Stability estimates for an inverse boundary value problem for biharmonic operators with first order perturbation from partial data

In this paper we study an inverse boundary value problem for the biharmonic operator with first order perturbation. Our geometric setting is that of a bounded simply connected domain in the Euclidean space of dimension three or higher. Assuming that the inaccessible portion of the boundary is flat, and we have knowledge of the Dirichlet-to-Neumann map on the complement, we prove logarithmic type stability estimates for both the first and the zeroth order perturbation of the biharmonic operator.


Introduction and Statement of Results
Let Ω ⊆ R n , n ≥ 3, be a bounded simply connected domain with smooth boundary ∂Ω.Consider a biharmonic operator with first order perturbation where D = i −1 ∇, A is a complex-valued vector field, called the magnetic potential, and q is a complex-valued function, called the electric potential.In practice, the operator L A,q arises when we consider the equilibrium configuration of an elastic plate hinged along the boundary.In physics and geometry, higher order operators occur in the study of the Kirchhoff plate equation in the theory of elasticity, the continuum mechanics of buckling problems, and the study of the Paneitz-Branson operator in conformal geometry, see [4,18,32].We also refer readers to [12,35] for more on the elasticity model and perturbed biharmonic operators.The operator L A,q equipped with the domain is an unbounded closed operator on L 2 (Ω) with purely discrete spectrum, see [20,Chapter 11].In this paper we shall make the following assumption about the eigenvalues of the operator L A,q .Assumption 1: 0 is not an eigenvalue of the perturbed biharmonic operator L A,q : D(L A,q ) → L 2 (Ω).
The eigenvalues of L A,q depend strongly on the coefficients A and q.For instance, if the L ∞ -norms of A and q are strictly positive, then L A,q satisfies Assumption 1.
Some properties of the eigenvalues and eigenfunctions of the polyharmonic operator (−∆) m , m ≥ 2, are discussed in [18,Chapter 3].For the perturbed biharmonic operator L A,q , when A = 0 and q is real-valued and compactly supported in a ball of radius R > 0, the Dirichlet eigenvalues of the operator L 0,q are real-valued and bi-Lipschitz equivalent to n 4/3 , see [28,Lemma 3.6].To the best of our knowledge, the study of eigenvalue problems for L A,q with A = 0 has not received any attention, and it is not within the scope of this paper.
Let us next describe the geometric framework considered in this paper.We assume that Ω ⊆ {x = (x 1 , . . ., x n ) ∈ R n : x n < 0}, n ≥ 3. We split the boundary ∂Ω in two non-empty subsets Γ 0 := ∂Ω ∩ {x ∈ R n : x n = 0}, called the inaccessible part, and its complement Γ = ∂Ω \ Γ 0 .That is, the inaccessible part of the boundary is flat.This geometric setting was first considered in [23] in the context of inverse problems.
We define the partial Dirichlet-to-Neumann map associated with the boundary value problem (1.2) on the open set Γ ⊂ ∂Ω as where supp f, supp g ⊂ Γ, and ν is the outer unit normal of ∂Ω.Let us also define the norm of Λ Γ A,q by Λ Γ A,q := sup{ Λ Γ A,q (f, g) H 2 (Γ) = 1}.The purpose of this paper is to establish log-type stability estimates for the magnetic potential A and electric potential q from the partial Dirichlet-to-Neumann map Λ Γ A,q .The corresponding uniqueness result was established in [42,Theorem 1.4].
The study of inverse boundary value problems concerning the first order perturbation of the polyharmonic operator (−∆) m , m ≥ 2, was initiated in [26], where unique identification results from the Dirichlet-to-Neumann map were obtained.We highlight that unique recovery of the first order perturbation appearing in higher order elliptic operators differs from analogous problems for second order operators, such as the magnetic Schrödinger operator.In the latter case, due to the gauge invariance of boundary measurements, one can only hope to uniquely recover dA, which is the exterior derivative of the first order perturbation A, see for instance [25,29,34,39] and the references therein among the extensive literature in this direction.Here, if we view A as a 1-form, then dA is a 2-form given by the formula (1.4) For the polyharmonic operator, which is of order 2m, a gauge invariance occurs when we attempt to recover perturbations of order 2m − 1 from boundary measurements, see a very recent result [37].Before stating the main results of this paper, let us introduce the following admissible sets for the coefficients A and q of the operator L A,q .Given a constant M > 0, we define the class of admissible magnetic potentials A(Ω, M) by Notice that we assumed a priori bounds for the H s -norm of A instead of the natural W 1,∞norm in the definition above.We shall explain the reason in Subsection 1.2.Let us also define the class of admissible electric potentials Q(Ω, M) in a similar fashion by The main results of this paper are as follows.First, we state a log-type stability estimate for the magnetic potential A.
