Determining rough first order perturbations of the polyharmonic operator

We show that the knowledge of Dirichlet to Neumann map for rough $A$ and $q$ in $(-\Delta)^m +A\cdot D +q$ for $m \geq 2$ for a bounded domain in $\mathbb{R}^n$, $n \geq 3$ determines $A$ and $q$ uniquely. The unique identifiability is proved using property of products of functions in Sobolev spaces and constructing complex geometrical optics solutions with sufficient decay of remainder terms.


Introduction and Statement of Results
If 0 is not in the spectrum of L A,q , it can be shown that the Dirichlet problem (1.1) has a unique solution u ∈ H 2m (Ω). We can then define the Dirichlet-to-Neumann N A,q map as The inverse boundary value problem for the perturbed polyharmonic operator L A,q is to determine A and q in Ω from the knowledge of the Dirichlet to Neumann map N A,q .
Here and in what follows, E ′ (Ω) = {u ∈ D ′ (R n : supp(u) ⊆Ω} and W s,p (R n ) is the standard L p based Sobolev space on R n , s ∈ R and 1 < p < ∞, which is defined via the Bessel potential operator.
We can also define the analogous spaces W s,p (Ω) for Ω a bounded open set with smooth boundary. The reader is referred to [1] for properties of these spaces.
The goal of this paper is to extend the results of [2] for the case m ≥ n. Moreover, even for m < n, we aim to improve the uniqueness result of [2] for a more general class of A and q.
As was observed in [32], for the case m = 1, there is a gauge invariance that prohibits uniqueness and therefore we can hope to recover A and q only modulo such a gauge transformation. It was shown in [32] that such uniqueness modulo a gauge invariance is possible provided that A ∈ W 2,∞ , q ∈ L ∞ and dA satisfy a smallness condition. There have been many successive papers which have weakened the regularity assumptions on A and q. The reader is referred to [14,21,25,30,34] for details.
Inverse problems for higher order operators have been considered in [2,3,20,22,35,36] where unique recovery actually becomes possible. Higher order polyharmonic operators occur in the areas of physics and geometry such as the study of the Kirchoff plate equation in the theory of elasticity, and the study of the Paneitz-Branson operator in conformal geometry; for more details see monograph [11].
Let us remark that the problem considered in this paper can be considered as a generalization of the Calderón's inverse conductivity problem [6], also known as electrical impedance tomography, for which the question of reducing regularity has been studied extensively. In the fundamental paper by Sylvester and Uhlmann [33] it was shown that C 2 conductivities can be uniquely determined from boundary measurements. Successive papers have focused on weakening the regularity for the conductivity; see [5,7,12,15,16,27] for more details.
1.2. Statement of Result. Throughout this paper we assume m ≥ 2 and n ≥ 3. Suppose that the first order perturbation A be in For a fixed δ with 0 < δ < 1 2 , suppose that the zeroth order perturbation q be in Before stating the main result, we consider the bi-linear forms B A and b q on H m (Ω) which are defined by where ·, · denotes the distributional duality on R n such that ·,· naturally extends L 2 (R n )-inner product, andũ,ṽ ∈ H m (R n ) are any extensions of u and v, respectively. In Appendix A, we show that these definitions are well defined i.e. independent of the choice of extensionsũ,ṽ. Using a property of multiplication of functions in Sobolev spaces, we show that the forms B A and b q are bounded on H m (Ω). We also adopt the convention that for any z > 1, the number z ′ is defined by z ′ = z/(z − 1).
Consider the operator D A , which is formally A · D where D j = −i∂ j , and the operator m q of multiplication by q. To be precise, for u ∈ H m (Ω), D A (u) and m q (u) are defined as where ·, · Ω is the distribution duality on Ω such that ·,· Ω naturally extends L 2 (Ω)-inner product. The operators D A and m q are shown in Appendix A to be bounded H m (Ω) → H −m (Ω) and hence, standard arguments show that the operator , consider the Dirichlet problem (1.1). If 0 is not in the spectrum of L A,q , it is shown in Appendix B that the Dirichlet problem (1.1) has a unique solution u ∈ H m (Ω). We define the Dirichlet-to-Neumann map N A,q weakly as follows is any extension of h so that γv h = h, and where ·, · ∂Ω is the distribution duality on ∂Ω such that ·,· ∂Ω naturally extends L 2 (∂Ω)-inner product. It is shown in Appendix B that N A,q is a well-defined bounded as an operator Our main result is as follows.
be a bounded open set with C ∞ boundary, and let m ≥ 2 be an integer. Let 0 < δ < 1/2. Suppose that A 1 , A 2 satisfy (1.2) and q 1 , q 2 satisfy (1.3) and 0 is not in the spectrums of L A 1 ,q 1 and Detailed explanation of the assumption δ > 0 is given in Remark 3.4.
For the proof of Theorem 1.1, we follow a similar approach as in [2]. The key ingredient in the proof of Theorem 1.1 is construction of complex geometric optics solutions for the operator L A,q with correct decay for the remained term. For this, we use the method of Carleman estimates which is based on the corresponding Carelman estimates for the Laplacian, with a gain of two derivatives, due to Salo and Tzou [31] and Proposition 2.2, which gives property of products of functions in various Sobolev spaces.
The idea of constructing such complex geometric optics solutions to elliptic operators goes back to the fundamental paper by Sylvester and Uhlmann [33] and has been extensively used to show unique recovery of coefficents in many inverse problems.
The rest of the paper is organized as follows. We construct complex geometrical optics solutions for the perturbed polyharmonic operator L A,q with A and q as defined in (1.2) and (1.3) respectively. The proof of Theorem 1.1 is given in section 3. In Appendix A, we study mapping properties of D A and m q . Appendix B is devoted to the well-posedness of the Dirichlet problem L A,q with A satisfying (1.2) and q satisfying (1.3). In Appendix C, we specify why we use Bessel potential to define fractional Sobolev spaces.
Proposition 2.1. Let φ be a limiting Carleman weight for −h 2 ∆ inΩ and let We now state a theorem on products of functions in Sobolev spaces; see Theorem 1 and Theorem 2 in [29,Section 4.4.4].
If u ∈ W s 1 ,p 1 (R n ) and v ∈ W s 2 ,p 2 (R n ), then uv ∈ W s 1 ,p (R n ). Moreover, the pointwise multiplication of functions is a continuous bi-linear map W s 1 ,p 1 (R n ) · W s 2 ,p 2 (R n ) ֒→ W s 1 ,p (R n ) and where the constant C depends only on the various indices.
We now derive Carleman estimate for the perturbed operator L A,q when A and q are as in (1.
Proof. Iterate the Carleman estimate in Proposition 2.1 m times with s = −3m/2 and a fixed ǫ > 0 sufficiently small and independent of h to get the estimate for all u ∈ C ∞ 0 (Ω) and 0 < h ≪ ǫ ≪ 1.
Let ψ ∈ C ∞ 0 (R n ). By duality and Proposition 2.2, we have for any m

