Uniform null controllability for a parabolic equations with discontinuous diffusion coefficient

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.

The main objective of this paper is to analyze the uniform null controllability of (1.2) with respect to the parameter σ, i.e., we are interested in the behavior of the controls (u σ ) σ1,σ2>0 as a function of the diffusive coefficient σ.
In this setting, we ask what happens when one of this medium is highly conductive, that is σ 1 → ∞, σ 2 a 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 fixed σ of the form (1.1), is known when ∂Ω 1 ∩∂Ω 2 is smooth and does not intersect the boundary (see [21,22,20]). We will be able to give precise results only in the following geometrical situation, that we assume from now on: (A1) Ω 1 Ω, Ω 2 = Ω \ Ω 1 , (A2) There exists a point x 0 ∈ Ω 1 such that (a) The domain Ω 1 is star shaped with respect to x 0 . (b) The set ω is such that ω ⊂ Ω 2 and there exists ε > 0 such that ω ⊃ {x ∈ Ω ; d(x, Γ 0 ) ε} where Γ 0 = {x ∈ ∂Ω : (x − x 0 ) · ν(x) > 0}. (1.5) The above geometrical assumptions are illustrated in Figure 1. We point out that, under assumptions (A1) and (A2), the control is applied only in a subset of Ω 2 . More precisely, in (1.3) we have that ω ∩ Ω 1 = ∅ 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 consider u 1,σ to be the null function, so the control function will now be denoted only u σ instead of (u 1,σ , u 2,σ ) as in (1.3). (The case ω ⊂ Ω 1 will be briefly discussed in Theorem 4.3.) Our main result is the following one, whose proof is given in Section 3.
It is then natural to ask what is the limit system of (1.2) obtained as σ 1 → ∞ and σ 2 > 0 is fixed. This can be guessed intuitively. Indeed, since σ 1 → ∞, in the limit, the medium in Ω 1 is perfectly conductive. Therefore, its temperature y * 1 should be independent of the space variable. On the other hand, integrating (1.2) on Ω 1 , we see that one should have: where y * 2 is the temperature in the medium in Ω 2 . Besides, according to (1.7), (u σ ) σ1 σ2 converges to u * in some sense that will be made precise in the next section.
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 time t → T − t as usual), that is the existence of a time T > 0 and a constant C = C(Ω, ω, T, σ 2 ) > 0 such that for all σ 1 σ 2 , the solution z σ of in Ω, (1.11) with z 0 ∈ H 1 0 (Ω) and σ as in (1.1) satisfies In fact, this uniform observability result will be deduced, using the transmutation technique borrowed from [11], from a uniform observability result for the corresponding wave equation: . 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 velocity σ is 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 [16].
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 σ 2 > 0 and assume Assumptions (A1) and (A2).
In fact, Theorem 1.4 has already been proved under Assumptions (A.1) and (A.2) in [24, Chapter 6] except for the uniformity of the observability constant with respect to σ 1 . 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 [26,27], 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 in L 2 (Ω) × H −1 (Ω). However, if we want to keep track of the dependence of the constants in terms of σ, one should be careful that the accurate norm used in H 1 0 (Ω) is √ σ∇ · L 2 (Ω) which depends on σ, and thus the space H −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 [12,11], 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 [15] and [23], based on Carleman estimates and which apply for conductivities σ ∈ W 1,∞ (Ω). In our case, the conductivity given by (1.1) does not belong to W 1,∞ (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 conductivity σ ∈ BV (Ω), based on Russell's method (see e.g. [29]). The work [1] later improved this result, still in 1d, to the case σ ∈ L ∞ (Ω) using the theory of quasiconformal maps, and the more recent work [25] proposed an alternative approach for rough coefficients, still in the 1-dimensional case.
The multi-d heat equation was analysed later for piecewise C 2 conductivities with a smooth surface of C 0 discontinuity in [10] 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 of BV coefficients in 1d was dealt with using the Fursikov-Imanuvilov approach in [19]. The case of piecewise smooth coefficients with a smooth surface of discontinuity was then studied under no geometric assumptions in the works [5,6,7,21,22,20], 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 limit σ 1 → ∞, 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 [12,11], in a somewhat dual version of the transmutation techniques in [26,27]. 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 [18,24], 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 [9] 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 [16] 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, except in the 1d context in [17,Chapter 5] where this model has been dealt with using the moment method, see also Section 4.1 for more comments. Still, the controllability result obtained in [28] 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 to σ given 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.