Theorem 1.1.Let Ω ⊆ {x ∈ R n : x n < 0}, n ≥ 3, be a bounded simply connected domain with smooth boundary ∂Ω.Assume that Γ 0 = ∂Ω ∩ {x ∈ R n : x n = 0} is nonempty, and let Γ = ∂Ω \ Γ 0 .Let M ≥ 0, and let A (j) ∈ A(Ω, M) and q j ∈ Q(Ω, M), j = 1, 2. Suppose that Assumption 1 holds for both operators L A (1) ,q 1 and L A (2) ,q 2 .Then there exists a constant C = C(Ω, n, M, s) > 0 such that ) We shall also establish a log-type estimate for the electrical potential q, which is given in the following theorem.Theorem 1.2.Under the same hypotheses as in Theorem 1.1, there exists a constant C = C(Ω, n, M, s) > 0 such that ) If the electric potentials q 1 and q 2 pose additional regularity and a priori bounds, we have the following corollary of Theorem 1.2.
Let A (j) ∈ A(Ω, M), and let q j ∈ L ∞ (Ω) be such that q j H s (Ω) ≤ M, j = 1, 2. Suppose that Assumption 1 holds for both operators L A (1) ,q 1 and L A (2) ,q 2 .Then there exists a constant C = C(Ω, n, M, s) > 0 such that Here the constants θ 2 and η are the same as in Theorem 1.2.
Remark 1.4.To the best of our knowledge, estimates (1.5) and (1.6) are the first stability estimates for partial data inverse boundary value problems for the biharmonic operator with first order perturbations.
1.1.Previous literature.The study of inverse problems concerning biharmonic or polyharmonic operators have attracted significant attentions in recent years.For the case that only the zeroth order perturbation is considered, unique identifiability results were obtained in [21,22].If we extend our consideration to include first order perturbations, it was proved in [26] that the Dirichlet-to-Neumann map determines sufficiently smooth first order perturbation of the polyharmonic operator uniquely.There are extensive subsequent efforts to recover the first order perturbation of the polyharmonic operator with lower regularity, see for instance [5,6,11,27] and the references therein.We remark that relaxing the regularity of coefficients is crucial in inverse problems, since it enables the imaging of rough medium.In addition to the aforementioned results in the Euclidean setting, we refer readers to [7,41] for some uniqueness results in the setting of Riemannian manifolds.
In all of the aforementioned results, boundary measurements are made on the entire boundary of a domain or a manifold.However, in many applications such as geophysics, it could be either impossible or extremely cost-consuming to perform measurements on the entire boundary of a medium.Hence, it is of great interest and significance to investigate partial data inverse problems, in which case measurements are made only on open subsets of the boundary.In the context of inverse boundary value problems for biharmonic operators, it was established in [24] that the partial Dirichlet-to-Neumann map, where the Dirichlet data is measured on the entire boundary but the Neumann data is measured on slightly more than half of the boundary, determines the first and the zeroth order perturbations of the biharmonic operator uniquely.Several partial data uniqueness results were obtained in [42] to recover the first and the zeroth order perturbation of the biharmonic operator from the set of Cauchy data, in bounded domains and infinite slabs.
Since polyharmonic operators are of order 2m, it is natural to consider recovering second or higher order perturbations from boundary measurements, and some progress have been made in this direction.For instance, the authors of [8,9] obtained unique determination results for perturbations of up to second order appearing in the polyharmonic operator.Beyond the second order perturbations, it was proved in [10] that the Dirichlet-to-Neumann map, with the Neumann data measured on roughly half of the boundary, determines anisotropic perturbations of up to order m appearing in polyharmonic operators.Also, recovery of multiple isotropic perturbations, given as either a function or a vector field, of a polyharmonic operator from partial boundary measurements was established in [19].We would like to emphasize that, to the best of our knowledge, unique determination of the first and the zeroth order perturbation of the polyharmonic operator from partial boundary measurements, where both the Dirichlet data and the Neumann data are measured only on open subsets of the boundary, remain an important open problem.The analogous problem for the magnetic Schrödinger operator was solved in [16].