(2.4)
Remark 2.4. Estimate (2.4) actually goes through even for δ = 0. For m < n, in Propostion 2.2, It is also easy to see that we in fact have a stronger decay of O(h m+2δ ) in (2.4).
By definition of dual norm, (2.5) For m > 2, by duality, we have

Now, we use Hölder's inequality and Sobolev Embedding Theorem to get
Thus, for any m ≥ 2, by definition of dual norm we have Combining this together with (2.3) and (2.5), for small enough h > 0 and m ≥ 2, we get (2.6) Since e −φǫ/h u = e −φ/h e −φ 2 /2ǫ u and φ is smooth, we obtain (2.2). A natural question would be why in particular has s been chosen so that s + 2m = m/2. If we choose s+2m < m/2 or s+2m > m/2 then in the former case we will have to take more regular A and q to ensure that we have the correct decay essentially as dictated by the hypotheses in Proposition 2.2 or in the latter case we can no longer ensure a decay of at least which is crucially used in the construction of complex geometric optics solutions.
We now use the above proved Carleman estimate to first establish an existence and uniqueness result for the inhomogeneous partial differential equation Note that the zeroth order coefficient of the adjoint operator L * A,q comprises of two termsq and D ·Ā. The Carleman estimate forq is the same as (2.4) asq lies in the same class as q.
m if m < n, p ′ > 2 if m = n or m = n + 2 and p ′ ≥ 2 otherwise and as mentioned in Remark 2.4, estimate (2.4) goes through for zeroth order pertubration belonging to this class too. Hence, estimate (2.2) is valid for L * φ , since −φ is a limiting Carleman weight as well. We now convert the Carleman estimate (2.2) for L * φ in to a solvability result for L φ . For s ≥ 0, we define semi-classical Sobolev norms on a smooth bounded domain Ω as . Proposition 2.6. Let A and q be as defined in (1.2) and (1.3) respectively and let φ be a limiting Carleman weight for −h 2 ∆ onΩ. If h > 0 is small enough, then for any .
Proof. Let D = L * φ (C ∞ 0 (Ω)) and consider the linear functional L : Hahn-Banach theorem ensures that there is a bounded linear functionalL : (Ω) . By the Riesz Representation theorem, there This finishes the proof.
We now wish to construct complex geometric optics solutions for the equation L A,q u = 0 in Ω with A and q as defined in (1.2) and (1.3) respectively using the solvability result Proposition 2.6. These are solutions of the form where ζ ∈ C n is such that ζ · ζ = 0, |ζ| ∼ 1, a ∈ C ∞ (Ω) is an amplitude, r is a correction term, and h > 0 is a small parameter.
Following [21], we shall consider ζ depending slightly on h, i.e ζ = ζ 0 + ζ 1 with ζ 0 independent of h and ζ 1 = O(h) as h → 0. We also assume that |Re ζ 0 | = |Imζ 0 | = 1. Then we can write (2.8) as Observe that (2.7) is a solution to L A,q = 0 if and only if and hence if and only if (Ω) norm on the right-hand side of (2.9). The terms h 2m D A a, h 2m−1 m A·(ζ 0 +ζ 1 ) a and h 2m m q a will eventually give us a decay of O(h m+m/2 ) provided m ≥ 2.

