Carleman commutator approach in logarithmic convexity for parabolic equations

In this paper we investigate on a new strategy combining the logarithmic convexity (or frequency function) and the Carleman commutator to obtain an observation estimate at one time for the heat equation in a bounded domain. We also consider the heat equation with an inverse square potential. Moreover, a spectral inequality for the associated eigenvalue problem is derived.

In a series of articles (see [PW1], [PW2], [PWZ], [BP] for parabolic equations) inspired by [Po] and [EFV], we were interested on the function t → Ω |u (x, t)| 2 e Φ(x,t) dx and its frequency function t → Ω |∇u (x, t)| 2 e Φ(x,t) dx Here c, K > 0 and β ∈ (0, 1). From the above observation at one time, many applications were derived as bang-bang control [PW2] and impulse control [PWX], fast stabilization [PWX] or local backward reconstruction [Vo]. In particular, we can also deduce the observability estimate for parabolic equations on a positive measurable set in time [PW2]. Recall that observability for parabolic equations have a long history now from the works of [LR] and [FI] based on Carleman inequalities. Furthermore, it was remarked in [AEWZ] that the observation estimate at one point in time is equivalent to the Lebeau-Robbiano spectral inequality on the sum of eigenfunctions of the Dirichlet Laplacian. Recall that the Lebeau-Robbiano spectral inequality, originally derived from Carleman inequalities for elliptic equations (see [JL], [LRL], [Lu]), was used in different contexts as in thermoelasticity (see [LZ], [BN]), for the Stokes operator [CL], in transmission problem and coupled systems (see [Le], [LLR]), for the Bilaplacian (see [Ga], [EMZ], [LRR3]), in Kolmogorov equation (see [LRM], [Z]). We also refer to [M].
In this paper, we study the equation solved by f (x, t) = u (x, t) e 1 2 Φ(x,t) for a larger set of weight functions Φ (x, t) and establish a kind of convexity property for t →ln f (·, t) 2 L 2 (Ω) . By such approach we make appear the Carleman commutator. The link between logarithmic convexity (or frequency function) and Carleman inequality has already appeared in [EKPV1] (see also [EKPV2], [EKPV3]).
Theorem 1.1 thus states both the observability for the heat equation and the spectral inequality for the Dirichlet Laplacian in a simple geometry. One can see how fast the constant cost blows up when the observation region ω becomes smaller. Notice that the constant K ε does not depend on the dimension n (see [BP,Theorem 4.2]).
Theorem 1.2 gives a spectral inequality for the Schrödinger operator −∆− µ |x| 2 under a quite strong assumption on µ < µ * where the critical coefficient is µ * = 1 4 (n − 2) 2 . Our first motivation was to be able to choose 0 / ∈ ω by performing localization with annulus. We believe that a similar analysis can be handle with more suitable weight function Φ than those considered here and may considerably improve the results presented here.
We have organized our paper as follows. Section 2 is the important part of this article. We present the strategy to get the observation at one point by studying the equation solved by f = ue Φ/2 for a larger set of weight functions Φ adapting the energy estimates style of computations in [BT] (see also [BP,Section 4]). The Carleman commutator appears naturally here. Section 3 is devoted to check different possibilities for the weight function Φ, and in particular for the localization with annulus. In Section 4, we prove Theorem 1.1. The proof of Theorem 1.2 is given in Section 5. In Appendix, we recall the useful link between the observation at one point and the spectral inequality.
I am happy to dedicate this paper to my friend and colleague Jiongmin Yong on the occasion of his 60th birthday. I am also grateful for his book [LY] in where I often found the answer on my questions.

The strategy of logarithmic convexity with the Carleman commutator
We present an approach to get the observation estimate at one point in time for a model heat equation in a bounded domain Ω ⊂ R n with Dirichlet boundary condition. We shall present this strategy step-by-step. Two different geometric cases are discussed: When Ω is convex or star-shaped, we can used a global weight function; For the more general C 2 domain Ω, we will use localized weight functions exploiting a covering argument and propagation of interpolation inequalities along a chain of balls (also called propagation of smallness).

