The higher order fractional Calderón problem for linear local operators : Uniqueness

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY-NC-ND 4.0 https://creativecommons.org/licenses/by-nc-nd/4.0/ The higher order fractional Calderón problem for linear local operators : Uniqueness © 2022 Published by Elsevier Inc. Accepted version (Final draft) Covi, Giovanni; Mönkkönen, Keijo; Railo, Jesse; Uhlmann, Gunther Covi, G., Mönkkönen, K., Railo, J., & Uhlmann, G. (2022). The higher order fractional Calderón problem for linear local operators : Uniqueness. Advances in Mathematics, 399, Article 108246. https://doi.org/10.1016/j.aim.2022.108246 2022 THE HIGHER ORDER FRACTIONAL CALDER ´ ON PROBLEM FOR LINEAR LOCAL OPERATORS: UNIQUENESS GIOVANNI COVI, KEIJO M ¨ ONKK¨ ONEN, JESSE RAILO, AND GUNTHER UHLMANN Abstract. We study an inverse problem for the fractional Schr¨odinger equation (FSE) with a local perturbation by a linear partial differential operator (PDO) of order smaller than the one of the fractional Laplacian. We show that one can uniquely recover the coefficients of the PDO from the exterior Dirichlet-to-Neumann (DN) map associated to the perturbed FSE. This is proved for two classes of coefficients: coefficients which belong to certain spaces of Sobolev multipliers and coefficients which belong to fractional Sobolev spaces with bounded derivatives. Our study generalizes recent results for the zeroth and first order perturbations to higher order perturbations. 1. Introduction Let s∈R+\Z, Ω ⊂Rna bounded open set where n≥1, Ωe=Rn\Ω its exterior and P(x, D) a linear partial differential operator (PDO) of order m∈N P(x, D) = X |α|≤m aα(x)Dα where the coefficients aα=aα(x) are functions defined in Ω. We study a nonlocal inverse problem for the perturbed fractional Schr¨odinger equation (1) ((−∆)su+P(x, D)u= 0 in Ω u=fin Ωe where (−∆)sis a nonlocal pseudo-differential operator (−∆)su=F−1(|·|2sˆu) in contrast to the local operator P(x, D). In the inverse problem, one aims to recover the local operator Pfrom the associated Dirichlet-to-Neumann map. We always assume that 0 is not a Dirichlet eigenvalue of the operator ((−∆)s+P(x, D)), i.e. If u∈Hs(Rn) solves ((−∆)s+P(x, D))u= 0 in Ω and u|Ωe= 0,then u= 0. Our data for the inverse problem is the exterior Dirichlet-to-Neumann (DN) map ΛP:Hs(Ωe)→ (Hs(Ωe))∗which maps Dirichlet exterior values to a nonlocal version of the Neumann boundary value (see section 2 and 3.1). The main question that we study in this article is whether the exterior DN map ΛPdetermines uniquely the coefficients aαin Ω. In other words, does ΛP1= ΛP2imply that a1,α =a2,α in Ω for all |α| ≤ m? We prove that the answer is positive under certain restrictions on the coefficients aαand the order of the PDOs. This gives a positive answer to the uniqueness problem [12, Question 7.5] posed by the first three authors in a previous work. The precise statement in [12] asks to prove uniqueness for the higher order fractional Calder´on problem in the case of a bounded domain with smooth boundary and PDOs with smooth coefficients (up to the boundary). The positive answer to this question follows from theorem 1.4. The study of the fractional Calder´on problem was initiated by Ghosh, Salo and Uhlmann in the work [18] where the uniqueness for the associated inverse problem is proved when m= 0, s∈(0,1) and a0∈L∞(Ω). Date: January 14, 2022. Key words and phrases. Inverse problems, fractional Calder´on problem, fractional Schr¨odinger equation, Sobolev multipliers. 1 We briefly note that by Peetre’s theorem any linear operator L:C∞ c(Ω) →C∞ c(Ω) which does not increase supports, i.e. spt(Lf)⊂spt(f) for all f∈C∞ c(Ω), is in fact a differential operator [38] (see also the original work [40]). Therefore our results apply to any local operator satisfying such properties and it is enough to study PDOs only. For a more general formulation of Peetre’s theorem on the level of vector bundles, see [39]. 