Preliminaries
In this section, we provide several existence and uniqueness results for heat and wave equations with discontinuous 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) as σ 1 goes to infinity towards those of (1.8). 5

The heat equation for general conductivities σ
We recall the functional setting corresponding to the Cauchy problem for the parabolic equation in Ω, for a general conductivity σ satisfying σ ∈ L ∞ (Ω) and there exists α > 0 such that σ(x) α, a. e. in Ω. (2.2) We start with the definition of the bilinear form for y a and y b in H 1 0 (Ω). From (2.2), it is clear that a σ defines a continuous bilinear form on H 1 0 (Ω) which is coercive on H 1 0 (Ω). This allows to define the operator A σ : H 1 0 (Ω) → H −1 (Ω) by the formula: As one easily checks using Lax-Milgram theorem and condition (2.2), this operator is a self-adjoint maximal operator on H −1 (Ω) with domain D(A σ ) = H 1 0 (Ω). Therefore, for f ∈ L 1 (0, T ; H −1 (Ω)) and y 0 ∈ H −1 (Ω), as a consequence of Hille-Yosida theorem, interpreting equation (2.1) as the abstract equation there exists a unique solution y of (2.1) in the class C 0 ([0, T ]; H −1 (Ω)).
Proof. This result is folklore in the literature, so we only briefly sketch it. Of course, the only difficulty is close to the interface ∂Ω 1 . The idea there is to work as in the proof of elliptic regularity close to the boundary (see e.g. [13], 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 to L 2 (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 operatorÃ σ defined on the Hilbert space L 2 (Ω) is given as follows:

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.
As before, we can define the bilinear form As one easily checks using Lax-Milgram theorem and condition (2.2), this operator is a self-adjoint maximal operator on V * with domain V * . As before, for (F, f 2 ) ∈ L 1 (0, T ; V * ) and (Y 0 , y 2,0 ) ∈ V * , as a consequence of Hille-Yosida theorem, interpreting equation (2.15) as the abstract equation there exists a unique solution (Y, y 2 ) of (2.15) in the class C 0 ([0, T ]; V * ). But the space V * is not easy to deal with, and we prefer to give a functional setting based on the usual L 2 space. We thus introduce the space which, endowed with the scalar product, is a Hilbert space. We then introduce the operatorÃ * ,σ2 defined on H * with domain and defined for (Y, y 2 ) ∈ D(Ã * ,σ2 ) bỹ One can check thatÃ * ,σ2 is a self-adjoint maximal operator on H * . Interpreting equation (2.15) as the abstract equation Hille-Yosida Theorem then yields that for any (Y 0 , y 2,0 ) ∈ H * and (F, Besides, the following energy estimates can be derived: in D (0, T ), Let us finally indicate that the solution (Y, y 2 ) of (2.19) can be interpreted as the unique element of L 2 (0, T ; H * ) such that for all (G, g 2 ) ∈ L 2 (0, T ; H * ), where (Z, z 2 ) is the solution of −(Z, z 2 ) +Ã σ (Z, z 2 ) = (G, g 2 ) in (0, T ) and (Z, z 2 )(T ) = 0. 8

A convergence result
In this section, we explain how to pass to the limit in systems (1.2) to derive equation (1.8).
In all this section, σ 2 > 0 is fixed and σ 1 1 is going to infinity. Since the conductivity σ in (1.1) is characterized by (σ 1 , σ 2 ), we will write the sequence of functions associated to the conductivity σ as (f σ ) σ1 1 .

26)
and for σ 1 1, introduce z σ the solution of we have the following convergence result: The proof of Lemma 2.3 will be given afterwards, and we first resume the proof of Theorem 2.2. The basic idea is to identify the limit y * by looking at the weak formulations (2.12) satisfied by y σ and pass to the limit in them. We thus take (G, g 2 ) ∈ L 2 (0, T ) × L 2 (0, T ; L 2 (Ω 2 )), and introduce z σ the corresponding solution of (2.27) for σ characterized by (σ 1 , σ 2 ) as in (1.1). Therefore, for all σ 1 1, we get Using the convergences (2.29) and (2.30), we can pass to the limit σ 1 → ∞ in these identities: where z * is given by (2.28).
These convergences finish the proof of Lemma 2.3.

Well-posedness issues for wave equation with discontinuous velocities
Here, we recall some facts on the well-posedness of the wave equation for a general σ ∈ L ∞ (Ω) satisfying (2.2) and initial data (w 0 , w 1 ) in H 1 0 (Ω) × L 2 (Ω). Note that, following the notation used in Section 2.1, (2.36) can be reformulated as Since we want to keep track of the dependences of the solutions with respect to the coefficient σ, we need to be slightly more precise than that in the following study.
3 Proofs of Theorem 1.1 and Corollary 1.3 In this section, we prove the main result of this article, i.e. Theorem 1.1 and its corollary, Corollary 1.3. Roughly speaking, the plan of the proof of Theorem 1.1 can be described as follows: 1. We focus on the proof of the uniform observability inequality of the wave equation with respect to σ 1 , i.e., we prove Theorem 1.4. This will be done using multiplier techniques.
2. We apply the transmutation method as in [11] to prove a uniform observability inequality for the heat equation where the observation is given in a part of the boundary.
3. Using an auxiliary cut-off function and a Cacciopoli inequality, we replace the norm of the boundary by the L 2 (0, T ; L 2 (ω)) norm in the previously obtained observability inequality.
For simplicity, the proof is divided into several steps that will be presented in different paragraphs. The proof of Corollary 1.3 is then presented in the last paragraph of the section.
Notations and convention. In the following, the coefficient σ 2 is assumed to be fixed to a positive constant. The coefficient σ 1 however is simply larger than σ 2 , and the constants are all independent of σ 1 .