Convex domain
Throughout this subsection, we assume that Ω ⊂ R n is a convex domain or a starshaped domain with respect to x 0 ∈ Ω. Let ·, · denote the usual scalar product in L 2 (Ω) and let · be its corresponding norm. Here, recall that u (x, t) = e t∆ u 0 (x) ∈ C ([0, T ] ; L 2 (Ω)) ∩ C ((0, T ] ; H 1 0 (Ω)) and we aim to check that The strategy to establish the above observation at one time is as follows. We decompose the proof into six steps.
Step 2.1.1. Symmetric part and antisymmetric part.
Let Φ be a sufficiently smooth function of (x, t) ∈ R n × R t and define We look for the equation solved by f by computing e Φ(x,t)/2 (∂ t − ∆) e −Φ(x,t)/2 f (x, t) . We find that and furthermore, f |∂Ω = 0. Introduce We can check that Step 2.1.2. Energy estimates.
Multiplying by f the above equation, integrating over Ω, we obtain that Introduce the frequency function t → N (t) defined by Thus, 1 2 Now, we compute the derivative of N and claim that: In the third line, we used 1 2 d dt f 2 + −Sf, f = 0; In the fourth line, multiplying the equation of f by Sf , and integrating over Ω, give Sf, f ′ = Sf 2 + Sf, Af ; In the fifth line, ∂ ν denotes the normal derivative to the boundary, and we used In the sixth line, we used Cauchy-Schwarz inequality.
Here we have followed the energy estimates style of computations in [BT] (see also [Ph,p.535]) The interested reader may wish here to compare with [EKPV1, Theorem 3].

Assume that
∂Ω ∂ ν f Af dσ ≥ 0 on (0, T ) by convexity or star-shaped property of Ω, and suppose that where Υ (t) = T − t + and > 0. Therefore the following differential inequalities hold.
By solving such system of differential inequalities, we obtain (see [BP,p.655]): For any In other words, we have Step 2.1.4.
Let ω be a nonempty open subset of Ω. We take off the weight function Φ from the integrals: Using the fact that u (·, T ) ≤ u (·, t) ≤ u (·, 0) ∀0 < t < T , the above inequality becomes Step 2.1.5. Special weight function.
Finally, we choose > 0 such that This ends to the desired inequality.

C 2 bounded domain
For C 2 bounded domain Ω, we will use localized weight functions exploiting a covering argument and propagation of smallness.
It suffices to prove the following result to get the desired observation inequality at one point in time for the heat equation with Dirichlet boundary condition in a C 2 bounded domain Ω (see [PWZ,Lemma 4 and Lemma 5 at p.493]).
Lemma 2.1. There is ω 0 a nonempty open subset of B x 0 ,r and constants c, K > 0 and β ∈ (0, 1) such that for any T > 0 and u 0 ∈ L 2 (Ω), The strategy to establish the above Lemma 2.1 is as follows. It will be divided into seven steps.
Multiplying by f the above equation and integrating over Ω ∩ B x 0 ,R 0 , we find that Introduce the frequency function t → N (t) defined by Thus, 1 2 d dt f 2 0 + N f 2 0 = e Φ/2 g, f 0 . Now, we compute the derivative of N and claim that: In the third line, we used 1 2 d dt f 2 0 − Sf, f 0 = f, e Φ/2 g 0 ; In the fifth line, multiplying the equation of f by Sf , and integrating over Ω ∩ B x 0 ,R 0 , give Sf, f ′ 0 = Sf, Sf + Af + e Φ/2 g 0 = Sf 2 0 + Sf, Af 0 + Sf, e Φ/2 g 0 ; In the sixth line, we used Cauchy-Schwarz inequality. Finally, recall that Step 2.2.3. Assumption on Carleman commutator.

Assume that
where Υ (t) = T − t + and > 0. Therefore, the following differential inequalities hold.
. By solving such system of differential inequalities, we have: Then we solve Finally, combining the case t 1 ≤ t ≤ t 2 and the case t 2 ≤ t ≤ t 3 , we have which implies the desired inequality.
Step 2.2.4. The rest term.

