Stability estimates in a partial data inverse boundary value problem for biharmonic operators at high frequencies

We study the inverse boundary value problems of determining a potential in the Helmholtz type equation for the perturbed biharmonic operator from the knowledge of the partial Cauchy data set. Our geometric setting is that of a domain whose inaccessible portion of the boundary is contained in a hyperplane, and we are given the Cauchy data set on the complement. The uniqueness and logarithmic stability for this problem were established in [ 37 ] and [ 7 ], respectively. We establish stability estimates in the high frequency regime, with an explicit dependence on the frequency parameter, under mild regularity assumptions on the potentials, sharpening those of [ 7 ].

Such fourth order operators arise in the context of modeling of hinged elastic beams and suspension bridges, see [13]. Associated to the equation (1.1) and the open portion Γ of ∂Ω, we introduce the partial Cauchy data set Here and in what follows, ν is the unit outer normal to ∂Ω, and H s (Ω) = {U | Ω : U ∈ H s (R n )}, s ∈ R, is the standard Sobolev space on Ω, see [1,Chapter 5].
In this paper we are concerned with the partial data inverse problem of recovering the potential q in Ω from the knowledge of the partial Cauchy data set C Γ q (k) at a fixed frequency k ≥ 0. The global uniqueness for this problem was established in [37]. Stability estimates of logarithmic type complementing the uniqueness result were obtained in [7], when k = 0, under the assumption that q ∈ H s (Ω), for some s > n 2 . Thanks to the work [33], logarithmic stability estimates are expected to be optimal for such inverse boundary value problems when the operator ∆ 2 is replaced by ∆ and k = 0, even when the full Cauchy data set C ∂Ω q (k) is given. We refer to the works [3], [4], [5], [9], [14], [19], [20], [27], [28], [29], [35], among others, for the study of inverse boundary value problems for perturbed biharmonic operators.
The logarithmic stability estimates established in [7] indicate that the inverse boundary value problem in question is severely ill-posed, which makes it most challenging to design reconstruction algorithms with high resolution in practice, since small errors in measurements may result in exponentially large errors in the reconstruction of the unknown potential. Nevertheless, it has been observed numerically that the stability may increase when the frequency k of the problem becomes large, see [10]. Stability estimates at high frequencies as well as the phenomenon of increasing stability in the high frequency regime for several fundamental inverse boundary value problems, have been studied rigorously in [18], [22] [23], [24], [25], [26], among others, in the full data case. In the case of partial data inverse boundary value problems, the question of deriving stability estimates at large frequencies and understanding the increasing stability phenomenon has only been studied in the works [8], [31], and [32], all in the case of Schrödinger operators, to the best of our knowledge.
The goal of this paper is to study the issue of deriving stability estimates in the high frequency regime for the partial data inverse problem formulated above, for the perturbed biharmonic operator. To state our results, let q 1 , q 2 ∈ (L ∞ ∩ H s )(Ω), for some 0 < s < 1/2. We define the distance between the partial Cauchy data sets as follows, The main result of this paper is as follows.
be a bounded open set with C ∞ boundary. Assume that Γ 0 = ∂Ω ∩ {x n = 0} is non-empty, and let Γ = ∂Ω \ Γ 0 . Let M > 0, 0 < s < 1/2, and let q 1 , q 2 ∈ (L ∞ ∩ H s )(Ω) be such that q j L ∞ (Ω) + q j H s (Ω) ≤ M , j = 1, 2. Then there is a constant C > 0 such that for all k ≥ 1, 0 < δ := dist(C Γ q1 (k), C Γ q2 (k)) < 1/e, we have Here 0 < α = (n−1)s (2s+n−1)(n+2) < 1 2 and C > 0 depends on Ω, M , s, but is independent of k. Remark 1.2. To the best of our knowledge, Theorem 1.1 seems to be the first stability estimate at high frequencies in the context of inverse boundary value problems for higher order elliptic PDE. Remark 1.3. Theorem 1.1 implies also a stability result for a fixed frequency k, which is sharper than the stability result of [7], in terms of the regularity of the potentials. Indeed, in [7] one assumes that the potentials are of class H s (Ω), s > n 2 , whereas in Theorem 1.1, the (L ∞ ∩ H s )(Ω)-regularity, 0 < s < 1/2, suffices. In particular, no continuity of the potentials in Theorem 1.1 is assumed, and the required Sobolev regularity assumptions are fairly mild and are independent of the dimension.