Proof of Theorem 1.4: Uniform observability of the wave equation
In this step, we study uniform observability properties for the wave equation (1.13), that is to say, we will prove the existence of a constant C = C(Ω, Γ 0 , T, σ 2 ) independent of σ 1 ( σ 2 ) such that for all w σ solution of (1.13) and E σ defined in (2.38). As said in the introduction, our proof follows the one in [24,Chapter 6], keeping track of the dependence of the constants in terms of σ 1 . By a usual density argument, it is sufficient to consider the case of initial conditions lying in D(Ã σ ) × D(Ã In order to simplify our notation, in this step we will ignore the dependence of w σ in σ, so we simply write w instead of w σ , but all the constants appearing next will be independent of σ 1 . In addition, we suppose without loss of generality that x 0 = 0 R d . Since for w ∈ D(Ã σ ), Lemma 2.1 implies that we have (w 1 , w 2 ) = (w| Ω1 , w| Ω2 ) ∈ H 2 (Ω 1 ) × H 2 (Ω 2 ), it is natural to consider the equation satisfied by w 1 in (0, T ) × Ω 1 and by w 2 in (0, T ) × Ω 2 : There, in view of the aforementioned regularity, we can develop the usual multiplier argument, which consists in multiplying the equation (3.2) by x·∇w 1 +(d−1)w 1 /2 and the equation (3.3) by x·∇w 2 +(d−1)w 2 /2 and do integration by parts.

Proof of Theorem 1.1, Step 2: From heat processes to waves
Our goal here is to prove the following result: Proposition 3.1. Let σ 2 > 0 and assume Assumptions (A1) and (A2). For all T > 0, there exists a constant C = C(Ω, Γ 0 , T, σ 2 ) > 0 such that for all σ 1 σ 2 , the solution z σ of (1.11) with z 0 ∈ H 1 0 (Ω) satisfies Proof. The argument here is based on [11, Section 3.1]. As explained there, according to [11,Proposition 3.2], setting we can construct a kernel K = K(t, s) solution of for (t, s) ∈ (0, T ) × (−S, S), K(0, s) = K(T, s) = 0, for s ∈ (−S, S), (3.10) and such that and for which we have the estimate: Now, let z 0 ∈ H 1 0 (Ω) and let z σ be the solution of (1.11). Then set Easy computations show that w σ satisfies the wave equation (3.14) and Applying the uniform observability obtained in Theorem 1.4 to W σ (τ ) = w σ (τ − S) for τ ∈ (0, 2 S), and using that the energy is constant, easily leads to the existence of a constant C independent of σ 1 σ 2 such that On one hand, as, for all µ 0, one has decomposing z σ (t) on the orthonormal basis formed by the eigenfunctions of the positive self-adjoint operator A σ easily leads to .
On the other hand, the right-hand side of (3.16) can be easily bounded as well since K is uniformly bounded by S in (−S, S) × (0, T ) (recall (3.11)): Combining the last estimates, we easily deduce the observability inequality (3.8).