Now we can estimate
, by regularizing effect, as follows.
Step 2.2.5. First assumption on the weight function.
Let ω 0 be a nonempty open subset of B x 0 ,r . Now by taking off the weight function from the integrals, we have that for any 0 < ≤ θmin C (ℓ,ϕ) , 1/ (2ℓ) , But, by Lemma 2.2, observe that Since u (·, T ) ≤ u (·, 0) , we can see that and conclude that Step 2.2.7. Second assumption on the weight function.
We construct ϕ (x) and choose ℓ > 1 sufficiently large in order that Consequently, there are C 1 > 0 and C 2 > 0 such that for any > 0 with ≤ θmin C (ℓ,ϕ) , 1/ (2ℓ) := θC 3 , On the other hand, for any ≥ θC 3 , 1 ≤ e C 2 C 3 1 θ e −C 2 1 . Therefore for any > 0, Finally, we choose > 0 such that As a consequence, we obtain that for some c > 0, But recall the definition of θ in Lemma 2.2 saying that Therefore, which gives for some K > 0, the following inequality and yields to the desired conclusion of Lemma 2.1.

The weight function
In the previous section, the observation estimate at one time was derived by using appropriate assumptions on the weight function Φ and by solving a system of differential inequalities. Now, our goal is to explore different explicit choices of weight function Φ.
The weight function Φ used in a series of results for the doubling property or frequency function for heat equations was based on the backward heat kernel (we also refer to [BP] for parabolic equations where the Euclidian distance is replaced by the geodesic distance). Precisely, It leads to the following differential inequalities (see [PWZ, Lemma 2 at p.487]): Define for z ∈ H 1 (0, T ; L 2 (Ω ∩ B x 0 ,R 0 )) ∩ L 2 (0, T ; H 2 ∩ H 1 0 (Ω ∩ B x 0 ,R 0 )) and t ∈ (0, T ], The following two properties hold. ii) When Ω ∩ B x 0 ,R 0 is star-shaped with respect to x 0 , The differential inequalities obtained with the Carleman commutator are given in Step 2.1.2 and Step 2.2.2 of the previous Section 2: The following two properties hold.
We will assume that ∂(Ω∩B x 0 ,R 0 ) ∂ ν f Af dσ ≥ 0 by the star-shaped property of Ω ∩ B x 0 ,R 0 . Now we focus our attention on the term − (S ′ + [S, A]) f, f 0 . We decompose our presentation into three parts.
We claim that: Proof of the claim .-First, S ′ f = ∂ t ηf . Next, we compute [S, A] f := SAf − ASf . Precisely, with standard summation notations, This implies that Therefore, we obtain that Furthermore, by one integration by parts we have Combining the above equalities yields the desired formula. Then the claim follows.
Example linked with the heat kernel .
by the star-shaped property of Ω ∩ B x 0 ,R 0 . Here and from now, Υ (t) := T − t + and − → ν is the outward unit normal vector to ∂ (Ω ∩ B x 0 ,R 0 ).

Part 3.2. A particular form of the weight function.
Assume that Φ (x, t) = ϕ (x) T − t + . Then, we can see that Indeed, Example of a weight function for localization with balls .-If by the star-shaped property of Ω∩B x 0 ,R 0 . One conclude that, with such weight function Φ, the assumptions of Step 2.2.3 of the previous Section 2 are satisfied and therefore Now we check the assumptions on ϕ (x) = − 1 4 |x − x 0 | 2 at Step 2.2.5 and Step 2.2.7 of the previous Section 2. We observe that by choosing ω 0 = B x 0 ,r ⋐ Ω with 0 < r < R and by taking ℓ > 1 sufficiently large.
Part 3.3. The weight function for localization with annulus.
Assume that ϕ (x) = −a |x − x 0 | 2 + b |x − x 0 | s − c for some a, b, c > 0 and 1 ≤ s < 2. We would like to check the assumptions of the previous Section 2 and find the adequate parameters a, b, c, s. First, we observe that the formula in the previous Part 3.2 We start to choose a = 1 4 . Next we treat the third line of the above formula by using Cauchy-Schwarz inequality, we find that In order that − (S ′ + [S, A]) f, f 0 − 1 Υ −Sf, f 0 ≤ 0, we can take n ≥ 3 and s = 1 with c ≤ (bs) 2 (2 − s) = b 2 . Another choice is n ≥ 3 and s = 4 3 which gives and finally, we can choose c 2 ≤ 1 6 4 3 b 3 and b 9 R 2/3 0 Finally, we check the assumptions on ϕ at Step 2.2.5 and Step 2.2.7 of the previous Section 2: We observe that 3 4 , 0 and by taking ℓ > 1 sufficiently large.

