Uniqueness in the Calderón problem and bilinear restriction estimates
Introduction
For , let be a bounded domain with Lipschitz boundary, and let denote the set of all functions satisfying in Ω for some . Throughout the paper, we assume . For and , we consider the Dirichlet problem: Let denote the outward normal derivative on the boundary ∂Ω. The Dirichlet-to-Neumann map is formally defined by . Since the boundary value problem (1.1) has a unique solution (for example, see [14, Theorem 2.52]), by the trace theorem and Green's formula, the operator can be formulated in the weak sense. Precisely, for and , where and . It is well known that is well defined and is continuous from to .
Calderón's problem Calderón's inverse conductivity problem concerns whether γ can be uniquely determined from , that is to say, whether the map is injective. The problem was introduced by Calderón [9] who showed uniqueness for the linearized problem. Afterwards, numerous works have been devoted to extending the function class for which the map is injective ([47]). Kohn and Vogelius [26] showed that if ∂Ω is smooth and then γ vanishes to infinite order at ∂Ω provided that (also, see [40]). Consequently, the mapping is injective if we choose to be the space of analytic functions on . Sylvester and Uhlmann in their influential work [39] proved that γ is completely determined by if for . They made use of the complex geometrical optics solutions which become most predominant tool not only in the Calderón problem but also in various related problems. Afterward, it has been shown that regularity on conductivity can be lowered further. The regularity assumption was relaxed to by Brown [5]. Päivärinta, Panchenko, and Uhlmann [37] showed global uniqueness of conductivities in , and results with conductivities in , were obtained by Brown and Torres [7]. Nguyen and Spirn [36] obtained a result with conductivities in for when . In two dimensions, the problem has different nature and uniqueness of conductivity was established by Astala and Päivärinta [2]. Their result is an extension of the previous ones in [35], [8]. Recently, Cârstea and Wang [11] obtained uniqueness of unbounded conductivities. (See [1] and references therein for related results.) For , the regularity condition was remarkably improved by Haberman and Tataru [21]. By making use of Bourgain's type spaces, they proved uniqueness when , or with the assumption that is small. This smallness assumption was later removed by Caro and Rogers [10].
As already mentioned before, for , most of the previous results were obtained under the assumption that γ has bounded gradient. Since the equation can be rewritten as with , it naturally relates to the unique continuation problem for u satisfying . Meanwhile, it is known that the unique continuation property holds with [48] and generally fails if for [25]. In this regards Brown [7] proposed a conjecture that uniqueness should be valid for , but no counterexample which shows the optimality of this conjecture has been known yet. Recently, Brown's conjecture was verified by Haberman [19] for , and he also showed that uniqueness remains valid even if ∇γ is unbounded when . More precisely, he showed that is injective if γ belongs to with and for , and and for .
For a given function q, let be the multiplication operator and let be the orthonormal group in . Most important part of the argument in Haberman [19] ([21]) is to show that there are sequences in and in such that and as . We refer the reader forward to Section 2 for the definition of the spaces and . If and has compact support, it is not difficult to show , see Remark 5. However, does not behave as nicely as . This is also related to the failure of the Carleman estimate of the form when . See [24], [3], [48], [22]. To get around the difficulty averaged estimates over and τ were considered ([21], [36], [19]). In view of Wolff's work [48] it still seems plausible to expect (1.2) or its variant holds with .
Restriction estimate Let 1 be a smooth compact hypersurface with nonvanishing Gaussian curvature and let dμ be the surface measure on S. The estimate , is known as the Stein-Tomas theorem. The range of r is optimal since the estimate fails if . The restriction estimate can be rewritten in its adjoint form: The restriction conjecture is to determine for which (1.3) holds. Even for most typical surfaces such as the sphere and the paraboloid, the conjecture is left open when . We refer the reader to [17], [18] for the most recent progress. There have been bilinear and multilinear generalizations of the linear estimate (1.3) under additional transversality conditions between surfaces ([43], [41], [4]), and these estimates played important roles in development of the restriction problem. To be precise, let be hypersurfaces in and let , be the surface measures on , , respectively. The following form of estimate is called bilinear (adjoint) restriction estimate: Under certain condition between and the estimate (1.4) remains valid for some with which (1.3) fails if . (See Theorem 2.5 and [42], [29] for detailed discussion.)