Turning attention to the issue of stability, the majority of known literature for polyharmonic operators considers only the zeroth order perturbation.A log-type estimate was obtained in [15] when measurements are made on the entire boundary.In the realm of partial data results, a log-type estimate was proved in [14] when the inaccessible part of the boundary is contained in a hyperplane.Additionally, a log-log-type estimate was established in [15] in the scenario that the Neumann data is measured on slightly more than half of the boundary.We remark that the stability estimates probed in [14,15] indicate that the inverse problems considered are severely ill-posed.This phenomenon means that small errors in measurements may result in exponentially large errors in the reconstruction of the unknown potential, which makes it very challenging to design reconstruction algorithms with high resolution in practice.However, as observed numerically in [17], if a frequency k is introduced to the operator, then stability may improve to a Hölder type or a Lipschitz type when k becomes large.For the perturbed biharmonic operator L A,q , with A = 0, the author derived a Hölder-type stability estimate for the potential q at high frequencies from partial boundary measurements in [30], where the inaccessible part of the boundary is flat.
Once stability estimates are established for the zeroth order perturbation of the polyharmonic operator, the natural next step is to derive estimates for higher order perturbations.To the best of our knowledge, stability estimates for higher order perturbations of the biharmonic operator were obtained only when full data is considered.We refer readers to see a very recent result [31] for log-type stability estimates for the first and zeroth perturbations of the biharmonic operator L A,q , as well as [3] for stability estimates concerning perturbations of up the second order of polyharmonic operators.
1.2.Outline of the proof of Theorem 1.1 and Theorem 1.2.The general strategy of the proof follows from the methods introduced in [2] using complex geometric optics (CGO) solutions.To elaborate, we shall construct a solution v ∈ H 4 (Ω) to the equation We shall accomplish this by employing the reflection argument originated in [23].This approach has been successfully implemented to solve many partial data inverse boundary value problems where the inaccessible part of the boundary is contained in a hyperplane or a sphere.We refer readers to see for instance [13,14,25,30,36,42] and the references therein.We would like to mention that we utilized three mutually orthogonal vectors in R n satisfying (3.3) to construct CGO solutions.This is only possible when the domain Ω has dimension n ≥ 3. Hence, the techniques used in this paper are not applicable if n = 2, see [33] for the methods to establish a uniqueness result for the Calderón problem in dimension two.
Another important component of the proof is the integral inequality (3.14), whose derivation requires knowledge of the partial Dirichlet-to-Neumann map.This integral inequality, in conjunction with CGO solutions (3.12) and (3.13), results in the Fourier transform of the difference of magnetic fields dA (2) − dA (1) , as well as some error terms.We then apply an estimate for the remainder terms of CGO solutions (3.10) and a quantitative version of the Riemann-Lebesgue lemma (Lemma 2.2) to control those error terms.Subsequently, we use the definition of the H −1 -norm in terms of the Fourier transform, as well as judicious choice of various parameters, to deduce an estimate for dA H −1 (Ω) .
The main difficulty to establish stability estimates for the magnetic potential A is as follows.We can only obtain an estimate for the magnetic field dA from the steps outlined above.However, as uniqueness results have been achieved for the magnetic potential A, we must also establish stability estimates for A in a suitable norm.To overcome this difficulty we shall utilize a specific Hodge decomposition of vector fields [38, Theorem 3.3.2]:For every vector field A, there exists a vector field A sol and a function ϕ, both of which are uniquely determined, such that A = A sol + ∇ϕ, where the solenoidal part A sol and the potential part ∇ϕ satisfy the properties div A sol = 0 and ϕ| ∂Ω = 0.Here div A sol is the divergence of A sol .Then an application of the Morrey's inequality yields . Notice that we need an estimate for dA L ∞ (Ω) in order to apply the Morrey's inequality, but we only established one for dA H −1 (Ω) from earlier parts of the proof.To medicate this, as we shall detail in estimate (3.40), we apply the Sobolev embedding theorem and the interpolation inequality [20,Theorem 7.22] to obtain an estimate for dA L ∞ (Ω) .We point out that this is the reason why we assumed an a priori bound for the norm A H s (Ω) , s > n 2 + 1, in the definition of A(Ω, M).This is also the reason why we have a log-type estimate for To prove an estimate for A L ∞ (Ω) , we still need to establish an estimate for ∇ϕ L ∞ (Ω) .This is accomplished in Lemma 3.4 by utilizing the CGO solutions (3.12) and (3.13) again.From here, estimate (1.5) follows immediately by combining the estimates for A sol and ∇ϕ L ∞ (Ω) .