Proof of Theorem 1.1
The observability estimate in Theorem 1.1 can be deduced from the observation inequality at one time (see [PW2] or the following Lemma 4.1). It was noticed in [AEWZ] that the spectral inequality in Theorem 1.1 is a consequence of the observation inequality at one time (see Lemma A in Appendix (see page 33)).
Lemma 4.1. Let ω be a nonempty open subset of Ω. Let p ∈ [1, 2], γ > 0, β ∈ (0, 1), γ and c, K > 0. Suppose that for any u 0 ∈ L 2 (Ω) and any T > 0, Then for any u 0 ∈ L 2 (Ω) and any T > 0, one has The above lemma is somehow standard, but we still give the proof here to make a self-contained discussion. that is, Replacing m by 2m, we can see that We write A m = e − (1+1/γ)K (z−1) γ T γ z γ(2m+2) and choose ε = A m , in order to get Our task is to have To this end, we take z 2γ 1 β = 1 + 1 β . It remains to sum the telescoping series from m = 0 to +∞ to complete the proof of Lemma 4.1 and to find that With the help of Lemma 4.1 and the analysis done in Section 2 for a convex domain Ω ⊂ R n or a star-shaped domain with respect to x 0 ∈ Ω, we are ready to show Theorem 1.1. It suffices to prove the observation at one point of Lemma 4.1 with γ = 1 and p = 2.

Proof of Theorem 1.2
Let n ≥ 3 and consider a C 2 bounded domain Ω ⊂ R n such that 0 ∈ Ω, and let ω ⊂ Ω be a nonempty open set. To simplify the presentation, we assume that 0 / ∈ ω, that can always be done, taking if necessary a smaller set. Let R 0 = 4 3 3/2 ≈ 1.53. We also assume that the unit ball B 0,R 0 is included in Ω and B 0,R 0 ∩ ω is empty. This can always be done by a scaling argument.
We look for the equation solved by f by computing It gives Then, it holds where ·, · 0 is the usual scalar product in L 2 (B 0,R 0 ) and · 0 will denote the corresponding norm. Furthermore, we have Multiplying by f the above equation, integrating over B 0,R 0 , it follows that 1 2 Introduce the frequency function t → N (t) defined by Then, we have 1 2 d dt f 2 0 + N f 2 0 = e Φ/2 g, f 0 , and the derivative of N satisfies (see Step 2.2.2 in Section 2): Notice that the boundary terms have vanished since The estimate of e Φ/2 g 2 0 f 2 0 can be obtained in a similar way than in Step 2.2.4 and Step 2.2.5 of Section 2. Indeed, first, we check that as follows: where in the fifth line we used Hardy inequality. Therefore, we have which implies that Lemma 2.2 is still true for any u solution of the heat equation with an inverse square potential.
Next we can estimate e Φ/2 g 2 0 f 2 0 as follows.
as long as T /2 ≤ T − θ ≤ t ≤ T , where in the third line we used the regularizing effect of a gradient term for the solution u of the heat equation with an inverse square potential. Therefore, the conclusion of Step 2.2.5 still holds: Under the assumptions of Step 2.2.5, we have e Φ/2 g (·, t) The difficulty with the heat equation with an inverse square potential comes with Notice also that the treatment far from the point 0 ∈ Ω where the inverse square potential have its singularities can be done in the same way than for the heat equation with a potential in L ∞ (Ω × (0, T )) (see [PWZ]). Our main task is to treat the assumptions of Step 2.2.5 and Step 2.2.7 of Section 2, carefully with a suitable choice of Φ (see also Part 3.3 of Section 3).
onto H, that A −1 is a linear compact operator in H and that Av, v > 0 ∀v ∈ D(A), v = 0. Introduce the set {λ j } ∞ j=1 for the family of all eigenvalues of A so that 0 < λ 1 ≤ λ 2 ≤ ·· ≤ λ m ≤ λ m+1 ≤ · · · and lim j→∞ λ j = ∞ , and let {e j } ∞ j=1 be the family of the corresponding orthogonal normalized eigenfunctions: Ae j = λ j e j , e j ∈ D(A) and e j , e i = δ i,j .
Below, H := L 2 (Ω) where Ω is a bounded open set of R n .
Choosing 0 < ǫ < 1 and optimizing with respect to λT by taking Setting β = 1 − ǫ, we finally have the desired observation estimate at one time.