Integral Identity
We first do a standard reduction to a larger domain. For the proof, similarly as in [2, Proposition 3.1], we follow [21, Proposition 3.2]. Proposition 3.1. Let Ω, Ω ′ ⊂ R n be two bounded open sets such that Ω ⊂⊂ Ω ′ and ∂Ω and ∂Ω ′ are smooth. Let A 1 , A 2 and q 1 , q 2 satisfy (1.2) and (1.3), respectively. If N A 1 ,q 1 = N A 2 ,q 2 , then N ′ A 1 ,q 1 = N ′ A 2 ,q 2 where N ′ A j ,q j denotes the set of the Dirichlet-to-Neumann map for L A j ,q j in Ω ′ , j = 1, 2.
Proof. Let f ′ ∈ m−1 j=0 H m−j−1/2 (∂Ω) and let v ′ 1 ∈ H m (Ω ′ ) be the unique solution (See Appendix B for justification of this statement) to 1 Ω ∈ H m (Ω) and let f = γv 1 . By the well-posedness result in Appendix B, we can guarantee the existence of a unique v 2 ∈ H m (Ω) so that L A 2 ,q 2 v 2 = 0 and We now show that L A 2 ,q 2 v ′ 2 = 0 in Ω ′ . Let ψ ∈ C ∞ 0 (Ω ′ ). We then have Since A 2 and q 2 are compactly supported inΩ and φ ∈ H m 0 (Ω), we can rewrite the above equality as Note that Hence, we have We get Using the fact A 1 and q 1 are compactly supported inΩ, we obtain Using exact same arguments, one can show that N ′ We now derive the following integral identity based on the assumption that N A 1 ,q 1 = N A 2 ,q 2 .

Proposition 3.2.
Let Ω ⊂ R n , n ≥ 3 be a bounded open set with smooth boundary. Assume that A 1 , A 2 and q 1 , q 2 satisfy (1.2) and (1.3), respectively. If N A 1 ,q 1 = N A 2 ,q 2 , then the following integral identity holds for any u 2 , v ∈ H m (Ω) satisfying L A 2 ,q 2 u 2 = 0 in Ω and L * A 1 ,q 1 v = 0 in Ω, respectively. Recall that L * A,q = LĀ ,q+D·Ā is the formal adjoint of L A,q .
To show A 1 = A 2 , we will need to use Poincare lemma for currents [28] which requires the domain to be simply connected. Therefore, we reduce the problem to larger simply connected domain, in particular to a ball.
Let B be an open ball in R n such that Ω ⊂⊂ B. According to Proposition (3.2), we know that N ′B A 1 ,q 1 = N ′B A 2 ,q 2 denotes the Dirichlet-to-Neumann map for L A j ,q j in B, j = 1, 2. Now by Proposition (3.2), the following integral identity holds Here and in what follows, by B B A 2 −A 1 and b B q 2 −q 1 we denote the bi-linear forms corresponding to A 2 − A 1 and q 2 − q 1 (as defined in Appendix A) in the ball B.

