa|∇β|²ψ) dxdt

−2sλ² Z _T

0

Z

S⁻

ϕa₁(∂_nψ)a₁|∇β₁|²ψdσdt+ 2sλ² Z _T

0

Z

S⁺

ϕa₀(∂_nψ)a₀|∇β₀|²ψdσdt

= 2sλ² ZZ

Q

ϕ|a∇β|²|∇ψ|²dxdt+ 2sλ² X1 l=0

Z _T

0

Z

Ω_l

ϕa_l∇(al|∇β|²)∇ψψdxdt +2sλ³

ZZ

Q

ϕa²∇β|∇β|²∇ψψdxdt

−2sλ² Z _T

0

Z

S⁻

ϕa1(∂_nψ)a1|∂_nβ1|²ψdσdt+ 2sλ² Z _T

0

Z

S⁺

ϕa0(∂_nψ)a0|∂_nβ0|²ψdσdt.

(67)

In (67), again we have used (58) and the fact that∇τβ = 0 onS. Finally, we have I13⁰ = 2sλ²

ZZ

Q

ϕ|a∇β|²|∇ψ|²dxdt+X2, (68)

where

X₂ = 2sλ² ZZ

Q

ϕa∇(a|∇β|²)∇ψψdxdt+ 2sλ³ ZZ

Q

ϕa²∇β|∇β|²∇ψψdxdt +2sλ²

Z _T

0

Z

S

ϕ(a∂_nψ)(a∂_nβ) [∂_nβ]_Sψdσdt.

(69)

The scalar product of the second term of M1ψwith the first one ofM2ψgives I₂₁0 =s²λ²

ZZ

Q

ϕ²|∇β|²a∂_tψψdxdt=−s²λ² ZZ

Q

ϕ∂_tϕ|∇β|²a|ψ|²dxdt. (70)

We now consider the scalar product between the second term ofM1ψwith the second one ofM2ψ. The following holds:

I₂₂0 =−2s³λ³ X1 l=0

Z _T

0

Z

Ω_l

ϕ³a²_l|∇β|²(∇β)∇ψψdxdt

= 3s³λ⁴ ZZ

Q

ϕ³a²|∇β|⁴|ψ|²dxdt+s³λ³ X1

l=0

Z _T

0

Z

Ω_l

ϕ³div(a²_l|∇β|²∇β)|ψ|²dxdt

−s³λ³ Z _T

0

Z

S⁻

ϕ³|a1∂_nβ1|²(∂_nβ1)|ψ|²dσdt+s³λ³ Z _T

0

Z

S⁺

ϕ³|a0∂_nβ0|²(∂_nβ0)|ψ|²dσdt

= 3s³λ⁴ ZZ

Q

ϕ³a²|∇β|⁴|ψ|²dxdt+s³λ³ Z _T

0

Z

S

ϕ³|a∂_nβ|²[∂_nβ]_S|ψ|²dσdt+X₃,

(71)

where

X₃=s³λ³ X1 l=0

Z _T

0

Z

Ω_l

ϕ³div(a²_l|∇β|²∇β)|ψ|²dxdt. (72)

Observe again that the last boundary integral in (71) is nonnegative because (66) holds.

Now we consider the third and the second terms of M1ψandM2ψ, respectively. Using (45) we can write I32⁰ =−2s²λ

X1 l=0

Z _T

0

Z

Ω_l

ϕ∂_tηa_l∇β· ∇ψψdxdt=s²λ ZZ

Q

ϕ∂_tηdiv(a∇β)|ψ|²dxdt +s²λ²

ZZ

Q

ϕ(∂_tη)a|∇β|²|ψ|²dxdt−s²λ² ZZ

Q

ϕ(∂_tϕ)a|∇β|²|ψ|²dxdt

−s²λ Z _T

0

Z

S⁻

ϕ∂_tη(a1∂_nβ1)|ψ|²dσdt+s²λ Z _T

0

Z

S⁺

ϕ∂_tη(a0∂_nβ0)|ψ|²dσ.

(73)

Since (32) holds we know that a₀∂_nβ₀|_S⁺−a₁∂_nβ₁|_S⁻ = 0, then we get I₃₂0 =s²λ

ZZ

Q

ϕ∂_tηdiv(a∇β)|ψ|²dxdt+s²λ² ZZ

Q

ϕ(∂_tη)a|∇β|²|ψ|²dxdt−s²λ² ZZ

Q

ϕ(∂_tϕ)a|∇β|²|ψ|²dxdt.

(74) The last integrals give

I23⁰ =−2s³λ⁴ ZZ

Q

ϕ³a²|∇β|⁴|ψ|²dxdt, (75)

I31⁰ =s ZZ

Q

∂_tηψ∂_tψdxdt=−1 2s

ZZ

Q

∂_t²η|ψ|²dxdt (76)

and

I₃₃0 =−2s²λ² ZZ

Q

ϕa∂_tη|∇β|²|ψ|²dxdt. (77)

Finally from (55), taking into account (56, 64, 67, 70, 71, 74–76) and (77) we deduce that (M1ψ , M2ψ) =sλ²

ZZ

Q

ϕ|a∇β|²|∇ψ|²dxdt+ 2sλ² ZZ

Q

ϕ(a∇ψ· ∇β)²dxdt +s³λ⁴

ZZ

Q

ϕ³a²|∇β|⁴|ψ|²dxdt+s³λ³ Z _T

0

Z

S

ϕ³|a∂_nβ|²[∂_nβ]_S|ψ|²dσdt +sλ

Z _T

0

Z

S

ϕ|a∂_nψ|²[∂_nβ]_S dσdt−sλ Z _T

0

Z

S

ϕ|∇τψ|²(a∂_nβ)[a]_Sdσdt

−sλ Z _T

0

Z

Γ

ϕ|a∂_nψ|²(∂_nβ) dσdt+X1+X2+I21⁰ +X3+I31⁰+I32⁰+I33⁰,

(78)