1.1. Main results. We denote by M(Hs−|α|→H−s) the space of all bounded Sobolev multipliers between the Sobolev spaces Hs−|α|(Rn) and H−s(Rn). We denote by M0(Hs−|α|→ H−s)⊂M(Hs−|α|→H−s) the space of bounded Sobolev multipliers that can be approximated by smooth compactly supported functions in the multiplier norm of M(Hs−|α|→H−s). We also write Hr,∞(Ω) for the local Bessel potential space with bounded derivatives. See section 2 for more detailed definitions. Our first theorem is a generalization of [44, Theorem 1.1] which considered the case m= 0 with s∈(0,1). It also generalizes [12, Theorem 1.5] which considered the higher order case s∈R+\Zwhen m= 0. Theorem 1.1. Let Ω⊂Rnbe a bounded open set where n≥1. Let s∈R+\Zand m∈Nbe such that 2s>m. Let Pj=X |α|≤m aj,αDα, j = 1,2, be linear PDOs of order mwith coefficients aj,α ∈M0(Hs−|α|→H−s). Given any two open sets W1, W2⊂Ωe, suppose that the exterior DN maps ΛPifor the equations ((−∆)s+Pj)u= 0 in Ωsatisfy ΛP1f|W2= ΛP2f|W2 for all f∈C∞ c(W1). Then P1|Ω=P2|Ω. In theorem 1.1 one can pick the lower order coefficients (|α|< s) from Lp(Ω) for high enough p (especially from L∞(Ω)) and higher order coefficients (s < |α|<2s) from the closure of C∞ c(Ω) in Hr,∞(Ω) for certain values of r∈R. In the following propositions, which are proved in Section 2, we give more examples of Sobolev spaces which belong to the space of multipliers M0(Hs−|α|→H−s): Proposition 1.2. Let Ω⊂Rnbe an open set and let t∈Rand r∈Rbe such that t > max{0, r}. The following inclusions hold: (i) e Hr0,∞(Ω) ⊂M0(H−r→H−t)whenever r0≥max{0, r}. (ii) Hr0,∞ 0(Ω) ⊂M0(H−r→H−t)whenever r0≥max{0, r}with r0/∈ {1 2,3 2,5 2, . . . }and Ωis a Lipschitz domain. (iii) e Hr0(Ω) ⊂M0(H−r→H−t)whenever r0≥tand r0> n/2. The same holds true for Hr0 Ω(Rn)if Ωis a Lipschitz domain, and for Hr0 0(Ω) when Ωis a Lipschitz domain and r0/∈ {1 2,3 2,5 2, . . . }. Note that the assumptions in theorem 1.1 satisfy the conditions of proposition 1.2 since then r=|α| − sand t=s. The next proposition gives examples of spaces of lower order coefficients (|α| ≤ s): Proposition 1.3. Let Ω⊂Rnbe an open set and t > 0. The following inclusions hold: (i) Lp(Ω) ⊂M0(H0→H−t)whenever 2≤p < ∞and p > n/t. Especially, if Ωis bounded, then L∞(Ω) ⊂M0(H0→H−t). (ii) e Hr(Ω) ⊂M0(H0→H−t)whenever r≥0and r > n/2−t. The same holds true for Hr Ω(Rn)if Ωis a Lipschitz domain, and for Hr 0(Ω) when Ωis Lipschitz domain and r /∈ {1 2,3 2,5 2, . . . }. 2 As mentioned above, we put t=s > 0 in theorem 1.1 and the condition in proposition 1.3 is satisfied. Note that under the assumption |α| ≤ swe have M0(H0→H−s)⊂M0(Hs−|α|→ H−s). Hence we can choose the lower order coefficients from a less regular space in theorem 1.1 (compare to proposition 1.2). We also note that when |α|= 0 the space of multipliers M0(Hs→ H−s) coincides with the one studied in [44]. It follows from lemma 2.5 that the space of multipliers is trivial for higher order operators, i.e. M(Hs−|α|→H−s) = {0}when s− |α|<−s. It would be possible to state theorem 1.1 for higher order PDOs, but that forces aα= 0 for all |α|>2s. For this reason we only consider PDOs whose order is m < 2s. See lemma 2.5 and the related remarks for more details. Our second theorem generalizes [8, Theorem 1.1] and [18, Theorem 1.1], where similar results are proved when m= 0,1 and s∈(0,1). It also generalizes [12, Theorem 1.5] where the case m= 0 and s∈R+\Zwas studied. Theorem 1.4. Let Ω⊂Rnbe a bounded Lipschitz domain where n≥1. Let s∈R+\Zand m∈Nbe such that 2s>m. Let Pj(x, D) = X |α|≤m aj,α(x)Dα, j = 1,2, be a linear PDOs of order mwith coefficients aj,α ∈Hrα,∞(Ω) where rα:= (0if |α| − s < 0, |α| − s+δif |α| − s∈ {1/2,3/2, ...}, |α| − sif otherwise (2) for any fixed δ > 0. Given any two open sets W1, W2⊂Ωe, suppose that the exterior DN maps ΛPifor the equations ((−∆)s+Pj(x, D))u= 0 in Ωsatisfy ΛP1f|W2= ΛP2f|W2 for all f∈C∞ c(W1). Then P1(x, D) = P2(x, D). Our first theorem is formulated for general bounded open sets and the second theorem for Lipschitz domains. The difference arises in the proof of the well-posedness of the forward problem. We note that theorem 1.4 holds for coefficients aαwhich are smooth up to the boundary (aα=g|Ωwhere g∈C∞(Rn)). The conditions (1.4) imply that one can choose aα∈L∞(Ω) for every αsuch that |α|< s. The case |α|=snever happens, as sis assumed not to be an integer. If |α|> s, we have aα∈H|α|−s,∞(Ω) when |α| − s6∈ {1/2,3/2, ...