(3.3)
Multiply by h throughout and let h → 0 to get Let us justify how we get (3.4). We use Proposition A.2 to show We also have for any m ≥ 2, using Proposition A.2, Thus, we see that after multiplying (3.3) by h, the latter 2 terms in (3.3) go to zero as h → 0.
To prove A 1 = A 2 , we consider A 1 − A 2 as a 1-current and using the Poincare lemma for currents, we conclude that there is a g ∈ D ′ (R n ) such that ∇g = A 1 − A 2 ; see [28]. Note that g is a constant outsideB since A 1 − A 2 = 0 in R n \B (also near ∂B). Considering g − c instead of g, we may instead assume g ∈ E ′ (B).
To show q 1 = q 2 , substitute A 1 = A 2 and a 1 = a 2 = 1 in to the identity (3.1) to obtain Let h → 0 to getq 1 (ξ) −q 2 (ξ) = 0 for all ξ ∈ R n . To justify this we need to show that as h → 0. We will only consider the term b B q 2 −q 1 (h m/2 r 1 , e ix·ξ ). The justification for the other two terms follows similarly. We have for any m ≥ 2 Sinceq 1 (ξ) −q 2 (ξ) = 0 for all ξ ∈ R n , we get q 1 = q 2 in B.
Remark 3.4. If we take δ = 0, then we see that all we can say using Propostion 2.2 is that |b B q 2 −q 1 (h m/2 r 1 , e ix·ξ )| = O(1). This is why we impose slightly higher regularity for q.
Appendix A. Properties of D A and m q Let Ω ⊂ R n , n ≥ 3, be a bounded open set with smooth boundary, and m ≥ 2 be an integer. Let A and q satisfy (1.2) and (1.3), respectively. As before, in what follows, W s,p is the standard L p based Sobolev space on R n , s ∈ R and 1 < p < ∞ defined using Bessel potential.
We start by considering the bi-linear forms The following result shows that the forms B R n A and b R n q are bounded on H m (R n ). The proof is based on a property of multiplication of functions in Sobolev spaces.
Proposition A.1. The bi-linear forms B R n A and b R n q on H m (R n ) are bounded and satisfy for any m ≥ 2, Proof. Using the duality between W − m 2 +δ,r ′ (R n ) and W m 2 −δ,r (R n ), we conclude from Proposition 2.2 that for allũ,ṽ ∈ H m (R n ) with m ≥ 2, ) We now give the proof for the bi-linear form B R n A . Using the duality between W − m 2 +1,p ′ (R n ) and W − m 2 +1,p (R n ) we conclude from Proposition 2.2 that for allũ,ṽ ∈ H m (R n ), for m > 2 we have . The proof is thus complete. Now, we show that the operators B A and b q defined in (1.4) are indeed well defined. Recall that whereũ,ṽ ∈ H m (R n ) are any extensions of u and v, respectively. We want to show that this definition is independent of the choice of extensionsũ,ṽ. Indeed, let u 1 , u 2 ∈ H m (R n ) be such that u 1 = u 2 = u in Ω, and let v 1 , v 2 ∈ H m (R n ) be such that v 1 = v 2 = v in Ω. It is enough to show that for all w ∈ H m (R n ), . Since A and q are supported inΩ and since u 1 = u 2 and v 1 = v 2 in Ω, we have The next result shows that the bi-linear forms B A and b q are bounded on H m (Ω).
Proposition A.2. The bi-linear forms B A and b q are bounded on H m (Ω) are bounded and satisfy for any m ≥ 2 Proof. This easily follows from the previous proposition in exactly the same way as in [2, Proposition A.2]. Now, for u ∈ H m (Ω), we define D A (u) and m q (u) for any v ∈ H m 0 (Ω) by The following result, which is an immediate corollary of Proposition A.2, implies that D A and m q are bounded operators from H m (Ω) → H −m (Ω). The norm on H −m (Ω) is the usual dual norm given by .
for all u ∈ H m (Ω).
Finally, we state the following identities which are useful for defining the adjoint of L A,q .
Proposition A.4. For any u, v ∈ H m (Ω), the forms B A and b q satisfy the following identities Proof. Since the proof repeats that of [2,Proposition A.4] almost word for word, we omit it. where γ is the Dirichlet trace operator γ : H m (Ω) → m−1 j=1 H m−j−1/2 (∂Ω) which is bounded and surjective; see [13,Theorem 9.5].
Therefore, the seqsuilinear form a is coercive on H m 0 (Ω). Compactness of the embedding H m 0 (Ω) ֒→ H −m (Ω) together with positivity of bounded operator L A,q + C 0 I : H m 0 (Ω) → H −m (Ω) imply that L A,q : H m 0 (Ω) → H −m (Ω) is Fredholm with zero index and hence Fredholm alternative holds for L A,q ; see [24,Theorem 2.33]. (B.2) thus has a unique solution u ∈ H m 0 (Ω) if 0 is outside the spectrum of L A,q . Now, consider the Dirichlet problem (B.1) and assume 0 is not in the spectrum of L A,q . We know that there is a w ∈ H m (Ω) such that γw = f . According to the Corollary (A.3), we have L A,q w ∈ H −m (Ω). Therefore u = v + w with v ∈ H m 0 (Ω) being the unique solution of the equation L A,q v = −L A,q w ∈ H −m (Ω) is the unique solution of the Dirichlet problem (B.1).
Under the assumption that 0 is not in the spectrum of L A,q , the Dirichlet-to-Neumann map is defined as follows: let f, h ∈ m−1 j=0 H m−j−1/2 (∂Ω). Set where u is the unique solution of the Dirichlet problem (B.1) and v h ∈ H m (Ω) is an extension of h, that is γv h = h. To see that this definition is independent of v h , let v h,1 , v h,2 ∈ H m (Ω) be such that γv h,1 = v h,2 = h. Since w = v h,1 − v h,2 ∈ H m 0 (Ω) and u solves the Dirichlet problem (B.1), we have