Carleman commutator approach in logarithmic convexity for parabolic equations

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Carleman commutator approach in logarithmic convexity for parabolic equations

Kim Dang Phung

To cite this version:

Kim Dang Phung. Carleman commutator approach in logarithmic convexity for parabolic equations.

2018. �hal-01710168�

Carleman commutator approach in

logarithmic convexity for parabolic equations

Kim Dang Phung^∗

Abstract

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, spectral inequality for the associated eigenvalue problem is derived.

1 Introduction and main results

When we mention the logarithmic convexity method for the heat equation in a bounded domain Ω⊂Rⁿ :





∂_tu−∆u= 0 , in Ω×(0, T) ,

u= 0 , on ∂Ω×(0, T) ,

u(·,0) =u0 ∈L²(Ω)\{0} ,

we have in mind that t 7→lnku(·, t)k²_L2(Ω) is a convex function by evaluating the sign of the derivative of t 7→

R

Ω|∇u(x, t)|²dx R

Ω|u(x, t)|²dx (see [AN], [Pa, p.11], [I, p.43], [Ve]). As a consequence, the following well-known estimate holds. For any 0≤t ≤T,

e^t∆u0

L²(Ω) ≤ e^T∆u0

^t/T

L²(Ω)ku0k¹_L⁻2(Ω)^t/T .

In a series of articles (see [PW1], [PW2], [PWZ], [BP] for parabolic equations) in- spired by [Po] and [EFV], we were interested on the function t7→R

Ω|u(x, t)|²e^Φ(x,t)dx and its frequency functiont 7→

R

Ω|∇u(x, t)|²e^Φ(x,t)dx R

Ω|u(x, t)|²e^Φ(x,t)dx whene^Φ(x,t) = ¹

(T−t+~)^n/2e^{− |x−x0|}

2 4(T−t+~)

∗Universit´e d’Orl´eans, Laboratoire MAPMO, CNRS UMR 7349, F´ed´eration Denis Poisson, FR CNRS 2964, Bˆatiment de Math´ematiques, B.P. 6759, 45067 Orl´eans Cedex 2, France. E-mail address:

kim dang phung@yahoo.fr.

with x₀ ∈ Ω, ~>0. It provides us with an observation estimate at one point in time:

For any T >0 and any ω nonempty open subset of Ω, e^T∆u0

L²(Ω) ≤ ce^KT

e^T∆u0

L²(ω)

β

ku0k¹_L⁻2(Ω)^β .

Here c, K > 0 and β ∈ (0,1). From the above observation at one time, many ap- plications 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 Dirich- let 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^12Φ(x,t) for a larger set of weight functions Φ (x, t) and establish a kind of convexity property for t 7→lnkf(·, t)k²_L2(Ω). By such approach we make appear the Carleman commutator.

The link between logarithmic convexity (or frequency function) and Carleman inequal- ity has already appeared in [EKPV1] (see also [EKPV2], [EKPV3]).

Choosing suitable weight functions Φ (not necessary linked to the heat kernel) we obtain the following new results:

Theorem 1.1. Suppose that Ω⊂Rⁿ is a convex domain or a star-shaped domain with respect to x0 ∈ Ω such that {x;|x−x0|< r} ⋐ Ω for some r ∈ (0,1). Then for any u₀ ∈L²(Ω),T >0, (a_j)_j≥1 ∈R, λ >0, ε∈(0,1), one has

e^T∆u0

L²(Ω)≤Kεe^{Kεrε T1} Z T

0

e^t∆u0

L²(|x−x0|<r)dt and

X

λj≤λ

|aj|² ≤4e^Kεrε√λ Z

|x−x0|<r

X

λj≤λ

ajej(x)

2

dx

where Kε>0is a constant only depending on ε,max

|x−x0|;x∈Ω . Here(λj, ej) denotes the eigenbasis of the Laplace operator with Dirichlet boundary condition.

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 Kε does not depend on the dimension n (see [BP, Theorem 4.2]).

Theorem 1.2. Let n≥3 and consider aC² bounded domain Ω⊂Rⁿ such that 0∈Ω.

Let ω ⊂Ω be a nonempty open set. Suppose that µ≤

7

2·3³ , if n= 3 ,

1

4 (n−1) (n−3) , if n≥4 .

