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Preprint submitted on 2 Nov 2020 (v1), last revised 10 Jun 2021 (v2)
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Uniform null controllability for a parabolic equations with discontinuous diffusion coefficients
Jérémi Dardé, Sylvain Ervedoza, Roberto Morales
To cite this version:
Jérémi Dardé, Sylvain Ervedoza, Roberto Morales. Uniform null controllability for a parabolic equa- tions with discontinuous diffusion coefficients. 2020. �hal-02986358v1�
Uniform null controllability for a parabolic equations with discontinuous diffusion coefficients
∗J´er´emi Dard´e† Sylvain Ervedoza‡ Roberto Morales§ November 2, 2020
Abstract
In this article, we study the null-controllability of a heat equation in a domain composed of two media of different constant conductivities. In particular, we are interested in the behavior of the system when the conductivity of the medium on which the control does not act goes to infinity, corresponding at the limit to a perfectly conductive medium. In that case, and under suitable geometric conditions, we obtain a uniform null-controllability result. Our strategy is based on the analysis of the controllability of the corresponding wave operators and the transmutation technique, which explains the geometric conditions.
1 Introduction
Main results. Let Ω be a smooth (C2) bounded domain inRd(d∈N∗), Ω1be a smooth strict subdomain of Ω, and set Ω2= Ω\Ω1.
Forσ1>0 andσ2>0, we introduce the conductivityσ=σ(x) given by σ(x) =
(σ1 ifx∈Ω1,
σ2 ifx∈Ω2, (1.1)
and we consider the controlled heat equation
∂tyσ−div(σ∇yσ) =uσ1ω, in (0, T)×Ω,
yσ= 0, on (0, T)×∂Ω,
yσ(0) =y0, in Ω.
(1.2)
Here,yσ denotes the state (for instance the temperature), and uσ denotes the control function, assumed to be acting on a non-empty open subsetωof Ω.
The main objective of this paper is to analyze the uniform null controllability of (1.2) with respect to the parameterσ, i.e., we are interested in the behavior of the controls (uσ)σ1,σ2>0 as a function of the diffusive coefficientσ.
The fact that the conductivityσ takes the form (1.1) means that the domain Ω is filled with a material made of two media, one located in Ω1 and of conductivity σ1, the other one being located in Ω2 and of
∗The second author has been supported by the Agence Nationale de la Recherche, Project IFSMACS, grant ANR-15-CE40- 0010. The first and second authors have been supported by the CIMI Labex, Toulouse, France, under grant ANR-11-LABX- 0040-CIMI and the MATH AmSud program ACIPDE. The third author has been supported by FONDECYT 3200830.
†Institut de Math´ematiques de Toulouse, UMR 5219, Universit´e de Toulouse, CNRS, Toulouse, France email:
jeremi.darde@math.univ-toulouse.fr
‡Institut de Math´ematiques de Bordeaux UMR 5251, Universit´e de Bordeaux, Bordeaux INP, CNRS, F-33400 Talence, France, e-mail: sylvain.ervedoza@math.u-bordeaux.fr.
§Departamento de Matem´atica, Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile e-mail:
roberto.moralesp@usm.cl
conductivityσ2. Also note that, with Ω1bΩ and Ω2= Ω\Ω1, system (1.2) can alternatively be written as a transmission problem as follows:
∂ty1,σ−σ1∆y1,σ =u1,σ1ω∩Ω1, in (0, T)×Ω1,
∂ty2,σ−σ2∆y2,σ =u2,σ1ω∩Ω2, in (0, T)×Ω2, y1,σ =y2,σ, on (0, T)×∂Ω1, σ1∂νy1,σ =σ2∂νy2,σ, on (0, T)×∂Ω1,
y2,σ= 0, on (0, T)×∂Ω,
(y1,σ, y2,σ)(0) = (y0,1, y0,2), in Ω1×Ω2,
(1.3)
with the correspondence yσ(t, x) =y1,σ(t, x)1Ω1(x) +y2,σ(t, x)1Ω2(x) for (t, x)∈(0, T)×Ω. Hereafter, we defineν to be the normal to∂Ω1 oriented from Ω2 to Ω1.
In this setting, we ask what happens when one of this medium is highly conductive, that is
σ1→ ∞, σ2a fixed positive constant. (1.4)
Our goal is to analyze precisely the behavior of the null-controllability properties of (1.2) in the limit (1.4), which, for a fixedσof the form (1.1), is known when∂Ω1∩∂Ω2is smooth and does not intersect the boundary (see [18, 19, 17]).
