Estimate in the region G

Here, we place ourselves in the region G, and prove a Carleman estimate for uG,j, and conse- quently for ΞGv.

We introduce a microlocal cut-off functionχG F ∈C∞

c (M^∗+), 0≤χG F ≤1, satisfying χG F = 1 on a neighborhood of supp(χG),

χ^G+χ^F = 1 on a neighborhood of supp(χ^{G F}). (3.75) We choose ζ² ∈C_c^∞(0, X0) such that 0≤ζ²≤1,ζ² = 1 on a neighborhood of supp(ζ¹) (withζ¹ defined in (3.61)), and such that ˜ζj = 1 on supp( φ⁻_j¹_∗

ζ_j²) where ζ_j²(x0, y) =ζ²(x0)ψj(y). As in (3.63) we set

χG F,j = ˜ζj φ⁻_j¹_∗ χG F,

and we define the associated tangential pseudo-differential operator ΞG F by ΞG F = P

j∈J

ΞG F,j, with ΞG F,j =φ^∗_jOp_T(χG F,j) φ⁻_j¹∗

ζ_j², j∈J,

Note that the local symbol (see Proposition 3.45) of ΞG F in each chart is equal to one in the support of that of ΞG.

We recall that the functionζ=ζ(xn)∈C∞

c ([0,2ε)) satisfiesζ(0) = 1 on [0, ε).

Making use of the Calder´on projector technique forP_ϕ,j^r and of the standard Carleman techniques forP_ϕ,j^l , we obtain the following partial estimate.

Proposition 3.28. Suppose that the weight functionϕsatisfies the properties listed in Section 3.3.1.

Then, for all δ0 >0, there exist C >0 andh0 >0 such that, for all0 < δ ≤δ0 and0 < h≤h0, v^r/l ∈C_c^∞((0, X0)×S×[0,2ε))andv^s∈C_c^∞((0, X0)×S)satisfying (3.47), we have

kΞGv^rk²1+h|ΞGv^r|xn=0⁺|²1+h|DxnΞGv^r|xn=0⁺|²0≤C

kP_ϕ^rv^rk²0+h²kv^rk²1+h⁴|Dxnv^r|xn=0⁺|²0

, (3.76)

3.4. Proof of the Carleman estimate in a neighborhood of the interface 121 and

hkΞGv^lk²1+h|ΞGv^l|^xn=0⁺|²1+h|DxnΞGv^l|^xn=0⁺|²0

≤C 1 + δ² h²

kζP_ϕ^rv^rk²0+h²kΞ^{G F}v^rk²1+h⁴|Dxnv^r|xn=0⁺|²0+h⁴kv^rk²1+h³|v^s|²1

+C

kP_ϕ^lv^lk²0+h²kv^lk²1+h|θ_ϕ^l|²1+δ²

h|θ^r_ϕ|²0+h|θ_ϕ^r|²1+h|Θ^s_ϕ|²0

. (3.77)

Proof. The function uG,j, defined in (3.64), satisfies (TC_•,j), with • = G. On the “r” side, the root configuration described in Lemma 3.24 (and represented in Figure 3.3) allows us to apply the Calder´on projector technique used in [LR97, LR10]. According to [LR10, Remark 2.5] and using Eqs. (2.59), (2.60), and (2.61) therein, applied withv^d replaced here byv_j^r, we have

ku^rG,jk¹+h¹²|γ0(u^rG,j)|¹+h¹²|γ1(u^rG,j)|⁰.kP_ϕ,j^r v^r_jk⁰+hkv_j^rk¹+h²|Dxnv_j^r|xn=0⁺|⁰, (3.78) which is a local version of (3.76).

Let us now explain how such local estimates can be patched together to yield (3.76). Concerning the first term on the left hand-side of (3.78), and using the definition of Sobolev norms given in (3.20)–(3.22), we have

kΞGv^rk¹. P

j∈Jku^rG,jk¹, |ΞGv^r|xn=0⁺|¹. P

j∈J|γ0(u^r_G,j)|¹ (3.79) by (3.65) and Lemma 3.53. Similarly we have DxnΞGv^r|^xn=0⁺ =P

jφ^∗_jγ1(u^rG,j) since φ^∗_j does not depend on the xn-variable. As a consequence, we obtain

|(DxnΞ^Gv^r)|xn=0⁺|⁰≤ P

j∈J|φ^∗_jγ1(u^rG,j)|⁰. P

j∈J|γ1(u^rG,j)|⁰, (3.80) by Lemma 3.9.

