4 Strong unique continuation for a residual stress system with
4.6 Proof of the main theorem
In this section, we want to prove Theorem 1.1. If U = (u, v, w)t satisfies (4.4.13) and the SUCP, then the SUCP holds for u, where u fulfills (4.3.1).
4.6.1 SUCP for U
In the following theorem, we will prove the SUCP for U .
Theorem 4.11. Suppose that the second order elliptic operators Pℓ satisfies (4.3.3), (4.3.4), (4.3.5) and (4.3.6) for ℓ = 1, 2. α > 0 satisfies (4.3.7) at x = 0 and s satisfies (4.3.8). Let P = P2P1 be a fourth order elliptic operator. Then the SUCP holds for the elliptic system
P U = ∑
|β|≤3
aβ∂βU (4.6.1)
provided the coefficients of Pℓ are in the Gevery class Gs.
Proof. The proof follows from [9] and section 1. To prove Theorem 4.1, there are two steps. First, Gevrey regularity of the elliptic system implies the solution U of (4.6.1) is in the Gevrey class Gs (see Proposition 2.13 in [5]). Use the vanishing order assumption and U∈ Gs, we have
|U| . e−|x|−γ, (4.6.2)
near x = 0 and for some constant γ > 0. Second, we can show that (4.6.2) implies U vanishes near 0 by using appropriate Carleman estimates. In addition, since U vanishes near 0, by the results in [63], we have U≡ 0 in Ω.
Remark 4.12. We give the Carleman estimates which were used in the proof of Theorem 4.1 in the following section.
4.6.2 Carleman Estimates
We are going to derive the Carleman estimates with the weight eτ|x|−α for the fourth order elliptic operator P = P2P1 in this section. The following Carleman estimates for the scalar case has been proven in [8] and [9]. Similar to the scalar elliptic equation, we can derive the following Carleman estimate for the special elliptic system.
Proposition 4.13. Let Pℓ(x, D) =∑
jkaℓjk(x)∂jk2 be a principally diagonal second order elliptic operator where aℓjk(x)∈ Gssatisfies(4.3.3), (4.3.4), (4.3.5) and (4.3.6) for ℓ = 1, 2. α > 0 satisfies (4.3.7) at x = 0 and s satisfies (4.3.8). Then there exist τ0 > 0 and r0> 0 such that for τ > τ0
and for all V ∈ C∞((Br0\{0}); R7), ℓ = 1, 2, the following inequality holds:
τ ˆ
|D2(|x|α/2eτ|x|−αV|2dx + τ3 ˆ
|x|−4−3αe2τ|x|−αV|2dx .
ˆ
|e2τ|x|−α(PℓV )|2dx.
Proof. Since Pℓis the principally diagonal second order elliptic operator for ℓ = 1, 2, we can directly follow the consequences in [9] and use the proof in [8]. For more details and classical results, we refer readers to [14, 59].
By using the integration by parts, we can get a stronger inequality in the following. For more details, we refer readers to [9] and section 3, then we have
∑2 j=0
τ3−2j ˆ
e2τ|x|−α|x|α|x|2(j−2)(1+α)|DjV|2dx
. ˆ
|e2τ|x|−α|PℓV|2dx,
with Pℓ satisfying all the assumptions in Proposition 4.2 for ℓ = 1, 2. Note that the right hand side of (4.4.3) and (4.4.4) involve second order derivatives of u, we cannot apply the Carleman estimates for the second order differential systems directly to get the SUCP for U . Since we have transformed (4.3.1) into a special fourth order elliptic system with the same leading operator, see (4.4.13), then we can derive the Carleman estimates for the operator P = P2P1.
Corollary 4.14. [9] Let
A =∑
jk
ajk(x)∂x2jx
k
be a second order strongly elliptic operator with aijin the Gevrey class Gs. Suppose α > 0 satisfying
(4.3.7) at x = 0. Then there exists τ0 such that for all|s|, k ≤ ν and τ ≥ τ0
k+2∑
j=0
τ3−2j ˆ
|x|α+2j(1+α)|x|2se2τ|x|−α|DjV|2dx (4.6.3)
.
∑k j=0
τ−2j ˆ
|x|2(2+j)(1+α)|x|2se2τ|x|−α|Dj(AV )|2dx.
Proof. See [9] and section 3. We can use the induction hypothesis to prove the Corollary 4.3.
For the fourth order elliptic operator P U = P2P1U is the product of two second order elliptic operators which satisfies (4.4.13), where U = (u, v, w)tand Pℓ(x, D)U =∑
jkaℓjk(x)∂jk2 U . Recall that aℓjk ∈ Gs and α > 0 satisfying (4.3.7) uniformly in x and for ℓ = 1, 2, then we have the following key estimates.
Proposition 4.15. We have the following Carleman estimates
∑4 j=0
τ6−2j ˆ
|x|−8−6α|x|2j(1+α)e2τ|x|−α|DjV|2dx≤ C ˆ
e2τ|x|−α|P V |2dx.
Proof. Apply the Corollary 4.2 iteratively, then we have
∑4 j=0
τ6−2j ˆ
|x|−8−6α|x|2j(1+α)e2τ|x|−α|DjV|2dx (4.6.4)
.
∑2 j=9
τ3−2j ˆ
|x|−4−3α|x|j(1+α)e2τ|x|−α|Dj(P1V )|2dx
. ˆ
e2τ|x|−α|(P2P1V )|2dx = ˆ
e2τ|x|−α|P V |2dx,
where the first inequality is obtained by (4.6.3) with k = 2, s =−4 −7
2α and the second inequality is obtained by (4.6.3) with k = 0, s =−2(1 + α). For more details, we refer reader to see [9].