Then, there exist constants c >0, K >0such that for any (aj)_j≥1 ∈R and any λ >0, we have

s X

λj≤λ

|aj|² ≤ce^K√λ Z

ω

X

λj≤λ

ajej(x) dx

where(λj, ej)denotes the eigenbasis of the Schr¨odinger operator−∆− ^µ

|x|² with Dirichlet boundary condition

−∆ej − ^µ

|x|²ej =λjej , in Ω , e_j = 0 , on ∂Ω .

Theorem 1.2 gives a spectral inequality for the Schr¨odinger operator−∆− ^µ

|x|² under a quite strong assumption on µ < µ^∗ where the critical coefficient is µ^∗ = ¹₄(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 pre- sented 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 byf =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.

2 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 ⊂Rⁿwith 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

C² 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).

2.1 Convex domain

Throughout this subsection, we assume that Ω ⊂ Rⁿ is a convex domain or a star- shaped domain with respect to x0 ∈ Ω. Let h·,·i denote the usual scalar product in L²(Ω) and let k·k be its corresponding norm. Here, recall that u(x, t) = e^t∆u0(x) ∈ C([0, T] ;L²(Ω))∩C((0, T] ;H₀¹(Ω)) and we aim to check that

ku(·, T)k ≤

ce^KT ku(·, T)k_L2(ω)

β

ku(·,0)k^1−β .

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)∈Rⁿ×R_t and define f(x, t) =u(x, t)e^Φ(x,t)/2 .

We look for the equation solved by f by computinge^Φ(x,t)/2(∂t−∆) e^−Φ(x,t)/2f(x, t) . We find that

∂tf −∆f − 1 2f

∂tΦ + 1 2|∇Φ|²

+∇Φ· ∇f +1

2∆Φf = 0 in Ω×(0, T) , and furthermore, f_|∂Ω = 0. Introduce

Af =−∇Φ· ∇f −¹₂∆Φf ,

Sf = ∆f +ηf where η= ¹₂ ∂tΦ + ¹₂|∇Φ|² . We can check that

hAf, gi=− hAg, fi ,

hSf, gi=hSg, fi for any g ∈H₀¹(Ω) . Furthermore, we have

∂tf− Sf − Af = 0 . Step 2.1.2. Energy estimates.

Multiplying by f the above equation, integrating over Ω, we obtain that 1

2 d

dtkfk²+h−Sf, fi= 0 .

Introduce the frequency function t 7→N(t) defined by N= h−Sf, fi

kfk² . Thus,

1 2

d

dtkfk²+Nkfk² = 0 . Now, we compute the derivative of N and claim that:

d

dtN≤ 1

kfk² h−(S^′+ [S,A])f, fi − 1 kfk²

Z

∂Ω

∂νfAf dσ . Indeed,

d dtN

= 1

kfk⁴ d

dth−Sf, fi kfk²+hSf, fi d dtkfk²

= 1

kfk² [h−S^′f, fi −2hSf, f^′i] + 2

kfk⁴ hSf, fi²

= 1

kfk² [h−S^′f, fi −2hSf,Afi] + 2 kfk⁴

− kSfk²kfk²+hSf, fi²

= 1

kfk²

h−(S^′+ [S,A])f, fi − Z

∂Ω

∂νfAf dσ

+ 2

kfk⁴

− kSfk²kfk² +hSf, fi²

≤ 1 kfk²

h−(S^′ + [S,A])f, fi − Z

∂Ω

∂νfAf dσ

.

In the third line, we used ¹_2dt^d kfk²+h−Sf, fi= 0; In the fourth line, multiplying the equation of f by Sf, and integrating over Ω, give hSf, f^′i=kSfk²+hSf,Afi; In the fifth line, ∂ν denotes the normal derivative to the boundary, and we used

2hSf,Afi =hSAf, fi − hASf, fi+ Z

∂Ω

∂_νfAf dσ

:=h[S,A]f, fi+ Z

∂Ω

∂νfAf dσ ; 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].

Step 2.1.3. Assumption on Carleman commutator.

Assume that Z

∂Ω

∂νfAf dσ≥0 on (0, T) by convexity or star-shaped property of Ω, and suppose that

h−(S^′ + [S,A])f, fi ≤ 1

Υh−Sf, fi

on (0, T) where Υ (t) = T − t+ ~ and ~ > 0. Therefore the following differential inequalities hold. 