Now concerning the right hand-side of (3.78), we directly have kv_j^rk¹=k φ⁻_j¹_∗

ζjv^rk¹=kζ¹ φ⁻_j¹_∗

ψjv^rk¹.k φ⁻_j¹_∗

ψjv^rk¹.kv^rk¹, (3.81) by the definition ofk.k¹ onM⁺, as well as

|Dxnv^r_j|^xn=0⁺|⁰.|Dxnv^r|^xn=0⁺|⁰. (3.82) Finally, we computeP_ϕ,j^r v^r_j = φ⁻_j¹∗

P_ϕ^rφ^∗_j φ⁻_j¹∗

ζjv^r= φ⁻_j¹∗

ζjP_ϕ^rv^r+ φ⁻_j¹∗

[P_ϕ^r, ζj]v^r. We have k φ⁻_j¹_∗

ζjP_ϕ^rv^rk⁰=kζ¹ φ⁻_j¹_∗

ψjP_ϕ^rv^rk⁰≤ kP_ϕ^rv^rk⁰, (3.83) and, using Lemma 3.9,

k φ⁻_j¹_∗

[P_ϕ^r, ζj]v^rk⁰.k[P_ϕ^r, ζj]v^rk⁰.hkv^rk¹, (3.84) since [P_ϕ^r, ζj]∈hD¹(M⁺). Finally combining all the estimates (3.79)-(3.84), together with the local inequalities (3.78) summed overj∈J, we obtain the sought global estimate (3.76) onM⁺.

To obtain Estimate (3.77) on the “l” side we first need a more precise estimate for the “r” side. For this, we introduce another microlocal cut-off function ˜χG F satisfying the same requirements (3.75) asχG F, and such thatχG F = 1 on a neighborhood of supp( ˜χG F). We chooseζ³∈C∞

c (0, X0) such that 0≤ζ³≤1,ζ³= 1 on a neighborhood of supp(ζ¹), and such thatζ²= 1 on a neighborhood of supp(ζ³). As in (3.63) we set

˜

χG F,j = ˜ζj φ⁻_j¹∗

˜ χG F,

and we define the associated tangential pseudo-differential operator ˜ΞG F by Ξ˜G F = P

j∈J

Ξ˜G F,j, with Ξ˜G F,j =φ^∗_jOp_T( ˜χG F,j) φ⁻_j¹∗

ζ_j³, ζ_j³=ζ³ψj, j∈J, According to [LR10, Remark 2.5] and using (2.60) and (2.61) therein, applied withv^d replaced by ζ(xn) φ⁻_j¹∗

ζjΞ˜G Fv^r, we have h¹²|γ0(Op_T(χG,j) φ⁻_j¹∗

ζjΞ˜G Fv^r)|¹+h¹²|γ1(Op_T(χG,j) φ⁻_j¹∗

ζjΞ˜G Fv^r)|⁰ .kP_ϕ,j^r ζ φ⁻_j¹∗

ζjΞ˜G Fv^rk⁰+hkζ φ⁻_j¹∗

ζjΞ˜G Fv^rk¹+h²|γ1 φ⁻_j¹∗

ζjΞ˜G Fv^r

|⁰. (3.85) We notice that the right hand-side of this inequality can directly be bounded by global quantities.

First, we have

kζ φ⁻_j¹∗

ζjΞ˜G Fv^rk¹.kΞ˜G Fv^rk¹ (3.86) Second, we estimate

γ1 φ⁻_j¹∗

ζjΞ˜G Fv^r₀≤ | DxnΞ˜G Fv^r

|^xn=0⁺|⁰, where

DxnΞ˜G Fv^r

|^xn=0⁺ = ˜ΞG FDxnv^r

|^xn=0⁺+ [Dxn,Ξ˜G F]

| {z }

∈hΨ⁰_T(M+)

v^r

|^xn=0⁺.