Finally, we would like to point out that the techniques utilized to prove the main results in this paper do not enable us to establish estimates (1.5) and (1.6) simultaneously.Due to the Du 2 term in the integral inequality (3.14), there will be a term of magnitude h −1 after the CGO solutions (3.12) and (3.13) are substituted into the left-hand side of (3.14).To medicate this issue, we need to multiply the left-hand side of (3.14) by h.This step is done in the proof of Proposition 3.2.It then follows from estimate (3.32) that the term involving the electric potential q on the left-hand side of (3.14) vanishes as h → 0, which prevents us from achieving simultaneous reconstruction of both potentials A and q.
This paper is organized as follows.In Section 2 we state a result pertaining the existence of CGO solutions, as well as a quantitative version of the Riemann-Lebesgue lemma, which will be applied in the proof.Section 3 is devoted to proving Theorem 1.1, the estimate for the magnetic potential.Finally, we verify the estimates concerning the electric potential, i.e., Theorem 1.2 and Corollary 1.3, in Section 4.
Acknowledgments.We would like to express our gratitude to the anonymous referees for their valuable feedback, which led to significant improvements of the paper.

Preliminary Results
In this short section we collect some results that are necessary to establish the stability estimates in this paper.Let us start by stating a result concerning the existence of CGO solutions to the equation L A,q u = 0 in Ω of the form Here ζ ∈ C n is a complex vector satisfying ζ • ζ = 0, h > 0 is a small semiclassical parameter, a is a smooth amplitude, and r is a correction term that vanishes in a suitable sense in the limit h → 0. We refer readers to see [24,26,42] for detailed discussions and proofs.In the following proposition, we equip the domain Ω with a semiclassical Sobolev norm 1) , where and ζ (1) = O(h) as h → 0. Then for all h > 0 small enough, there exists a solution u ∈ H 4 (Ω) to the equation L A,q u = 0 in Ω of the form where a ∈ C ∞ (Ω) satisfies the transport equation and r H 1 scl (Ω) = O(h) as h → 0. We also need the following quantitative version of the Riemann-Lebesgue lemma to control the error terms involving Fourier transforms.This result was originally proved in [14, Lemma 2.4].

Estimate for the Magnetic Potential
The goal of this section is to prove Theorem 1.1.We shall start this section by constructing CGO solutions vanishing on the inaccessible part Γ 0 of the boundary, followed by establishing an estimate for the magnetic field dA.Finally, we utilize a special Hodge decomposition to verify estimate (1.5).
Let µ (1) , µ (2) be vectors in R n such that Then direct computations show that We now set where 0 < h ≪ 1 is a small semiclassical parameter such that 1 given in Proposition 2.1, we have (2) and ζ (0) 2 = µ (1) − iµ (2) .In order to fulfill the conditions in (3.1), similar to [23,25,36,42], we reflect Ω with respect to the hyperplane {x n = 0} and denote this reflection by We also reflect the magnetic potential A = (A 1 , . . ., A n ) and the electric potential q with respect to {x n = 0}.Specifically, we shall make even extensions for A j , j = 1, . . ., n − 1, and make an odd extension for A n .That is, we set In the same way, we extend electric potential q evenly to write As explained in [42, Section 5], we have (

3.7)
There also exists a solution u 2 ∈ H 4 (B) to the equation L A (2) , q 2 u 2 = 0 in B given by Here the amplitudes a j ∈ C ∞ (B), j = 1, 2, solve the transport equations ((µ (1) + iµ (2) ) • ∇) 2 a 1 (x, µ (1) + iµ (2) ) = 0 and ((µ (1) − iµ (2) respectively, and the correction terms r j ∈ H 4 (B), j = 1, 2, satisfy the estimate By the interior elliptic regularity, we have v, u 2 ∈ H 4 (Ω ∪ Ω * ) and We now proceed to construct CGO solutions in Ω satisfying the conditions in (3.1).To that end, we set , where v and u 2 are CGO solutions given by (3.7) and (3.8), respectively.Then we have v, u 2 ∈ H 4 (Ω).Furthermore, direct computations yield that v and u 2 satisfy the conditions in (3.1).Hence, the explicit forms of the desired CGO solutions v and u 2 in Ω are given by and 2 2 3.2.Derivation of an estimate for the magnetic field.The goal of this subsection is to utilize the CGO solutions (3.12) and (3.13) to derive an estimate for the magnetic field d(A (2) − A (1) ).Our starting point is the following integral inequality.