 1 2

d

dtkf(·, t)k²+N(t)kf(·, t)k² = 0 , d

dtN(t)≤ 1

Υ (t)N(t) .

By solving such system of differential inequalities, we obtain (see [BP, p.655]): For any 0< t1 < t2 < t3 ≤T,

kf(·, t2)k²1+M

≤ kf(·, t1)k²M

kf(·, t3)k² where

M = −ln (T −t3+~) + ln (T −t2+~)

−ln (T −t2+~) + ln (T −t1+~) . In other words, we have

Z

Ω

|u(x, t2)|²e^Φ(x,t2)dx 1+M

≤ Z

Ω

|u(x, t1)|²e^Φ(x,t1)dx MZ

Ω

|u(x, t3)|²e^Φ(x,t3)dx .

Step 2.1.4.

Letω be a nonempty open subset of Ω. We take off the weight function Φ from the integrals:

Z

Ω

|u(x, t2)|²dx 1+M

≤exp

−(1 +M) min

x∈Ω

Φ (x, t2) +Mmax

x∈Ω

Φ (x, t1)

× Z

Ω

|u(x, t1)|²dx MZ

Ω

|u(x, t3)|²e^Φ(x,t3)dx and Z

Ω

|u(x, t₃)|²e^Φ(x,t3)dx = Z

ω

|u(x, t₃)|²e^Φ(x,t3)dx+ Z

Ω\ω

|u(x, t₃)|²e^Φ(x,t3)dx

≤exph

maxx∈ωΦ (x, t3)i Z

ω

|u(x, t3)|²dx +exp

"

max

x∈Ω\ω

Φ (x, t₃)

# Z

Ω

|u(x, t₃)|²dx. Therefore, we obtain that

Z

Ω

|u(x, t2)|²dx 1+M

≤exp

−(1 +M) min

x∈ΩΦ (x, t2) +Mmax

x∈ΩΦ (x, t1) + max

x∈ωΦ (x, t3)

× Z

Ω

|u(x, t₁)|²dx MZ

ω

|u(x, t₃)|²dx +exp

"

−(1 +M) min

x∈ΩΦ (x, t2) +Mmax

x∈ΩΦ (x, t1) + max

x∈Ω\ω

Φ (x, t3)

#

× Z

Ω

|u(x, t1)|²dx MZ

Ω

|u(x, t3)|²dx .

Using the fact that ku(·, T)k ≤ ku(·, t)k ≤ ku(·,0)k ∀0 < t < T, the above inequality becomes

ku(·, T)k^21+M

≤exp

−(1 +M) min

x∈ΩΦ (x, t2) +Mmax

x∈ΩΦ (x, t1) + max

x∈ωΦ (x, t3)

× ku(·,0)k²MZ

ω

|u(x, t3)|²dx +exp

"

−(1 +M) min

x∈ΩΦ (x, t2) +Mmax

x∈ΩΦ (x, t1) + max

x∈Ω\ω

Φ (x, t3)

#

× ku(·,0)k²1+M

. Step 2.1.5. Special weight function.

Assume that Φ (x, t) = ϕ(x)

T −t+~, we get that ku(·, T)k^1+M ≤exp¹₂

−_T^1+M_−t

2+~min

x∈Ωϕ(x) + _T−^M_t

1+~max

x∈Ωϕ(x) + _T−¹_t

3+~max

x∈ωϕ(x)

× ku(·,0)k^M ku(·, t3)k_L2(ω)

+exp¹₂

"

−_T^1+M_−t2+~min

x∈Ωϕ(x) + _T−^M_t1+~max

x∈Ωϕ(x) + _T−t¹₃₊~ max

x∈Ω\ω

ϕ(x)

#

× ku(·,0)k^1+M .