}. Thus the conditions (1.4) coincide with [8, 18] when m= 0,1 and s∈(0,1). Our article is roughly divided into two parts. The first part of the article (theorem 1.1 and section 3) generalizes the study of the uniqueness problem for singular potentials in [44] and the second part (theorem 1.4 and section 4) generalizes the uniqueness problem for bounded first order perturbations in [8]. The approach to prove theorems 1.1 and 1.4 is the following. First one shows that the forward problem is well-posed and the corresponding bilinear forms are bounded. This leads to the boundedness of the exterior DN maps and an Alessandrini identity. By a unique continuation property of the higher order fractional Laplacian one obtains a Runge approximation property for equation (1). Using the Runge approximation and the Alessandrini identity for suitable test functions one proves the uniqueness of the inverse problem. 1.2. On the earlier literature. Equation (1) and theorems 1.1 and 1.4 are related to the Calder´on problem for the fractional Schr¨odinger equation first introduced in [18]. There one tries to uniquely recover the potential qin Ω by doing measurements in the exterior Ωe. This is a nonlocal (fractional) counterpart of the classical Calder´on problem arising in electrical impedance tomography, where one obtains information about the electrical properties of some bounded domain by doing voltage and current measurements on the boundary [49, 50]. In [44] the study of the fractional Calder´on problem is extended for “rough” potentials q, i.e. potentials which are in general bounded Sobolev multipliers. First order perturbations were studied in [8] 3 assuming that the fractional part dominates the equation, i.e. s∈(1/2,1), and that the perturbations have bounded fractional derivatives. A higher order version (s∈R+\Z) of the fractional Calder´on problem was introduced and studied in [12]. These three articles [8, 12, 44] motivate the study of higher order (rough) perturbations to the fractional Laplacian (−∆)sin equation (1). The natural restriction for the order of P(x, D) in theorems 1.1 and 1.4 is then 2s>m, so that the fractional part governs the equation (1). The fractional Calder´on problem for s∈(0,1) has been studied in many settings. We refer to the survey [46] for a more detailed treatment. In the work [44] stability was proved for singular potentials, and in [43] the related exponential instability was shown. The fractional Calder´on problem has also been solved under single measurement [17]. The perturbed equation is related to the fractional magnetic Schr¨odinger equation which is studied in [10, 30, 31, 32]. See also [5] for a fractional Schr¨odinger equation with a lower order nonlocal perturbation. Other variants of the fractional Calder´on problem include semilinear fractional (magnetic) Schr¨odinger equation [24, 25, 30, 32], fractional heat equation [26, 45] and fractional conductivity equation [11] (see also [7, 16] for equations arising from a nonlocal Schr¨odinger-type elliptic operator). In the recent work [12], the first three authors of this article studied higher order versions (s∈R+\Z) of the fractional Calder´on problem and proved uniqueness for the Calder´on problem for the fractional magnetic Schr¨odinger equation (up to a gauge). This article continues these studies by showing uniqueness for the fractional Schr¨odinger equation with higher order perturbations and gives positive answer to a question posed in [12, Question 7.5]. 1.3. Examples of fractional models in the sciences. Equations involving fractional Laplacians like (1) have applications in mathematics and natural sciences. Fractional Laplacians appear in the study of anomalous and nonlocal diffusion, and these diffusion phenomena can be used in many areas such as continuum mechanics, graph theory and ecology just to mention a few [2, 6, 15, 34, 41]. Another place where the fractional counterpart of the classical Laplacian naturally shows up is the formulation of fractional quantum mechanics [27, 28, 29]. See [42] and references therein for possible applications of higher order fractional Laplacians. For more applications of fractional mathematical models, see [6, 37] and the references therein. 