Proof.We begin the proof by recalling the Green's formula, which holds for all u, v ∈ H 4 (Ω), see [20]: (3.15)Here L * A,q is the L 2 -adjoint of L A,q , and ν is the outward unit normal to the boundary ∂Ω.Let u j , j = 1, 2, be solutions to the boundary value problem (1.2) with coefficients A (j) , q j , respectively.By a direct computation, we see that the function u := u 1 − u 2 solves the following equation Multiplying both sides of the first equation in (3.16) by v and applying the Green's formula (3.15), we obtain the integral identity where we have also used supp v, supp (∆v) ⊆ Γ.
Let us now estimate the right-hand side of (3.17).To that end, we apply the Cauchy-Schwartz inequality and the trace theorem to deduce To proceed, we need to estimate the H s -norms of u 2 , v, and their derivatives appearing in the inequality above.Since Ω is a bounded domain, there exists a constant R > 0 such that Ω ⊆ B(0, R).Thus, we have the inequality e From here, we obtain the claimed estimate (3.14) from the inequality above by using the fact that 1  h ≤ e R h .This completes the proof of Lemma 3.1.
We are now ready to state and prove an estimate for magnetic field.
Proof.We shall derive (3.20) by estimating the Fourier transform of d(A (2) − A (1) ).To that end, we extend A (j) and q j , j = 1, 2, by zero to R n \ Ω and denote the extensions by the same letters.It follows from [1, Chapter 1, Theorem 3.4.4]that A (j) ∈ W 1,∞ (R n ; C n ).For convenience, we shall denote A := A (2) − A (1) and q := q 2 − q 1 for the remainder of this section.
We now substitute the CGO solutions v and u 2 , given by (3.12) and (3.13), respectively, into the left-hand side of the integral inequality (3.14), multiply the resulted expression by h, and let h → 0. This requires us to compute the quantities vDu 2 and vu 2 , which contain expressions of the form where Hence, we write the first term on the left-hand side of (3.14) as ) where ) .We now investigate the limit of each integral in (3.23) as h → 0. To analyze I 1 , we recall the form of ζ 2 in (3.4) and the definition of A from (3.5), in conjunction with making a change of variables, to deduce that By replacing the vector µ (2) in (3.24) by −µ (2) , we have Hence, in view of (3.24) and (3.25), we see that for any µ ∈ Span{µ (1) , µ (2) } and all ξ ∈ R n satisfying (3.3).By choosing a 1 = a 2 = 1, which clearly satisfy the respective transport equation in (3.9), we get where d A is a 2-form defined by (1.4).
Turning attention to I 2 , we argue similarly as above to obtain (3.28) for any µ ∈ Span{µ (1) , µ (2) } and all ξ ∈ R n that satisfy (3.3).By Lemma 2.2 and the definition of ξ ± given by (3.22), there exist constant C > 0, ε 0 > 0, and 0 < α < 1 such that the estimate Finally, we estimate I 3 .Note that estimate (3.11) yields Since the amplitudes a j ∈ C ∞ (Ω), j = 1, 2, we utilize estimates (3.10) and (3.30), in conjunction with the Cauchy-Schwartz inequality, to deduce that We next proceed to estimate the second term on the left-hand side of the integral inequality (3.14).To that end, after substituting the CGO solutions v and u 2 given by (3.12) and (3.13), respectively, into (3.14),we have we apply estimate (3.10) and the Cauchy-Schwartz inequality to conclude that the following estimate is valid  .33)We are now ready to derive a stability estimate for dA H −1 (Ω) .With a parameter ρ > 0 to be specified later, let us consider the set Then an application of the Parseval's formula gives us Let us estimate the terms on the right-hand side of (3.35).For the second term, by the Plancherel theorem, we have (3.36) We now turn to estimate the first term.Observe that the inequality |ξ| 2 ≤ 2ρ 2 holds for all ξ ∈ E(ρ).Hence, we get is of zero Lebesgue measure, we estimate the first term on the right-hand side of (3.35) as follows: where B ′ (0, ρ) is a ball in R n−1 centered at the origin with radius ρ.