Using Proposition 3.50 and the trace formula (3.23), we have the estimate h²|γ1 φ⁻_j¹∗

ζjΞ˜^{G F}v^r

|⁰.h²|Dxnv^r|xn=0⁺|⁰+h⁵²kv^rk¹. (3.87) Concerning the term with P_ϕ,j^r in the right hand-side of (3.85), we can proceed as in (3.83)-(3.84) to obtain

kP_ϕ,j^r ζ φ⁻_j¹∗

ζjΞ˜^{G F}v^rk⁰=k φ⁻_j¹∗

P_ϕ^rζζjΞ˜^{G F}v^rk⁰.kP_ϕ^rζΞ˜^{G F}v^rk⁰+hkΞ˜^{G F}v^rk¹ (3.88) Moreover, using Proposition 3.48, we have ˜ΞG F(1−ΞG F)∈h^∞Ψ^−∞_T (M⁺), as their local symbols in every chart have disjoint supports by Proposition 3.52, because of the supports ofζ³ and ˜χG F. We then obtain with Proposition 3.50

hkΞ˜G Fv^rk¹.hkΞ˜G FΞG F v^rk¹+hkΞ˜G F(1−ΞG F)v^rk¹.hkΞG Fv^rk¹+h²kv^rk¹. (3.89) We also have

kP_ϕ^rζΞ˜G Fv^rk⁰.kΞ˜G FζP_ϕ^rv^rk⁰+k[P_ϕ^r,Ξ˜G Fζ]v^rk⁰. (3.90) Arguing as above with Propositions 3.48 and 3.52, and also Corollary 3.49, we have

[P_ϕ^r,Ξ˜G Fζ] = [P_ϕ^r,Ξ˜G Fζ]

| {z }

∈hΨ¹(M+)

ΞG F + [P_ϕ^r,Ξ˜G Fζ](1−ΞG F)

| {z }

∈h^∞Ψ^−∞(M+)

so that (3.90) now reads with Proposition 3.50

kP_ϕ^rζΞ˜G Fv^rk⁰.kζP_ϕ^rv^rk⁰+hkΞG Fv^rk¹+h²kv^rk¹. (3.91) The three estimates (3.88), (3.89) and (3.91) give

kP_ϕ,j^r ζ φ⁻_j¹∗

ζjΞ˜G Fv^rk⁰.kζP_ϕ^rv^rk⁰+hkΞG Fv^rk¹+h²kv^rk¹. (3.92) Combining (3.85) together with (3.86)–(3.87), (3.89) and (3.92), we finally have

h¹²|γ0(Op_T(χG,j) φ⁻_j¹∗

ζjΞ˜G Fv^r)|¹+h¹²|γ1(Op_T(χG,j) φ⁻_j¹∗

ζjΞ˜G Fv^r)|⁰

.kζP_ϕ^rv^rk⁰+hkΞG Fv^rk¹+h²|Dxnv^r|^xn=0⁺|⁰+h²kv^rk¹. (3.93) Then, we need the following lemma to come back to the variableu^rG,j = Op_T(χG,j) φ⁻_j¹_∗

ζjv^r on the left hand-side of (3.93).

3.4. Proof of the Carleman estimate in a neighborhood of the interface 123 Lemma 3.29. There existsR∈h^∞Ψ^−∞_T (M⁺), such that

Op_T(χG,j) φ⁻_j¹_∗

ζjΞ˜G Fv^r=u^rG,j+ φ⁻_j¹_∗ Rv^r. This lemma is proven in Appendix 3.8.6. As a consequence we have

h¹²|γ0(u^rG,j)|¹.h¹²|γ0(Op_T(χG,j) φ⁻_j¹∗

ζjΞ˜G Fv^r)|¹+h¹²|γ0( φ⁻_j¹∗

Rv^r)|⁰ .h¹²|γ0(Op_T(χG,j) φ⁻_j¹∗

ζjΞ˜G Fv^r)|¹+h²kv^rk¹ with the trace formula (3.23). This, together with Estimate (3.93) give