Now, we want to prove the SUCP for (4.3.1). Here we prove the theorem 2.2.
Proof of Theorem 4.1: The operator P = P2P1 is strongly elliptic in the Gevrey class Gs, then U is also in the Gevrey class Gs. Therefore, we have the vanishing of infinite order implies that
|u| . e−|x|−γ
for some γ > α. Let χ∈ C0∞(R3) be such that χ≡ 1 for |x| ≤ R and χ ≡ 0 for |x| ≥ 2R (R > 0 is
small enough). Then we can apply (4.6.4) to the function χU , which means
C
∑4
|β|=0
τ6−2|β|
ˆ
|x|<R|x|(2|β|−6)(1+α)−2e2τ|x|−α|DβU|2dx (4.6.5)
≤ ˆ
e2τ|x|−α|P U|2dx
≤ ˆ
|x|<R
e2τ|x|−α|P U|2dx + ˆ
|x|>R
e2τ|x|−α|P (χU)|2
≤ ˆ
|x|<R
e2τ|x|−α|
∑3 m=0
Edm(U )|2dx + ˆ
|x|>R
e2τ|x|−α|P (χU)|2,
by using the reduction elliptic system (4.4.13).
If τ is large and R is sufficiently small, then (4.6.5) implies
C
∑4
|β|=0
τ6−2|β|
ˆ
|x|<R|x|(2|β|−6)(1+α)−2e2τ|x|−α|DβU|2dx (4.6.6)
≤ ˆ
|x|>R
e2τ|x|−α|P (χU)|2,
for some constant C > 0. Notice that eτ|x|−α≥ eτ R−α for|x| < R and eτ|x|−α ≤ eτ R−α for|x| > R.
Therefore, we can use (4.6.6) to obtain
C
∑4
|β|=0
τ6−2|β|
ˆ
|x|<R|x|(2|β|−6)(1+α)−2|DβU|2dx
≤ ˆ
|x|>R|P (χU)|2.
Let τ → ∞, we get U = 0 in {|x| < R} for R small, which implies u = 0 in {|x| < R}. Furthermore, by using the unique continuation principal in [63], we can obtain u≡ 0 in Ω, then we are done.
Chapter 5
Future work
5.1 Fundamental solutions for the anisotropic Maxwell sys-tem
The last chapter of this thesis is going to list out related opening problems. We list some future works which related to this thesis in the following. In the above chapters, we have already men-tioned the enclosure-type method for the anisotropic medium. We gave reconstruction algorithms for both anisotropic elliptic equation (Chapter 2) and anisotropic Maxwell system (Chapter 3).
Recall that for the enclosure-type method, we have two tools: One is to define a suitable indicator function and the other is to construct appropriate special solutions for the mathematical model.
In Chapter 2 and 3, we have constructed oscillating-decaying (OD) solutions for both anisotropic elliptic equation and anisotropic Maxwell system. The drawback of this special type solutions is that we need to use the Runge approximation property to find a sequence of solutions defined on the whole domain and to satisfy the same equation which approximates to OD solutions. It looks like the Runge approximation property used in the thesis is not constructive. If we can make the proof in a constructive way, then this may be useful if one tries to implement the method numerically.
The Runge approximation has a constructive version. Indeed, we can use the density property of the single layer operator between appropriate Sobolev spaces (as L2-spaces) and the well-posedness of the forward problem. This argument can be used as soon as we have the corresponding fun-damental solution and the unique continuation property of the Maxwell model. If ϵ and µ are isotropic, this is of course possible. In the anisotropic cases, we need the construction of the fundamental solution (and justify its type of singularities) in addition to the unique continuation property. We could not find these properties in the literature, so we need to do more construc-tive work for the fundamental solutions. One of our future work is to construct the fundamental solutions for the anistropic Maxwell system.
5.2 More L
pestimates for the anisotropic Maxwell system
For the anisotropic Maxwell system in Chapter 3, we only consider the electric permittivity to be allowed to have the jump and the anisotropy and the magnetic permeability µ is a scalar function.
It is known that the same method would work when the role of the two parameters exchange.
Recall that we have defined the impedance map as ΛD: ν× H|∂Ω→ ν × E|∂Ωand we can allow ϵ to be anisotropic and to have jump discontinuity. If we exchange ϵ and µ, we need to use the other impedance map fΛD : ν× E|∂Ω → ν × H|∂Ω. Indeed, we needed to assume that the other coefficients are smooth. The technique is due to the type of Lp estimates we are using. Then, we are able to remove these assumptions by using the Layer potential techniques. and we could test this idea for the scalar divergence form PDE model and it works. Hence, we do hope that this idea can go smoothly to Maxwell as well. We are working on it.
5.3 Strong unique continuation for the general second order elliptic system
In many literature, we know the strong unique continuation property (SUCP) holds for the scalar elliptic equation case. However, for more general elliptic system, we do not know much about the result. In Chapter 4, we gave a very special method to derive the SUCP for the residual stress system with Gevrey coefficients. We use the “product” of two elliptic operators to derive the SUCP for this system. Our future work is try to use similar method to derive more SUCP for more general elliptic system with Gevrey coefficients. In [8], the authors derived the SUCP for the second order elliptic operator P (x, D) with complex coefficients. We want to generalize their ideas to second order elliptic system without any assumptions.
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