We will be able to give precise results only in the following geometrical situation, that we assume from now on:
(A1) Ω1bΩ, Ω2= Ω\Ω1,
(A2) There exists a pointx0∈Ω1 such that
(a) The domain Ω1 is star shaped with respect tox0.
(b) The set ω is such thatω⊂Ω2 and there existsε >0 such that
ω⊃ {x∈Ω ;d(x,Γ0)6ε}where Γ0={x∈∂Ω : (x−x0)·ν(x)>0}. (1.5) The above geometrical assumptions are illustrated in Figure 1.
Figure 1: Geometrical situation given by(A1)and (A2), where ω is represented by the grey region.
We point out that, under assumptions (A1) and (A2), the control is applied only in a subset of Ω2. More precisely, in (1.3) we have that ω∩Ω1 =∅ and therefore the second equation in (1.3) is controlled directly by the action of the control, while the first one is being controlled indirectly, through the transmission conditions. In other words, we can consideru1,σ to be the null function, so the control function will now be denoted onlyuσ instead of (u1,σ, u2,σ) as in (1.3). (The caseω⊂Ω1will be briefly discussed in Theorem 4.1.)
Our main result is the following one, whose proof is given in Section 3.
Theorem 1.1. Let σ2>0 and assume(A1)and(A2).
For all T > 0, for all y0 ∈ L2(Ω) and for all σ1 > σ2, for σ as in (1.1), there exists a control uσ ∈ L2((0, T)×ω)such that the solutionyσ of (1.2)satisfies
yσ(T) = 0 inΩ. (1.6)
Moreover, the sequence of controls (uσ)σ1>σ2 is uniformly bounded, i.e. there exists a constant C = C(Ω, ω, T, σ2)>0such that for all σ1>σ2,
kuσkL2((0,T)×ω)6Cky0kL2(Ω). (1.7)
Estimates (1.7) state the uniform null controllability of (1.2) with respect toσ1>σ2, whereσ2 >0 is a fixed constant.
It is then natural to ask what is the limit system of (1.2) obtained as σ1→ ∞andσ2>0 is fixed. This can be guessed intuitively. Indeed, since σ1 → ∞, in the limit, the medium in Ω1 is perfectly conductive.
Therefore, its temperaturey1∗ should be independent of the space variable. On the other hand, integrating (1.2) on Ω1, we see that one should have:
d dt
Z
Ω1
y∗1(t, x)dx
=− Z
∂Ω1
σ2∂νy∗2(t, x)dγ,
wherey2∗ is the temperature in the medium in Ω2. Besides, according to (1.7), (uσ)σ1>σ2 converges tou∗ in some sense that will be made precise in the next section.
We should thus expect, and this will be proved afterwards (see Theorem 2.2), that the limit system of (1.2) in the limit (1.4) should be:
∂ty2∗−σ2∆y∗2 =u∗1ω, in (0, T)×Ω2, y2∗(t, x) = 0, on (0, T)×∂Ω, y2∗(t, x) =Y∗(t), on (0, T)×∂Ω1, y1∗(t, x) =Y∗(t), in (0, T)×Ω1,
∂tY∗+ σ2
|Ω1| Z
∂Ω1
∂νy∗2(t)dγ= 0, in (0, T), y2∗(0,·) =y0,2(·), in Ω2, Y∗(0) =Y0,
(1.8)
where the initial datumY0is given by
Y0= 1
|Ω1| Z
Ω1
y0,1dx.
Remark 1.2. Note that, in fact the function y∗1 is now reduced to a function Y∗ depending only on the time variable, so that the system (1.8) can be thought as an equation on (Y∗, y∗2) only, with for all t ∈ (0, T), Y∗(t)∈R, andy∗2(t)∈L2(Ω2).
Passing to the limit σ1→ ∞ in Theorem 1.1, we will thus deduce the following controllability result for (1.8), see Section 3.5 for its proof.