Choose t3 =T, t2 =T −ℓ~,t1 =T −2ℓ~ with 0<2ℓ~< T and ℓ >1, and denote Mℓ = ln (ℓ+ 1)

ln ^2ℓ+1_ℓ+1 . Therefore, we have

ku(·, T)k^1+Mℓ ≤exp₂¹_~

−^1+M_1+ℓ^ℓmin

x∈Ω

ϕ(x) + _1+2ℓ^Mℓ max

x∈Ω

ϕ(x) + max

x∈ωϕ(x)

× ku(·,0)k^Mℓku(·, T)k_L2(ω)

+exp₂¹~

"

−^1+M_1+ℓ^ℓmin

x∈Ωϕ(x) + _1+2ℓ^Mℓ max

x∈Ωϕ(x) + max

x∈Ω\ω

ϕ(x)

#

× ku(·,0)k^1+Mℓ . Step 2.1.6. Assumption on weight function.

We construct ϕ(x) and choose ℓ >1 sufficiently large in order that

"

−1 +M_ℓ 1 +ℓ min

x∈Ω

ϕ(x) + M_ℓ 1 + 2ℓmax

x∈Ω

ϕ(x) + max

x∈Ω\ω

ϕ(x)

#

<0 .

Consequently, there are C1 >0 and C2 >0 such that for any ~>0 with 0<2ℓ~< T, ku(·, T)k^1+Mℓ ≤e^C11~ku(·,0)k^Mℓku(·, T)k_L2(ω)+e^−C21~ku(·,0)k^1+Mℓ .

Notice that ku(·, T)k ≤ ku(·,0)k and for any 2ℓ~ ≥ T, 1 ≤ e^C22ℓTe^−C21~. We deduce that for any ~>0,

ku(·, T)k^1+Mℓ ≤e^C1~1ku(·,0)k^Mℓku(·, T)k_L2(ω)+e^C22ℓTe^−C21~ku(·,0)k^1+Mℓ . Finally, we choose ~>0 such that

e^C22ℓTe^−C21~ku(·,0)k^1+Mℓ = 1

2ku(·, T)k^1+Mℓ , that is,

e^C21~ := 2e^C22ℓT

ku(·,0)k ku(·, T)k

1+Mℓ

in order that

ku(·, T)k^1+Mℓ ≤2 2e^C22ℓT

ku(·,0)k ku(·, T)k

1+Mℓ!^C_C¹

2

ku(·,0)k^Mℓku(·, T)k_L2(ω) , that is,

ku(·, T)k ≤2^1+CC12e^C12ℓT

ku(·,0)k ku(·, T)k

Mℓ+(1+Mℓ)^C_C¹

2 ku(·, T)k_L2(ω) . This ends to the desired inequality.

2.2 C² bounded domain

ForC² bounded domain Ω, we will use localized weight functions exploiting a covering argument and propagation of smallness.

Let 0 < r < R, x0 ∈ Ω and δ ∈ (0,1]. Denote R0 := (1 + 2δ)R and Bx0,r :=

{x;|x−x0|< r}. Assume that Bx0,r ⋐ Ω and Ω∩Bx0,R0 is star-shaped with respect to x₀. Let h·,·i₀ denote the usual scalar product in L²(Ω∩B_x0,R0) and let k·k₀ be its corresponding norm.

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 C² bounded domain Ω (see [PWZ, Lemma 4 and Lemma 5 at p.493]).

Lemma 2.1. There is ω0 a nonempty open subset ofBx0,r and constants c, K >0 and β ∈(0,1)such that for any T >0 and u₀ ∈L²(Ω),

e^T∆u₀

L²(^Ω∩Bx0,R) ≤

ce^KT e^T∆u₀

L²(ω0)

β

ku₀k^1−β .

The strategy to establish the above Lemma 2.1 is as follows. It will be divided into seven steps.

Step 2.2.1. Localization, symmetric and antisymmetric parts.

Let χ∈ C₀^∞(Bx0,R0), 0 ≤χ ≤ 1, χ = 1 on {x;|x−x0| ≤(1 + 3δ/2)R}. Introduce z =χu. It solves

∂tz−∆z =g :=−2∇χ· ∇u−∆χu ,

and furthermore, z|^∂(^Ω∩Bx0,R0) = 0. Let Φ be a sufficiently smooth function of (x, t) ∈ Rⁿ×R_t depending onx0. Set

f(x, t) =z(x, t)e^Φ(x,t)/2 .