h¹²|γ0(u^rG,j)|¹.kζP_ϕ^rv^rk⁰+hkΞG Fv^rk¹+h²kv^rk¹+h²|Dxnv^r|xn=0⁺|⁰. (3.94) Lemma 3.29 also yields

h¹²|γ1(u^rG,j)|⁰.h¹²|γ1(Op_T(χG,j) φ⁻_j¹∗

ψjΞ˜G Fv^r)|⁰+h¹²|γ1( φ⁻_j¹∗

Rv^r)|⁰, .h¹²|γ1(Op_T(χG,j) φ⁻_j¹_∗

ψjΞ˜G Fv^r)|⁰+h²kv^rk¹+h²|Dxnv^r|^xn=0⁺|⁰, (3.95) Combining (3.93) together with (3.95), we finally obtain

h¹²|γ1(u^rG,j)|⁰.kζP_ϕ^rv^rk⁰+hkΞG Fv^rk¹+h²kv^rk¹+h²|Dxnv^r|xn=0⁺|⁰. (3.96)

On the “l” side, we apply the Carleman method. With the properties of the weight function of Section 3.3.1 and in particular by (3.56), and by Lemma 2 in [LR95], we then have

hku^lG,jk²1+Re

hB^l(u^lG,j) +h² (Dnu^lG,j+L^l₁u^lG,j)|xn=0⁺, L^l₀u^lG,j|xn=0⁺

0

.kPϕ,ju^lG,jk²0, (3.97) for 0< h≤h0, h0 sufficiently small, whereL^l₁∈D¹

T,L^l₀∈Ψ⁰_T. The quadratic formB^l is given by B^l(ψ) =

2∂xnϕ^l_j|xn=0⁺ B^l₁ B₁^l′ B^l₂

! γ1(ψ) γ0(ψ)

, γ1(ψ)

γ0(ψ)

!

0

, supp(ψ)⊂(0, X0)×U˜j×[0,2ε), (3.98) whereB₁^l,B₁^l′ ∈D¹

T, with principal symbolsσ(B₁^l) =σ(B^l₁^′) = 2 φ⁻_j¹∗

q^l₁|xn=0⁺andB₂^l ∈D²

T, with σ(B^l₂) =−2∂xnϕ^l_j φ⁻_j¹_∗

q₂^l|^xn=0⁺. Observe that we have

(Dnu^lG,j+L^l₁u^lG,j)|^xn=0⁺, L^l₀u^lG,j|^xn=0⁺

0

.|γ1(u^lG,j)|²0+|γ0(u^lG,j)|²1. (3.99) and

|B^l(u^lG,j)|.|γ0(u^lG,j)|²1+|γ1(u^lG,j)|²0. (3.100) Now, using (3.97), together with the estimates (3.99) and (3.100), we have,

hku^l_G,jk²1.kP_ϕ,j^l u^l_G,jk²0+h|γ0(u^l_G,j)|²1+h|γ1(u^l_G,j)|²0. (3.101) It remains to estimate the traces on the “l” side by the traces on the “r” side, through the transmission conditions (TC_•,j) :











γ0(u^lG,j) =γ0(u^rG,j) +θG^l,j−θ^rG,j

γ1(u^lG,j) = δc^s_j

h i c^l_jP_ϕ,j^s γ0(u^rG,j)−θ^rG,j

−βγ1(u^rG,j) +kγ0(u^rG,j)−G˜1, u^sG,j =γ0(u^rG,j)−θG^r,j.

As a consequence,γ0(u^l_G,j) andγ1(u^l_G,j) can be estimated as follows





|γ0(u^lG,j)|¹≤ |γ0(u^rG,j)|¹+|θ^lG,j|¹+|θG^r,j|¹,

|γ1(u^l_G,j)|⁰.|γ1(u^r_G,j)|⁰+δ

h|P_ϕ,j^s γ0(u^r_G,j)|⁰+ δ

h|P_ϕ,j^s θ^r_G,j|⁰+|γ0(u^rG,j)|⁰+|G˜1|⁰. (3.102) We now prove that, on the support of χG,j, the operator P_ϕ,j^s is of order 0. For this, let ˜χ ∈ C∞