Corollary 1.3. Let σ2 > 0 and assume (A1) and (A2). Then for all T > 0 and for all (Y0, y0,2) ∈ R×L2(Ω2)there exists a control functionu∗∈L2((0, T)×ω) such that the solutiony∗ of (1.8)satisfies
y2∗(T) = 0inΩ2, and Y∗(T) = 0. (1.9) Moreover, there exists a constantC=C(Ω, ω, T, σ2)>0 such that
ku∗kL2((0,T)×ω)6Ck(Y0, y0,2)kR×L2(Ω2). (1.10) In addition, for (Y0, y0,2)∈R×L2(Ω2), the sequence of controls (uσ)σ1>σ2 of minimalL2(0, T;L2(ω))- norm of (1.2) corresponding to an initial datumy0=Y01Ω1+y0,21Ω2 weakly converges up to a subsequence asσ1→+∞to a null control u∗∈L2(0, T;L2(ω))for (1.8)with initial data(Y0, y0,2).
To prove Theorem 1.1, we first remark that it is equivalent to prove a uniform observability result for the adjoint heat equation (on which we perform the change of timet→T −tas usual), that is the existence of a timeT >0 and a constantC=C(Ω, ω, T, σ2)>0 such that for allσ1>σ2, the solution zσ of
∂tzσ−div(σ∇zσ) = 0, in (0, T)×Ω,
zσ= 0, on (0, T)×∂Ω,
zσ(0) =z0, in Ω,
(1.11)
withz0∈H01(Ω) andσas in (1.1) satisfies
kzσ(T)kL2(Ω)6CkzσkL2((0,T)×ω). (1.12) In fact, this uniform observability result will be deduced, using the transmutation technique borrowed from [10], from a uniform observability result for the corresponding wave equation:
∂t2wσ−div(σ∇wσ) = 0, in (0, T)×Ω,
wσ= 0, on (0, T)×∂Ω,
(wσ, ∂twσ)(0) = (w0, w1), in Ω,
(1.13)
with (w0, w1)∈H01(Ω)×L2(Ω). This strategy is responsible for the geometric conditions (A1)and (A2), since some geometric conditions are required for the controllability of the wave equation (1.13). This is of course already the case when the velocityσis smooth (this is the celebrated Geometric Control Conditions of [2]) but when it is discontinuous as in (1.1), the situation is much more intricate and we refer to the recent results [3], [4] and [14].
In fact, the uniform observability result we prove for (1.13) is the following one, see Section 3.1 for its proof:
Theorem 1.4. Let σ2>0 and assume Assumptions(A1)and(A2).
Then for allT >0 satisfying
√σ2T >2 sup
Ω
{ |x−x0|},
there existsC >0 such that for all(w0, w1)∈H01(Ω)×L2(Ω)and for allσ1>σ2, the solutionwσ of (1.13) with initial datum(w0, w1)∈H01(Ω)×L2(Ω) andσas in (1.1)verifies
Z
Ω
σ|∇w0|2+|w1|2
dx6C Z T
0
Z
Γ0
|∂νw|2dγdt. (1.14)
The proof of Theorem 1.4 is based on a multiplier argument. This is where the geometric conditions(A1) and(A2)appear naturally.
In fact, Theorem 1.4 has already been proved under Assumptions (A.1) and (A.2) in [21, Chapter 6]
except for the uniformity of the observability constant with respect toσ1. As this is a critical argument in our proof, we will present the complete proof of Theorem 1.4 in Section 3.1 for completeness.
Remark 1.5. There are a priori several paths based on transmutation techniques to deduce Theorem 1.1 from Theorem 1.4. In particular, one could try to follow the approach in [23, 24], which would consist in first deducing from Theorem 1.4 a uniform controllability result for the wave equation, and then use a transmutation technique to deduce a uniform controllability result for the corresponding heat equation. This is in principle possible, but in our case this would be delicate since the observability estimate (1.14) implies the controllability of the corresponding wave equation with a control bounded by the norm of the initial datum inL2(Ω)×H−1(Ω). However, if we want to keep track of the dependence of the constants in terms ofσ, one should be careful that the accurate norm used inH01(Ω) is k√
σ∇ · kL2(Ω) which depends on σ, and thus the spaceH−1(Ω)should be endowed with the corresponding dual norm. To avoid these difficulties and follow the dependence in σ more clearly, we have chosen to use the transmutation technique of [11, 10], and to avoid the use of negative Sobolev spaces.
Related references. Our investigations are part of the broad question of null-controllability for parabolic equations. The pioneering results for this question are the articles [13] and [20], based on Carleman estimates and which apply for conductivities σ ∈ W1,∞(Ω). In our case, the conductivity given by (1.1) does not belong toW1,∞(Ω) and these results thus do not apply.
