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Inverse Scattering and Stability Estimates for The Biharmonic Operator

We study an inverse scattering problem of a perturbed biharmonic operator. we show that the high-frequency asymptotic of scattering amplitude of the biharmonic operator uniquely determine the curl A and {V -1/2 nablaA}. We also study the near-field scattering problem and show that the high-frequency asymptotic expansion up to a certain error in terms of frequency lambda recovers the same two above quantities with no additional information about A and V. We also prove stability estimates for curl A and {V -1/2 nablaA}.


Introduction
Consider the following biharmonic equation: where ∆ is Laplacian and · is the dot product a · b in R n . Here A is a vector-valued function representing the magnetic field and V is a scalar-valued function representing the potential function, with both A and V regular enough and compactly supported.
The scattering and the inverse scattering problems for the Schrödinger operators have long history, see [Lax-Reed]. One major application of the operator ( 1.1 1.1) is in the study of the theory of vibrations of beams and the elasticity theory, see 1 [1] for the case of the linear beam equation and 7 [7] for the nonlinear scattering problems. In a work by Tyni and Serov 15 [15], a Saito's type formula has been proved and it is shown that one can uniquely recover V − 1 2 ∇ · A, where A ∈ W 1 p,2δ (R n ) and V ∈ L p 2δ (R n ). Our goal in this paper is to find out what information about A and V can be recovered from the highfrequency asymptotic of scattering amplitude. We do not consider the whole amplitude at zero or near zero frequencies (Calderon's problem). In the case of inverse boundary valued problem at zero frequency, one can fully reconstruct the potential V and magnetic field A, and in particular there is no gauge invariance (see 4,3,5,14 [4,3,5,14]). The main result of this work is the following. Theorem 1.1. Let V and A ∈ C k (R 3 ) for k large enough. Then the high-frequency asymptotic expansion of scattering amplitude a(ω, θ, λ) up to O(λ −3 ) recovers curl A and V − 1 2 ∇ · A uniquely. In other words, for another pair (Ã,Ṽ ) with scattering amplitudeã so that a =ã + O(λ −3 ), then curl A = curlÃ and V − 1 2 ∇ · A =Ṽ − 1 2 ∇ ·Ã. We prove the following high-frequency approximation of the scattering amplitude a(ω, θ, λ) = iλθ ·Â(λ(ω − θ)) +V (λ(ω − θ)) + O(λ −1 ) as λ → ∞. We also study the near-field scattering problem and show that knowing the high-frequency asymptotic expansion up to error of order λ −3 recovers the same two above quantities but contains no additional information about A and V . Our recovery process is constructive and explicit, and in principal stable, but we do not formally study stability.
For the well-studied scattering problem of the Schrödinger equations, it has been shown that one can fully recover curl A and V and there is a gauge invariance, that is, for any two compactly supported magnetic fields with the same curl, the measurement cannot distinguish between them. This paper is organized as follows: we state some preliminary definitions and results in section two. In section three, the asymptotic solution of the biharmonic equation has been formulated explicitly and appropriate error estimates have been established. Section four is devoted to near-field scattering problem and the proof of our main results is given in section five. The appendix is a collection of some technical formula and statements from 15 [15].  Acknowledgments. The author would like to express his gratitude for Prof. P. Stefanov for suggesting this topic and many useful discussions throughout this work. The author thanks Prof. V. Serov and T. Tyni for carefully reading the manuscript and their comments which helped in improving the manuscript significantly.

Preliminarily Results
Our goal is to find a special solution of equation (   1.1 1.1) and corresponding amplitude which are called scattering solution and scattering amplitude. We first define the outgoing resolvent and solution which are fundamental notations.
Definition 2.1. We define the outgoing free resolvent operator R 0 (λ) := (∆ 2 − λ 4 ) −1 from C ∞ 0 to C ∞ as the analytic continuation of the operator for k large enough. We denote the outgoing resolvent operator by Given λ ∈ C, we say that the function u is λ-outgoing, if there exists c > 0 and f ∈ E ′ such that u | |x|>c = R 0 (λ)f | |x|>c . In applications, the constant c is larger than the radius of the support of the perturbations.
We now formulate the scattering amplitude where the derivation mainly follows 16 [16]. Let u = e iλx·θ + u sc be a solution for equation (