Further comments and open problems 4.1 A spectral approach in the 1d case
In the 1-dimensional case, Theorem 1.1 can alternatively be proved using the moment method, i.e. a spectral approach based on suitable properties of the eigenvalues and eigenvectors of the underlying operator, the difficulty being to get uniform estimates with respect to the parameter going to infinity.
Note that, strictly speaking the system of equation (4.1) does not correspond to the system considered in (1.3) in 1d, but it clearly has the same flavor and can be dealt with similarly to the price of some minor additional technical difficulties.
We will not provide a detailed proof of Theorem 4.1 since it is quite lengthy and could also be done quicker using the observability of the corresponding wave equation and the transmutation technique as we did for the proof of Theorem 1.1.
We will nevertheless briefly sketch it below to explain the key steps to prove Theorem 4.1 using the moment method.
Sketch of the proof of Theorem 4.1. The moment method is a spectral approach, relying on a good knowledge of eigenvalues and eigenvectors. Thus, for all ε ∈ (0, 1), we define the operator A ε defined on L 2 (−1, 0) × L 2 (0, 1) by with y L (−1) = y R (1) = 0, y L (0) = y R (0), and ∂ x y L (0) = 1 Similarly as in Section 2.1, it is classical to check that these operators are self-adjoint, positive definite and with compact resolvent. Therefore, their spectrum is made by a sequence of strictly positive eigenvalues (λ k,ε ) k∈N (increasingly ordered) and of corresponding eigenvectors (Φ k,L,ε , Φ k,R,ε ) k∈N .
Proof of estimate (4.4). In order to prove the estimate (4.4), according to the above discussion, it is sufficient to prove that there exists a constant α > 0 such that for ε > 0 small enough, two positive consecutive root of the function f ε (defined in (4.7)) are at a distance α. This is not difficult to check, following the arguments hereafter.
If we first assume that ε cannot be written under the form (2k + 1)/(2 + 1) for (k, ) ∈ N 2 , then there are no common roots of cos and cos(ε·), so the equation reduces to the analysis of the roots of g ε . Since g ε is a strictly increasing function apart from the singularities of tan and tan(ε·), it is clear that g ε has exactly one roots between each of the singularities of tan and tan(ε·).
If ω is a root of g ε which belongs to an interval of the form (−π/2 + kπ, π/2 + kπ) with k ∈ N such that tan(ε·) has no singularity in this interval, then we are in one of the following cases: if ω < π/4 + kπ, the next root being after the next singularity of g ε , it necessarily is at a distance at least π/4.
We have thus proved the uniform spectral gap condition (4.4) when ε is small enough and not of the form (2k + 1)/(2 + 1) for (k, ) ∈ N 2 . In fact, when ε is of the form (2k + 1)/(2 + 1) for (k, ) ∈ N 2 , the same arguments as above can be adapted locally around the common roots of cos and cos(ε·) with almost no change, so we leave it to the reader.
This concludes the proof of property (4.11) and property (4.5).
Conclusion. To conclude the proof of Theorem 4.1, for all ε ∈ (0, ε 0 ), we can then follow the classical strategy (see e.g. [30] or [8,Chapter IV]), which consists in building biorthogonal families to the exponential functions (t → exp(−λ k,ε t)) k∈N and constructing a control u ε by a suitable combination of these biorthogonal families. The only point to check is the uniform bound on the controls u ε constructed that way, which will follow from the application of [8, Theorems IV.1.8 and IV.1.9], whose assumptions are satisfied due the uniform bounds in (4.4) and (4.5). In particular, one easily checks that condition (4.4) implies, by setting that for all r > 0 and ε ∈ (0, ε 0 ), so that all families (λ k,ε ) k∈N have a uniform remainder function. The other conditions of [8, Theorems IV.1.8 and IV.1.9] follow easily from the condition (4.4) and the fact that the first eigenvalues λ 1,ε are uniformly bounded from below.

More general geometric conditions
A very interesting problem is to discuss if the geometric conditions (A1) and (A2) are really needed or not to get uniform controllability results when the conductivity in one of the two medium goes to infinity. We emphasize that here, these geometric conditions arise because our strategy of proof relies on the controllability of the corresponding wave equation, and are thus probably only technical.
It would be very natural to try to develop suitable Carleman estimates in the spirit of the works [5,6,7,21,22,20] to derive uniform observability estimates (1.12) for the solutions of (1.11) for conductivities σ of the form (1.1) when σ 1 → ∞ under the only condition that the control set ω is non-empty and ω ⊂ Ω 2 . This is so far an open problem, even when considering the restrictive geometric setting of [10].

The case of a control set in the strongly conductive material
When considering a control acting in the strongly conductive material, we claim the following result: Let Ω be a smooth bounded domain of R d , d 2, Ω 1 be a smooth non-empty subdomain of Ω with Ω 1 Ω, and Ω 2 = Ω \ Ω 1 , ω ⊂ Ω 1 , σ 2 > 0, and T > 0. Then there is no constant C such that for all σ 1 σ 2 , any solution z σ of (1.11) satisfies the observability inequality z σ (T ) L 2 (Ω) C z σ L 2 ((0,T )×ω) . (4.12) In particular, the systems (1.3) are not uniformly controllable as σ 1 → ∞.

4.4
The case of a strongly insulating material σ 1 → 0 It is also interesting to briefly discuss the case σ 1 → 0 as it corresponds to the limit of a strongly insulating material.
To better understand what happens in this case, let us first present the following convergence result, whose proof is left to the reader: Proposition 4.4. Let Ω be a smooth (C 2 ) bounded domain in R d (d ∈ N * ), Ω 1 Ω be a smooth bounded domain, and Ω 2 = Ω \ Ω 1 .
Still, an interesting open question is the analysis of the rate at which the cost of controllability blows up as σ 1 goes to 0.