This regularity assumption on the conductivity was then studied in details. It was proved that, in 1d, the heat equation (1.2) is still null-controllable for a conductivityσ ∈ BV(Ω), based on Russell’s method (see e.g. [26]). The work [1] later improved this result, still in 1d, to the caseσ∈L∞(Ω) using the theory of quasiconformal maps, and the more recent work [22] proposed an alternative approach for rough coefficients, still in the 1-dimensional case.
The multi-d heat equation was analysed later for piecewiseC2conductivities with a smooth surface ofC0 discontinuity in [9] using Carleman estimates, when the control is supported in the region where the diffusion coefficient is the lowest, under some geometric conditions which are less restrictive than our assumptions (A1)and (A2). Later on, the case ofBV coefficients in 1d was dealt with using the Fursikov-Imanuvilov approach in [16]. The case of piecewise smooth coefficients with a smooth surface of discontinuity was then studied under no geometric assumptions in the works [5, 6, 7, 18, 19, 17], and null-controllability of (1.2) was proved in those cases.
Let us point out that, despite these numerous results, to our knowledge, the behavior of the controllability of the models (1.2) for conductivities of the form (1.1) has not been studied so far in the limitσ1→ ∞, even in the 1d case.
Our approach is quite different from the ones based on Carleman estimates, as it relies on the observability of the corresponding wave equation, inspired by the transmutation techniques developed in [11, 10], in a somewhat dual version of the transmutation techniques in [23, 24]. There, the idea is to associate solutions of the wave equations to solutions of the heat equation through a time kernel, see Section 3.2 for more details.
The advantage of this technique is that our problem then reduces to the study of the observability of the corresponding wave operators, for which other techniques are available. Here, we shall follow the classical multiplier approach, introduced in [15, 21], which allows to deal with conductivities of the form (1.1) under appropriate geometric conditions, namely(A1)and(A2). These multiplier conditions are known to be very robust with respect to the regularity of the coefficients, and we refer for instance to the recent work [8] dealing with conductivities which are continuous and satisfy some suitable growth conditions in the direction of the multiplier, and to the references therein.
In fact, for waves with discontinuous conductivities, observability properties can be derived from Carleman estimates [3, 4] or microlocal analysis [14] under appropriate geometric conditions, but here again, keeping track of how it depends on the conductivity coefficients is, to our knowledge, a challenging problem.
Let us also mention that, to our knowledge, properly speaking, the controllability of the limit system (1.8) has not been dealt with in the literature. Still, the controllability result obtained in [25] on a very close system indicates that the null controllability of the limit system (1.8) can be proved directly by Carleman estimates without any geometric assumption.
Outline. The rest of the paper is organized as follows. Section 2 is dedicated to study the existence and uniqueness results concerning problems (1.2), (1.8) and (1.13) and their dependence with respect toσgiven by (1.1) as well as some general cases. In Section 3, the proofs of Theorem 1.1 and Corollary 1.3 are given.
Finally, in Section 4 additional comments and open questions are presented.
2 Preliminaries
In this section, we provide several existence and uniqueness results for heat and wave equations with discon- tinuous diffusion / velocity coefficients of the form (1.1). We also study the limit system (1.8) and show the convergences of the solutions of (1.3) asσ1 goes to infinity towards those of (1.8).
2.1 The heat equation for general conductivities σ
We recall the functional setting corresponding to the Cauchy problem for the parabolic equation
∂ty−div(σ∇y) =f, in (0, T)×Ω,
y= 0, on (0, T)×∂Ω,
y(0) =y0, in Ω,
(2.1)
for a general conductivityσsatisfying
σ∈L∞(Ω) and there existsα >0 such thatσ(x)>α, a. e. in Ω. (2.2) We start with the definition of the bilinear form
aσ(ya, yb) = Z
Ω
σ∇ya· ∇ybdx, (2.3)
forya andyb in H01(Ω). From (2.2), it is clear thataσ defines a continuous bilinear form onH01(Ω) which is coercive onH01(Ω).
This allows to define the operatorAσ:H01(Ω)→H−1(Ω) by the formula:
∀(ya, yb)∈(H01(Ω))2, hAσya, ybiH−1(Ω),H01(Ω)=aσ(ya, yb). (2.4) As one easily checks using Lax-Milgram theorem and condition (2.2), this operator is a self-adjoint maximal operator onH−1(Ω) with domainD(Aσ) =H01(Ω).