c (T^∗(Rⁿ)), be equal to one on a neighborhood of the supp(χG,j|^xn=0⁺). We then have γ0(u^rG,j) = Op_T(χG,j)v_j^r|^xn=0⁺= Op_T( ˜χ) Op_T(χG,j)v^r_j|^xn=0⁺+ Op_T(1−χ) Op˜ _T(χG,j)

| {z }

∈h^∞Ψ^−∞_T

v_j^r|xn=0⁺,

which yields

P_ϕ,j^s γ0(u^rG,j) = P_ϕ,j^s Op_T( ˜χ)

| {z }

∈Ψ⁰_T

γ0(u^rG,j) +P_ϕ,j^s Op_T(1−χ) Op˜ _T(χG,j)

| {z }

∈h^∞Ψ^−∞_T

v_j^r|^xn=0⁺.

This, together with the trace formula (3.23) gives the estimate, δ

h|P_ϕ,j^s γ0(u^rG,j)|⁰≤Cδ

h|γ0(u^rG,j)|⁰+CNδh^Nkv_j^rk¹, N ∈N.

Similarly, we have the estimate δ

h|P_ϕ,j^s θ^rG,j|⁰≤Cδ

h|θ^rG,j|⁰+CNδh^N|θ_ϕ,j^r |⁰. δ h|θ_ϕ,j^r |⁰. The last two estimates and the second equation of (3.102) yield,

|γ1(u^lG,j)|⁰.|γ1(u^rG,j)|⁰+ 1 + δ h

|γ0(u^rG,j)|⁰+δ

h|θ_ϕ,j^r |⁰+|G˜1|⁰+CNδh^Nkv^r_jk¹, N∈N. Using estimates (3.94) and (3.96) to bound the traces on the “r” side, we obtain

h¹²|γ1(u^lG,j)|⁰. 1 + δ h

kζP_ϕ^rv^rk⁰+hkΞG Fv^rk¹+h²kv^rk¹+h²|Dxnv^r|^xn=0⁺|⁰ + δ

h¹²|θ_ϕ,j^r |⁰+h¹²|G˜1|⁰,

for 0< h≤h0, and, using (3.74) to estimate the remainder, we have h¹²|γ1(u^lG,j)|⁰. 1 + δ

h

kζP_ϕ^rv^rk⁰+hkΞ^{G F}v^rk¹+h²kv^rk¹+h²|Dxnv^r|xn=0⁺|⁰ +h³²|v^s_j|¹+h¹²|θ^r_ϕ,j|⁰

+h¹²|Θ^s_ϕ,j|⁰+h¹²|θ^l_ϕ,j|⁰, (3.103) We observe now that the first line of (3.102) together with (3.94) yields

h¹²|γ0(u^lG,j)|¹.kζP_ϕ^rv^rk⁰+hkΞ^{G F}v^rk¹+h²kv^rk¹+h²|Dxnv^r|xn=0⁺|⁰

+h¹²|θ^l_ϕ,j|¹+h¹²|θ^r_ϕ,j|¹. (3.104) Combining (3.68), with (3.101), (3.103) and (3.104) we obtain

hku^lG,jk²1+h|γ0(u^lG,j)|²1+h|γ1(u^lG,j)|²0

. 1 + δ² h²

kζP_ϕ^rv^rk²0+h²kΞG Fv^rk²1+h⁴kv^rk²1+h⁴|Dxnv^r|^xn=0⁺|²0+h³|v_j^s|²1

+h|θ^l_ϕ,j|²1+δ²

h|θ^r_ϕ,j|²0+h|θ_ϕ,j^r |²1+h|Θ^s_ϕ,j|²0+kP_ϕ,j^l v_j^lk²0+h²kv^l_jk²1. (3.105) This is a local version of (3.77). Patching together onM⁺ the local Carleman estimates (3.105) as we did in (3.79)-(3.84) yields (3.77). This concludes the proof of Proposition 3.28.

3.4. Proof of the Carleman estimate in a neighborhood of the interface 125

Reξn

Imξn

ρ^l,+

P_ϕ^l

ρ^l,− Reξn

Imξn

ρ^r,− ρ^r,+

P_ϕ^r

Figure 3.5 – Root configuration in the regionF.

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