We look for the equation solved by f by computinge^Φ(x,t)/2(∂t−∆) e^−Φ(x,t)/2f(x, t) . It gives

∂tf−∆f−1 2f

∂tΦ + 1 2|∇Φ|²

+∇Φ· ∇f+1

2∆Φf =e^Φ/2g in (Ω∩Bx0,R0)×(0, T) , and furthermore, f|^∂(^Ω∩Bx0,R0) = 0. Introduce

Af =−∇Φ· ∇f −¹₂∆Φf ,

Sf = ∆f +ηf where η= ¹₂ ∂tΦ + ¹₂|∇Φ|² .

It holds

hAf, vi₀ =− hAv, fi₀ ,

hSf, vi₀ =hSv, fi₀ for any v ∈H₀¹(Ω∩Bx0,R0) . Furthermore, one has

∂_tf − Sf− Af =e^Φ/2g . Step 2.2.2. Energy estimates.

Multiplying by f the above equation and integrating over Ω∩Bx0,R0, we find that 1

2 d

dtkfk²₀+h−Sf, fi₀ =

f, e^Φ/2g

0 . Introduce the frequency function t 7→N(t) defined by

N= h−Sf, fi₀ kfk²₀ . Thus,

1 2

d

dtkfk²₀+Nkfk²₀ =

e^Φ/2g, f

0 . Now, we compute the derivative of N and claim that:

d

dtN≤ 1

kfk²₀ h−(S^′+ [S,A])f, fi₀− 1 kfk²₀

Z

∂(^Ω∩Bx0,R0)∂νfAf dσ+ 1 2kfk²₀

e^Φ/2g ²

0 .

Indeed, d dtN

= 1

kfk⁴₀

−d

dthSf, fi₀kfk²₀ +hSf, fi₀ d dtkfk²₀

= −1

kfk²₀ [hS^′f, fi₀+ 2hSf, f^′i₀] + 2 kfk⁴₀

hSf, fi²₀+hSf, fi₀

f, e^Φ/2g

0

= −1

kfk²₀ [hS^′f, fi₀+ 2hSf, f^′i₀] + 2 kfk⁴₀

"

hSf, fi₀+ 1 2

f, e^Φ/2g

0

2

− 1 2

f, e^Φ/2g

0

2#

= −1

kfk²₀ [hS^′f, fi₀+ 2hSf,Afi₀] + −2 kfk²₀

kSfk²₀+

Sf, e^Φ/2g

0

+ 2 kfk⁴₀

"

Sf +1

2e^Φ/2g, f

0

2

− 1 2

f, e^Φ/2g

0

2#

≤ −1

kfk²₀ [hS^′f, fi₀+ 2hSf,Afi₀] + −2 kfk²₀

kSfk²₀+

Sf, e^Φ/2g

0

+ 2 kfk⁴₀

Sf +1 2e^Φ/2g

2 0

kfk²₀

= −1

kfk²₀ [hS^′f, fi₀+ 2hSf,Afi₀] + 2 kfk²₀

1

2e^Φ/2g

2 0

. In the third line, we used ¹_2dt^d kfk²₀− hSf, fi₀ =

f, e^Φ/2g

0; In the fifth line, multiplying the equation of f by Sf, and integrating over Ω∩Bx0,R0, give

hSf, f^′i₀ =

Sf, Sf +Af+e^Φ/2g

0

=kSfk²₀+hSf,Afi₀ +

Sf, e^Φ/2g

0 ; In the sixth line, we used Cauchy-Schwarz inequality. Finally, recall that

2hSf,Afi₀ :=h[S,A]f, fi₀+ Z

∂(^Ω∩B_x0,R0)∂νfAf dσ . Step 2.2.3. Assumption on Carleman commutator.

Assume that Z

∂(^Ω∩Bx0,R0)∂νfAf dσ ≥ 0 on (0, T) by the star-shaped property of Ω∩Bx0,R0, and suppose that

h−(S^′+ [S,A])f, fi₀ ≤ 1

Υh−Sf, fi₀

on (0, T) where Υ (t) = T − t+~ and ~ > 0. Therefore, the following differential inequalities hold.









 1

2 d

dtkf(·, t)k²₀+N(t)kf(·, t)k²₀ ≤

e^Φ/2g(·, t)

0kf(·, t)k₀ , d

dtN(t)≤ 1

Υ (t)N(t) +

e^Φ/2g(·, t) ²

0

kf(·, t)k²₀ .