Therefore, for f ∈L1(0, T;H−1(Ω)) andy0∈H−1(Ω), as a consequence of Hille-Yosida theorem, inter- preting equation (2.1) as the abstract equation
y0+Aσy=f, in (0, T), y(0) =y0, (2.5)
there exists a unique solutionyof (2.1) in the class C0([0, T];H−1(Ω)).
In fact, we could also have defined the operator ˜Aσ on the Hilbert spaceL2(Ω) with domain
D( ˜Aσ) ={y∈H01(Ω) s.t. div (σ∇y)∈L2(Ω)}, (2.6) and defined by
A˜σy=−div (σ∇y). (2.7)
Note that Aσ is the self-adjoint extension of ˜Aσ to H−1(Ω). The only difficulty when working with ˜Aσ is that, without additional assumption onσ, its domain cannot be made more explicit than (2.6).
Still, one can check that ˜Aσ is a self-adjoint maximal operator on L2(Ω). Interpreting equation (2.1) as the abstract equation
y0+ ˜Aσy=f, in (0, T), y(0) =y0, (2.8) Hille-Yosida Theorem then yields that for any y0 ∈ L2(Ω) and f ∈ L1(0, T;L2(Ω)), the solution of (2.8) belongs toC0([0, T];L2(Ω)).
Besides, the following energy estimates can be derived: inD0(0, T), d
dt 1
2 Z
Ω
|y(t)|2dx
+ Z
Ω
σ|∇y(t)|2dx= Z
Ω
f(t)y(t)dx, (2.9)
and
d dt
Z
Ω
σ|∇y(t)|2dx
+ Z
Ω
|∂ty(t)|2+|div (σ∇y(t))|2 dx=
Z
Ω
|f(t)|2dx. (2.10) In particular, for all timeT >0, there exists a constant C independent of σ such that, ify0 ∈ L2(Ω) and f ∈L2(0, T;L2(Ω)),
Z T 0
Z
Ω
t |∂ty(t)|2+|div (σ∇y(t))|2
dxdt6C Z
Ω
|y0|2dx+C Z T
0
Z
Ω
|f|2dxdt. (2.11)
Let us finally indicate that the solution y of (2.8) can also be characterized as the unique element of L2(0, T;L2(Ω)) such that for allg∈L2(0, T;L2(Ω)),
Z T 0
Z
Ω
yg dxdt= Z T
0
Z
Ω
f z dxdt+ Z
Ω
y0z(0)dx, (2.12)
wherezis the solution of −z0+ ˜Aσz=g in (0, T) andz(T) = 0.
2.2 The case σ as in (1.1): additional regularity
In our present geometrical configuration, that is Ω1 bΩ and Ω2 = Ω\Ω1 having smooth boundaries, and with σ as in (1.1) with constant (σ1, σ2), the domain of ˜Aσ is slightly more explicit. Indeed, we have the following regularity result:
Lemma 2.1. Let Ω be a smooth bounded domain of Rd, σ be as in (1.1), where Ω1 bΩ and Ω2 = Ω\Ω1
have smooth boundaries. Then, for anyf ∈L2(Ω), the solutiony∈H01(Ω) of
−div (σ∇y) =f in Ω, y= 0on∂Ω, is such thaty1=1Ω1y andy2=1Ω2y satisfy:
y1∈H2(Ω1) and y2∈H2(Ω2), (2.13)
with
k(y1, y2)kH2(Ω1)×H2(Ω2)6CkfkL2(Ω), (2.14) for a constantC >0depending on(σ1, σ2).
Proof. This result is folklore in the literature, so we only briefly sketch it. Of course, the only difficulty is close to the interface∂Ω1. The idea there is to work as in the proof of elliptic regularity close to the boundary (see e.g. [12], page 317), that is:
• Flatten the interface by a change of variables,
• Multiply the equations by divided differences which are approximations of the second order derivatives of the solution in the tangential variables, to deduce that the second order derivatives of the solution in the tangential variables belong toL2(Ω),
• Recover estimates on the second order derivatives in the normal variables from the equation directly.
Details are left to the reader.