By duality, in order to get estimate for , we consider the bilinear operator which is given by Compared with the previous results the main new input in [19] was the -Fourier restriction theorem for the sphere which is due to Tomas [45] and Stein [38] (Theorem 2.3). This is natural in that the multiplier which defines has mass concentrated near the surface given by (2.1) while the restriction estimate provides estimates for functions of which Fourier transform concentrates near hypersurface. The use of the bilinear restriction estimate instead of the linear one has a couple of obvious advantages. The bilinear restriction estimate not only has a wider range of boundedness but also naturally fits with the bilinear operator .
In this paper we aim to improve Haberman's results by making use of the bilinear restriction estimate (1.4) for the elliptic surfaces (see Definition 2.2 and Theorem 2.5). However, the bilinear estimates outside of the range of the restriction estimate are only true under the extra separation condition between the supports of Fourier transforms of the functions (see Corollary 2.6). Such estimates cannot be put in use directly. This leads to considerable technical involvement. The following is our main result.
Theorem 1.1 Let and Ω be a bounded domain with Lipschitz boundary. Then the map is injective if for , where Here, is the Sobolev-Slobodeckij space (cf. Fig. 1).
In particular, uniqueness holds for if and , and for if and . Since if is small enough, this result is not covered by the result in [10].
Even though our estimates are stronger than those in [19], the estimates do not immediately yield improved results in every dimension. As is to be seen later in the paper, our estimates for the low frequency part are especially improved but this is not the case for the high frequency part since we rely on the argument based on the properties of spaces ([21], [19]).
The argument based on the complex geometrical optics solutions shows that the Fourier transforms of , , are identical as long as . As was indicated in [19] this approach has a drawback when we deal with less regular conductivity. In order to use the Fourier transform one has to extend to the whole space such that on . Such extension is possible by exploiting the trace theorem ([30, Theorem 1]) but only under the condition . This additional restriction allows new results only for in Theorem 1.1.2 The same was also true with the result in [19]. However, as is mentioned in [19], if we additionally impose the condition on the boundary ∂Ω, then Theorem 1.1 can be extended to higher dimensions . In fact, by [30, Theorem 1], the restriction can be relaxed so that , which is valid for for and . See Remark 4 for the value of . However, the additional condition on the boundary is not known to be true under the assumption for as in Theorem 1.1. (In [7] Brown and Torres proved that if , then on ∂Ω for , , and .)
If we had the endpoint bilinear restriction estimate (i.e., the estimate (1.4) with , see Remark 1), the argument in this paper would allow us to obtain the uniqueness result with and when , and with and when . Unfortunately the endpoint bilinear restriction estimate is still left open.
Inverse problem for the Schrödinger operator For , let be a bounded domain with boundary. We now consider the Dirichlet problem:
Let us set Here is the Bessel potential space, see Notations for its definition. Since is defined by Fourier multipliers, the space is more convenient for dealing with various operators which are defined by Fourier transform. If we disregard ϵ–loss of the regularity, the spaces and are essentially equivalent because and provided . (See [46, Section 2.3] for more details.) Thus, the statement of Theorem 1.2 does not change if is replaced by .
Let with satisfying (1.8). We assume that zero is not a Dirichlet eigenvalue of . Then, by the standard argument ([31]) we see that there is a unique solution for every . In fact, this can be shown by a slight modification of the argument in [27, Appendix A] (also see the proof of Lemma 6.4 where is controlled while and ).
For let denote the Dirichlet-to-Neumann map given by where u is the unique solution to (1.6) and with . As in the Calderón problem, one may ask whether is injective. As is well known the problem is closely related to the Calderón problem. In fact, the Calderón problem can be reduced to the inverse problem for with (see [39]). The problem of injectivity of was originally considered with , but it is not difficult to see that we may consider with . (See, for example, Brown-Torres [7].) Since u, , it is natural to impose . In fact, is well defined provided that with The standard argument shows that is continuous.
The injectivity of the mapping was shown with , by Sylvester and Uhlmann [39]. The result was extended to include unbounded potential by Jerison and Kenig (see Chanillo [12]). The injectivity for q contained in the Fefferman-Phong class with small norm was shown by Chanillo [12] and the result for was announced by Lavine and Nachman in [34].3 Their result was recently extended to compact Riemannian manifolds by Dos Santos Ferreira, Kenig, and Salo [13]. Also see [27] for extensions to the polyharmonic operators.