1.1)
, where e iλx·θ is the harmonic plane wave with incoming direction θ ∈ S 2 (i.e. incident wave which is neither outgoing nor incoming), and u sc is the scattered solution which is assumed to be outgoing. To formulate the scattering amplitude a, we first formulate the the Lippmann-Schwinger integral equation. One has Since u sc is an outgoing scattered solution, one may invert the operator (∆ 2 − λ 4 ) to have an explicit formula for the scattering solution u sc using equation ( Note that since A and V are compactly supported, by definition R 0 (λ)(A · ∇ + V ) is outgoing and therefore u sc is unique. On the other hand, where R(λ) is defined by Definition 2.2. We need to justify that the right hand side of the last equation is also outgoing. Assume that the resolvent exists for some λ. As it is shown in [Corollary 4.4, 15 [15]], by the resolvent identity, one has Using Agmon's estimates one can show the above series converges in H 1 −δ and hence a unique solution exists. Therefore, for a compactly supported function, R(λ) is a well-defined outgoing operator and one has the following important identity We are particularly interested in an outgoing solution of (  [11]), for any (ω, θ, λ) ∈ S 2 × S 2 × R + , there exists a function a = a(ω, θ, λ) such that |x| is an outgoing direction. The scattering amplitude a(ω, θ, λ) is given by The scattering amplitude a measures scattering in direction θ, for a plane wave at frequency λ propagating in direction ω. Next section provides necessary tools to proof the main result. We use the above representation of scattering solution to find appropriate estimates for our results.

high-frequency Asymptotic Expansion of Biharmonic Solutions
Consider the following ansatz expansion for the solution of biharmonic equation ( Since the wave before entering the support is just a plane wave e iλx·θ propagating in direction θ ∈ S 2 , we assume the following initial condition where a 0 (x, θ) | x·θ≪0 = 1 and a i (x, θ) | x·θ≪0 = 0 for i = 1, 2, 3, . . . .
The proposition below gives explicit expression for the coefficient a i for the measurement up to O(λ −3 ). 1.1), then for i = 0, 1, . . . , 5, the coefficient a i solves the following zero-initial condition system of equations: Moreover, for (1 +ã), the following estimates hold; Proof. Let u(x, θ, λ) = e iλx·θ a be solution of biharmonic equation ( Rearranging all terms with respect to the power of λ and equating singular coefficients, we get the following transport equations for a i : The first transport equation above and the initial condition a 0| x·θ≪0 = 1 implies that a 0 ≡ 1. To compute a 1 , by the second equation above we have Integrating the last equation along the flow x + tθ yields which is the X-ray transform of the magnetic field A along the lines x · θ = t. Notice that for i = 3, 4, 5 all the coefficients a i depend on the potential and magnetic fields V and A can be computed recursively, by considering the transport equation corresponding to O(λ −i ) and integrating along the flow as above. Our next goal is to establish an estimate for the error term E(x, θ, λ) in ( Expanding the right hand side of above equation and using transport equations ( Notice that although the scattering solution u sc = u − e iλ·θ is outgoing (see Definition 2.2), the above error term E is not outgoing as E has infinite support. To apply the resolvent R(λ), we first need to localize the right hand side of above error in L 2 (R 3 ): Let the compact set K denotes the support of perturbation ∇ · A + V , and χ ∈ C ∞ 0 (R 3 ) be a smooth cut-off function such that χ(x) = 1 near the support K. We define E χ (x, θ, λ) := u(x, θ, λ) − e iλx·θ (1 + χã), where u = e iλx·θ + u sc solves the biharmonic equation ( For all x away from K, χ(x) = 0, and therefore E = u sc is outgoing. On the other hand, for λ ≫ 1, E is outgoing as χ(x) = 1 for x ∈ K. We have ) Using the Lie bracket notation and the fact that χA = A and χV = V Now we may apply the resolvent R(λ) to both sides of the above equation as the right hand side is compactly supported. Our goal is to show that E χ L 2 = O(λ −4 ), for λ > 0 large enough. Note that sinceã = O(λ −2 ), the problematic terms in establishing the estimate for E χ in the right hand side of above equation will be as the rest of above terms are of O(λ −1 ) which combined with the resolvent estimates given by Lemma 6.1 gives the desired estimates for E χ . Consider the first term in ( 3.3