Remark 1.4. The partial data inverse problem in the particular setting considered in this paper, i.e. when the inaccessible portion of the boundary of the domain Ω is a part of the hyperplane and the measurements are performed on its complement, was first studied in [21] in the context of the Schrödinger equation. The uniqueness result of [21] was complemented by the logarithmic stability estimates of [16] for potentials of class H s (Ω), s > n 2 , see also [6]. High frequency stability results were obtained in [8], for potentials of the same class, and increasing stability estimates were obtained in [32] for potentials of class C 1 (Ω), assuming also the potentials agree near the flat portion of the boundary. We would like to emphasize that the method of the proof of Theorem 1.1 allows one to improve the regularity assumptions on the potentials to (L ∞ ∩ H s )(Ω), 0 < s < 1/2, in all of the stability and increasing stability results for the Schrödinger equation. This is accomplished by deriving and exploiting a quantitative version of the Riemann-Lebesgue lemma, under mild regularity assumptions on the potentials, whereas the aforementioned works rely on a quantitative version of the Riemann-Lebesgue lemma valid for L 1 potentials with the L 1 -modulus of continuity of Hölder type, established in [16].
The strategy of the proof of Theorem 1.1, going back to [21], consists of using a reflection argument to construct complex geometric optics solutions to the biharmonic equation, which vanish on the flat portion of the boundary. A fundamental role in [21] is also played by the Riemann-Lebesgue lemma, which should be sharpened to a quantitative statement when deriving stability estimates. Here we establish an accurate version of the Riemann-Lebesgue lemma for compactly supported elements of H s (R n ), 0 < s < 1/2. The advantage of working with potentials of such Sobolev regularity, which we exploit, comes from the well known fact that the operator of extension by zero takes H s (Ω) to H s (R n ), 0 < s < 1/2, boundedly, see [1, Theorem 5.1.10] and [11]. It is thanks to this observation that we are able to establish Theorem 1.1 under fairly mild regularity assumptions.
Assuming that the potentials q 1 and q 2 enjoy some additional regularity properties and a priori bounds, we obtain the following corollary of Theorem 1.1.
This paper is organized as follows. Section 2 contains a construction of complex geometric optics solutions to Helmholtz type equations for perturbed biharmonic operators by means of a Fourier series approach, extending the result of [17] in the Schrödinger case. Let us remark that the original approach to the construction of such solutions in [36] proceeds globally, by constructing the Faddeev Green function and exploiting techniques of the Fourier transformation in all of R n . See also [29] for an extension of such constructions to the case of polyharmonic operators with (unbounded) potentials. In Section 3 we establish a quantitative version of the Riemann-Lebesgue Lemma for compactly supported functions in H s (R n ), 0 < s < 1/2. The proofs of Theorem 1.1 and Corollary 1.5 are given in Section 4.

2.
Complex geometric optics solutions to Helmholtz type equations for perturbed biharmonic operators. We have the following result in the spirit of [17], see also [12], for the Helmholtz type equations for perturbed biharmonic operators. This result is useful here since all the constants are independent of the frequency k.
and k ≥ 0. Then there are constants C 0 > 0 and C 1 > 0, depending on Ω and n only, such that for all ζ ∈ C n , ζ · ζ = k 2 , and |Imζ| ≥ max{C 0 q L ∞ (Ω) , 1}, the equation has a solution of the form where r ∈ L 2 (Ω) satisfies Proof. We shall follow [17], see also [12]. Let ζ ∈ C n be such that ζ · ζ = k 2 . Then we have Writing ζ = w 1 + iw 2 , where w 1 , w 2 ∈ R n , and using the fact that ζ · ζ = k 2 , we see that w 1 · w 2 = 0. Performing an orthogonal transformation, we may therefore assume that w 1 = |w 1 |e 1 and w 2 = |w 2 |e 2 , where e 1 and e 2 are the first two vectors in the standard basis of R n . In order to solve the equation (2.3), let us first solve the following equation, where f ∈ L 2 (Ω). In doing so let us assume for simplicity that Ω ⊂ Q := [−π, π] n . In the following everything works without this extra assumption if we replace the set Ω by its image under the map R n x → κx ∈ R n for some fixed κ > 0 sufficiently small. Let us extend f by zero outside of Ω into Q, and let us denote this extension again by f . Thus, it suffices to solve the following equation, To that end, let v l (x) = e i(l+ 1 2 e2)·x , l ∈ Z n . The set (v l ) l∈Z n is an orthonormal basis in L 2 (Q), see [12], and therefore, f can be written as a series The reason for considering a shifted integer lattice rather than the standard integer coordinate lattice is due to the fact that the symbol of the operator in (2.5) is non-vanishing along the shifted lattice, see (2.6) below, and the equation (2.5) can therefore be solved by division.