As a consequence, when σ is of the form (1.1), the domain of the operator ˜Aσ defined on the Hilbert spaceL2(Ω) is given as follows:
D( ˜Aσ) ={y∈H01(Ω) s.t.yj=1Ωjy∈H2(Ωj), j= 1,2, withσ1∂νy1=σ2∂νy2 on∂Ω1}.
2.3 The limit system (1.8)
In this section, we focus on the analysis of the Cauchy problem for (1.8), in which for simplicity of notation, we remove the∗ superscript.
∂ty2−σ2∆y2=f2, in (0, T)×Ω2,
y2(t, x) = 0, on (0, T)×∂Ω,
y2(t, x) =Y(t), on (0, T)×∂Ω1,
∂tY + 1
|Ω1| Z
∂Ω1
σ2∂νy2(t)dγ=F, in (0, T), y2(0,·) =y0,2(·), in Ω2, Y(0,·) =Y0.
(2.15)
As before, we can define the bilinear form
a∗,σ2 = ((Ya, y2,a),(Yb, y2,b)) = Z
Ω2
σ2∇y2,a· ∇y2,bdx,
on the setV∗={(Y, y2)∈R×H1(Ω2), with (y2)|∂Ω= 0 and (y2)|∂Ω1 =Y}. We can thus define the operator A∗,σ2 fromV∗ toV∗0 by
∀((Ya, y2,a),(Yb, y2,b))∈V∗2, hAσ(Ya, y2,a),(Yb, y2,b)iV∗0,V∗=aσ(y2,a, y2,b). (2.16) As one easily checks using Lax-Milgram theorem and condition (2.2), this operator is a self-adjoint maximal operator onV∗0 with domainV∗.
As before, for (F, f2) ∈ L1(0, T;V∗0) and (Y0, y2,0) ∈ V∗0, as a consequence of Hille-Yosida theorem, interpreting equation (2.15) as the abstract equation
(Y, y2)0+A∗,σ2(Y, y2) = (F, f2), in (0, T), (Y, y2(0)) = (Y0, y2,0), (2.17) there exists a unique solution (Y, y2) of (2.15) in the classC0([0, T];V∗0).
But the spaceV∗0 is not easy to deal with, and we prefer to give a functional setting based on the usual L2space. We thus introduce the space
H∗=R×L2(Ω2), (2.18)
which, endowed with the scalar product,
(Y, y2),( ˜Y ,y˜2)
H∗
=|Ω1|YY˜ + Z
Ω2
y2y˜2dx.
is a Hilbert space. We then introduce the operator ˜A∗,σ2 defined onH∗with domain D( ˜A∗,σ2) ={(Y, y2)∈V∗ s.t.y2∈H2(Ω2)},
and defined for (Y, y2)∈ D( ˜A∗,σ2) by
A˜∗,σ2(Y, y2) = σ2
|Ω1| Z
∂Ω2
∂νy2dγ, −σ2∆y2
.
One can check that ˜A∗,σ2 is a self-adjoint maximal operator on H∗. Interpreting equation (2.15) as the abstract equation
(Y, y2)0+ ˜A∗,σ2(Y, y2) = (F, f2), in (0, T), (Y, y2)(0) = (Y0, y2,0), (2.19) Hille-Yosida Theorem then yields that for any (Y0, y2,0) ∈ H∗ and (F, f2) ∈ L1(0, T;H∗), the solution of (2.19) belongs toC0([0, T];H∗).
Besides, the following energy estimates can be derived: inD0(0, T), 1
2 d dt
|Ω1||Y(t)|2+ Z
Ω2
|y2(t)|2dx
+ Z
Ω2
σ2|∇y2(t)|2dx=|Ω1|F(t)Y(t) + Z
Ω
f2(t)y2(t)dx. (2.20) Let us finally indicate that the solution (Y, y2) of (2.19) can be interpreted as the unique element of L2(0, T;H∗) such that for all (G, g2)∈L2(0, T;H∗),
|Ω1| Z T
0
Y G dt+ Z T
0
Z
Ω
y2g2dxdt=|Ω1| Z T
0
F Z dt+ Z T
0
Z
Ω
f2z2dxdt+|Ω1|Y0Z(0) + Z
Ω
y0,2z2(0)dx, (2.21) where (Z, z2) is the solution of−(Z, z2)0+ ˜Aσ(Z, z2) = (G, g2) in (0, T) and (Z, z2)(T) = 0.