3.3). By the resolvent identity
Similarly, for the second term in ( 3.3

3.3), one has
Now we are ready establish estimates for I 1 , I 2 , J 1 , and J 2 . Note that the integrands in above free-resolvent operators are compactly supported and therefore, all integrals above are well-defined (see equation (

2.1)).
Estimating I 2 , J 2 . By Lemma 6.1, , where we used Lemma 6.1 and the fact that A, V are compactly supported. Similarly, , Estimating I 1 , J 1 . We mainly follow the idea in 13 [13] to estimate I 1 , J 1 . Let Γ be the set where the derivatives of χ is supported, and Then for any x ∈ B(0, R) and y ∈ Γ ∩ S the kernel G 0 (x, y, λ), given by (
. Consider the operator ∇R(λ) : Note that since R(λ) is the resolvent with constant coefficient, the gradient and the resolvent commute. Therefore, the problematic terms in (   3.4 3.4) will be the same as ones discussed in ( 3.4) with stationary phase similar to I 1 and J 1 , where an integration by part argument has been used to establish estimates. We recall that we do not use Agmon's estimates to find H 1 δ estimates. Applying the gradient directly to K(x, y, λ, θ), one has K(x, y, λ, θ) = λK(x, y, λ, θ) whereK is smooth bounded away from diagonal {x = y}. Now integration by parts argument establishes the desired H 1 δ estimates of O(λ −N ), where N depends on the regularity of magnetic field A and potential function V . This proves the proposition.

Near-field Scattering of Biharmonic Solutions
In this section we study the near-field scattering problem. Let u be the biharmonic solution of equation ( 1.1 1.1) and B R be a ball with radius R > 0 large enough containing the perturbation A and V (supp A ∪ supp V ⊂ B R ). To study the near-field scattering we only consider u | x·θ=R as our scattering data and do not study the scattering amplitude to reconstruct the high-frequency asymptotic expansion of the solution. In the following proposition, we demonstrate that all terms up to order O(λ −3 ) contains no additional information.
which is the X-ray transform of the magnetic field A along the lines x + tθ, see 10 [10]. The function δu = u −ũ has near-field data up to error of order O(λ −3 ), hence δa 2 ≡ 0 implies that there exists a compactly supported function φ such that δA = dφ. This shows that the scattering data recovers curl A. To show the scattering data recovers V − 1 2 ∇ · A, we recall that To have an explicit formula for a 3 we need to invert the Radon transform as follows: by integrating above equation along the flow and using the fact that θ · ∇ = ∂ t and (θ · ∇)a 2 (x) = 1 4 θ · A(x) we have Again since δu = u −ũ has near-field data up to error of order O(λ −3 ) and δA vanishes outside of the support, δa 3 ≡ 0 implies that 0 −∞ 2δV (x+sθ) − ∆φ(x+sθ) ds = 0, as δa 2 (x) = 1 4 φ(x). For x · θ = R, the standard arguments for inverting the X-ray transform implies that 2δV = 2(V −Ṽ ) = ∆φ = ∇ · dφ = ∇ · δA. This shows that the scattering data up to error of order O(λ −3 ) recovers V − 1 2 ∇ · A. ii) By part i) we know that δA = dφ and δV = 1 2 ∆φ. Therefore, point-wise we have , which shows that the scattering data contains no additional information on A and V up to error of O(λ −4 ). In what follows, we calculate δa 4 and δa 5 , and show that due to non-linearity, the Radon and inverse Fourier transform techniques do not provide any insight in how to show that φ = 0.
The nonlinear term δa 4 can be rewritten as where we used the equations for δa 2 , δa 3 , δA, and the fact that 1 4 (θ · A) = (θ · ∇a 2 ). By a simple calculation

Proof of the Main Results
In this section we prove the main result. We first derive the asymptotic expansion of scattering amplitude which is known as Born approximation, see also 16 [16] .
with the remainder uniform in θ, ω.
We now are ready to prove our main result.
Proposition 5.1. For j = 1, 2, let a j be the amplitude with corresponding pair of a priori bounded magnetic and potentials fields (A j , V j ), i.e. A j C k < C 0 and V j C k < C 0 for some constant C 0 > 0 and k > 0.
Now α is arbitrary, so one can stably recover all the components of the curl A with the following estimates sup ξ | curl(δA)(ξ) ξ | < ǫ.