The regularity requirement for q can be relaxed. Results in this direction were obtained by Brown [5], Päivärinta, Panchenko, and Uhlmann [37], Brown and Torres [7] in connection with the Calderón problem. Those results can be improved to less regular q. In fact, Haberman's result implies that the injectivity holds with when (see [20]). It seems natural to conjecture that the same is true in any higher dimensions. Interpolating this conjecture with the result due to Lavine and Nachman [34] () leads to the following:
Conjecture 1 Let , and Ω be a bounded domain with Lipschitz boundary. Suppose and . If , then .
We define . For , set and, for , set
The following is a partial result concerning Conjecture 1 when . Theorem 1.2 Let and . The mapping (1.9) is injective if .
When , Theorem 1.2 shows that for or , where Conjecture 1 is verified except for the critical case . Similarly, when injectivity of (1.9) holds if and . We illustrate our result in Fig. 2.
Organization of the paper In Section 2, we recall basic properties of the spaces , and obtain estimates which rely on linear and bilinear restriction estimates. In Section 3 which is the most technical part of the paper, we make use of the bilinear restriction estimates and a Whitney type decomposition to get the crucial estimates while carefully considering orthogonality among the decomposed pieces. In Section 4 we take average of the estimates from Section 3 over dilation and rotation, which allows us to exploit extra cancellation due to frequency localization. Combining the previous estimates together we prove key estimates in Section 5. We prove Theorem 1.1 and Theorem 1.2 in Section 6.
Notations We list notations which are used throughout the paper.
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For we write if with some constant . We also use the notation if and .
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The orthogonal group in is denoted by .
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Let . For a function and a matrix we define . If U is the identity matrix, we denote by .
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The Fourier and inverse Fourier transforms: For an integrable function , we write and .
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For a measurable function a with polynomial growth, let .
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For and , we write .
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For and , we denote by the δ-neighborhood of E in , i.e., .
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We set . Also, for and , .
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If and are a pair of orthonormal vectors in , we write , .
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For we sometimes write , .
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For and , we denote by the Bessel potential space which is endowed with the norm .
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For and , we denote by , , the dual space of .
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We use and to denote the inner product and the bilinear pairing between distribution and function, respectively.
Section snippets
spaces and linear and bilinear restriction estimates
In this section, we recall basic properties of the spaces and linear and bilinear restriction estimates, which are to be used later.
Bilinear estimates
As mentioned in the introduction, we regard as a bilinear operator and attempt to obtain estimates while . In order to make use of the restriction estimates and its variants we work in frequency local setting after rescaling . This enables us to deal with instead of which varies along τ. In this section we use the estimates in the previous section to obtain estimates for in terms of .
Average over rotation and dilation
In this section, we consider the average of over and . The projection operators engaged in the definition of break the Fourier support of f into small pieces. Average over and makes it possible to exploit such smallness of Fourier supports. This gives considerably better bounds which are not viable when one attempts to control for a fixed e with .
For an invertible matrix U, let us define the projection operator
Key estimates: asymptotically vanishing averages
In this section, we assemble the various estimates in the previous sections and obtain the estimates that are the key ingredients for the proofs of Theorem 1.1 and Theorem 1.2.
Proposition 5.1 Let , , , and let with . Suppose (3.47) holds. Then, we have
Therefore, by Proposition 3.8, the estimate (5.1) holds provided that , , , , and . Moreover, when
Proof of Theorem 1.1 and Theorem 1.2
Once we have the key estimates in the previous sections, we can prove Theorem 1.1 and Theorem 1.2 following the argument in [19], which also relies on the basic strategy due to Sylvester-Uhlmann [39], and subsequent modifications due to Haberman-Tataru [21] and Nguyen-Spirn [36]. We begin with recalling several basic theorems which we need in what follows.
Acknowledgement
This work was supported by NRF-2017R1C1B2002959 (S. Ham), a KIAS Individual Grant (MG073702) at Korea Institute for Advanced Study, NRF-2020R1F1A1A0107352012 (Y. Kwon), and NRF-2021R1A2B5B02001786 (S. Lee).
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