We look for a solution of (2.5) in the form r = l∈Z n r l v l , and therefore, (2.5) leads to the following equation, We have Assume that |w 2 | = 0. Letting we see that and therefore, r L 2 (Q) ≤ 1 |w2| 2 f L 2 (Q) . Thus, we have shown that for any ζ ∈ C n such that ζ · ζ = k 2 and |Im ζ| = 0, the equation (2.4) has a solution operator, such that G ζ L(L 2 (Q),L 2 (Q)) ≤ 1 |Im ζ| 2 . Let us now return to the equation (2.3) and look for a solution in the form r = G ζ r, where r ∈ L 2 (Q) is to be determined. Then we get (I + qG ζ ) r = −q in L 2 (Q).
3. Quantitative version of the Riemann-Lebesgue lemma. The goal of this section is to prove a quantitative version of the Riemann-Lebesgue lemma for functions f ∈ H s (R n ), 0 < s < 1, with supp (f ) compact. Another closely related version of the Riemann-Lebesgue lemma is established in [16], where it is applied to zero extensions of Hölder continuous functions defined on smooth bounded domains.
The following result is a quantitative version of the Riemann-Lebesgue lemma.
Using Young's inequality, we see that f τ − f L 2 (R n ) ≤ 2 f L 2 (R n ) for τ ∈ (0, ∞). Combining this with the fact that supp (f ) is compact and using Lemma 3.1, we get for all 0 < τ < 1, This completes the proof of Proposition 3.2.
Multiplying the first equation in (4.2) by v, using the Green formula (4.1) and (4.4), we get Combining (4.3) and (4.6), we get Letting and therefore, (4.8) This completes the proof of Lemma 4.1.
An application of the trace theorem gives the following corollary of Lemma 4.1, see [1,Theorem 5.1.7].
We shall next construct complex geometric optics solutions to the equations which satisfy the following conditions, (4.12) u j | Γ0 = (∆u j )| Γ0 = 0, j = 1, 2. Following [21], in order to fulfill the condition (4.12), we reflect Ω with respect to the plane x n = 0 and denote this reflection by where x = (x 1 , . . . , x n−1 ). Let us denote by q even j the even extension of q j | R n − to R n + , By [1, Theorem 3.5.1], we see that q even Let B = B(0, R) be a ball in R n , centered at 0, of radius R ≥ 1, such that Ω∪Ω * ⊂⊂ B. By Proposition 2.1, there are constants C 0 > 0 and C 1 > 0, depending on B and n only, such that for |Imζ j | = a ≥ max{C 0 √ M , 1}, the equation has a solution of the form where r j ∈ L 2 (B) satisfies By the interior elliptic regularity, we have u j ∈ H 4 (Ω ∪ Ω * ), and in view of (4.14), we have for all k ≥ 1, see [15,Theorem 6.29]. It follows from (4.17) that Now let We have u j ∈ H 4 (Ω) and u j satisfies (4.11) and (4.12).
Letting δ := dist(C Γ q1 (k), C Γ q2 (k)), and using the fact that 0 < δ < 1 e , we finally choose This choice of a is motivated, on the one hand, by the constraint (4.34), and on the other hand by the requirement that the right hand side of the resulting bound should vanish as δ → 0 + , to be able to recover the uniqueness result.
It follows therefore from (4.33) that where ·, · Ω is the distributional duality on Ω. This completes the proof of Theorem 1.1.