We next follow the arguments in [14, Section 3] closely to estimate the integral To that end, by choosing ε > 0 such that ε = √ h and using the polar coordinates, we obtain where we have made a change of variables r = h Furthermore, since 0 < h ≪ 1, n−1 2 ≥ 1, and 0 < α < 1, we have Let us now choose ρ such that ρ n h α = 1 ρ 2 , i.e., ρ = h − α n+2 .Then we obtain .
Since nα n+2 < 2, we get From here, the claimed estimate (3.20) follows immediately from (3.39) by an application of the inequality This completes the proof of Proposition 3.2.
For later purposes we need an estimate for dA L ∞ (Ω) .Recall that we assumed an a priori bound for A H s (Ω) , s > n 2 + 1, in the definition of the admissible set A(Ω, M).Thus, there exists a constant η > 0 such that s = n 2 + 2η.Hence, we apply the Sobolev embedding theorem and the interpolation theorem to derive the following estimate: .
(3.40) 3.3.Estimating the magnetic potential.In this subsection we verify estimate (1.5), thus complete the proof of Theorem 1.1.The key step is to apply a special Hodge decomposition of a vector field [38,Theorem 3.3.2].Lemma 3.3.Let Ω ⊆ R n , n ≥ 3, be a bounded domain with smooth boundary ∂Ω, and let A ∈ W 1,∞ (Ω, C n ) be a complex-valued vector field.Then there exist uniquely determined vector field Furthermore, the following inequalities are valid, where the constant C is independent of A.
From here, we have the claimed estimate (3.44) by similar computations as in estimate (3.40).This completes the proof of Lemma 3.4.
We are now ready to verify estimate (1.5).In view of estimates (3.43) and (3.44), we have .52) where we have used the inequality 1 in the last step.

.53)
With this choice of h, we have Hence, by substituting (3.53) into (3.52),we get the estimate Thus, we have verified estimate (1.5).

Estimates for the Electric Potential
This section is devoted to proving the estimates for the electric potential, namely, (1.6) and (1.7).For convenience, in what follows we shall again denote A = A (2) − A (1) and q = q 2 − q 1 .We begin this section by recalling the integral identity (3.17): Here v is a solution to the equation L * A (1) ,q 1 v = 0 in Ω, and u j , j = 1, 2, satisfies the equation L A (j) ,q j u j = 0 in Ω.
Let us now estimate the right-hand side of the integral identity above.To that end, we apply Lemma 3.1 to obtain To estimate the left-hand side of the integral identity, by the Cauchy-Schwartz inequality, estimate (3.19), and the inequality 1  h ≤ e R h , we have Hence, by combining the previous two estimates, we obtain We next substitute the CGO solutions (3.12) and (3.13) into the left-hand side of (4.1) and pass to the limit h → 0. To that end, by a direct computation, we get where ξ ± is given by (3.22), and ).Let us now analyze each integral in (4.2) in the limit h → 0. We extend q by zero on R n \ Ω and denote the extension by the same letter.For I 3 , since q ∈ Q(Ω, M), and the amplitudes a j ∈ C ∞ (Ω), j = 1, 2, we utilize estimate (3.11) to deduce Turning attention to I 1 , by choosing a 1 = a 2 = 1 and making a change of variables, we have where the function q is the even extension of q given by (3.6).We finally estimate I 2 .To that end, we set a 1 = a 2 = 1 again to obtain q(−e −ix•ξ + − e −ix•ξ − )dx = −F ( q)(ξ + ) − F ( q)(ξ − ).
From here, the claimed estimate (1.6) follows from the estimate above and the inequality q H −1 (Ω) ≤ q H −1 (R n ) .

. 1 )
Let us first introduce some notations.In what follows we shall denote x