1.4. The organization of the article. In section 2 we introduce the notation and give preliminaries on Sobolev spaces and fractional Laplacians. We also define the spaces of rough coefficients (Sobolev multipliers) and discuss some of the basic properties. In section 3 we prove theorem 1.1 in detail. Finally, in section 4 we prove theorem 1.4 but as the proofs of both theorems are very similar we do not repeat all identical steps and we keep our focus in the differences of the proofs. Acknowledgements. G.C. was partially supported by the European Research Council under Horizon 2020 (ERC CoG 770924). K.M. and J.R. were supported by Academy of Finland (Centre of Excellence in Inverse Modelling and Imaging, grant numbers 284715 and 309963). G.U. was partly supported by NSF, a Walker Family Endowed Professorship at UW and a Si Yuan Professorship at IAS, HKUST. G.C. would like to thank Angkana R¨uland for helpful discussions and her hospitality during his visit to Max Planck Institute for Mathematics in the Sciences. J.R. and G.U. wish to thank Maarten V. de Hoop, Rice University and Simons Foundation for providing the support to participate 2020 MATH + X Symposium on Inverse Problems and Deep Learning, Mitigating Natural Hazards, where this project was initiated. 2. Preliminaries In this section we recall some basic theory of Sobolev spaces, Fourier analysis and fractional Laplacians on Rn. We also introduce the spaces of Sobolev multipliers and prove a few properties for them. Some auxiliary lemmas which are needed in the proofs of our main theorems are given as well. We follow the references [1, 18, 36, 35, 48, 51] (see also section 2 in [12]). 4 2.1. Sobolev spaces. The (inhomogeneous) fractional L2-based Sobolev space of order r∈R is defined to be Hr(Rn) = {u∈S0(Rn) : F−1(h·irˆu)∈L2(Rn)} equipped with the norm kukHr(Rn)= F−1(h·irˆu) L2(Rn). Here ˆu=F(u) is the Fourier transform of a tempered distribution u∈S0(Rn), F−1is the inverse Fourier transform and hxi= (1 + |x|2)1/2. We define the fractional Laplacian of order s∈R+\Zas (−∆)sϕ=F−1(|·|2sˆϕ) where ϕ∈S(Rn) is a Schwartz function. Then (−∆)s extends to a bounded operator (−∆)s:Hr(Rn)→Hr−2s(Rn) for all r∈Rby density of S(Rn) in Hr(Rn) [18, Lemma 2.1] (see also [12, Section 2.2]). It would be possible to define the fractional Laplacian in many other ways, according to the intended application (check e.g. [13, 23, 33]). In particular, our global definition of the fractional Laplacian using Fourier transform will be different from the spectral definition used in [21] (see for example [13, 47]). Let Ω ⊂Rnbe an open set and F⊂Rna closed set. We define the following Sobolev spaces Hr F(Rn) = {u∈Hr(Rn) : spt(u)⊂F} e Hr(Ω) = closure of C∞ c(Ω) in Hr(Rn) Hr(Ω) = {u|Ω:u∈Hr(Rn)} Hr 0(Ω) = closure of C∞ c(Ω) in Hr(Ω). It trivially follows that e Hr(Ω) ⊂Hr 0(Ω) and e Hr(Ω) ⊂Hr Ω(Rn). Further, we have ( e Hr(Ω))∗= H−r(Ω) and (Hr(Ω))∗=e H−r(Ω) for any open set Ω and r∈R[9, Theorem 3.3]. If Ω is in addition a Lipschitz domain, then we have e Hr(Ω) = Hr Ω(Rn) for all r∈Rand Hr 0(Ω) = Hr Ω(Rn) when r≥0 such that r /∈ {1 2,3 2,5 2. . . }[36, Theorems 3.29 and 3.33]. More generally, let 1 ≤p≤ ∞ and r∈R. We define the Bessel potential space Hr,p(Rn) = {u∈S0(Rn) : F−1(h·irˆu)∈Lp(Rn)} equipped with the norm kukHr,p(Rn)= F−1(h·irˆu) Lp(Rn). We also write F−1(h·irˆu) =: Jruwhere the Fourier multiplier J= (Id−∆)1/2is called the Bessel potential. We have the continuous inclusions Hr,p(Rn),→Ht,p(Rn) whenever r≥t[4, Theorem 6.2.3]. By the Mikhlin multiplier theorem one can show that (−∆)s:Hr,p(Rn)→Hr−2s,p(Rn) is continuous whenever s≥0 and 1 <p<∞(see [18, Remark 2.2] and [1, Theorem 7.2]). The local version of the space Hr,p(Rn) is defined as earlier by the restrictions Hr,p(Ω) = {u|Ω:u∈Hr,p(Rn)} where Ω ⊂Rnis any open set. This space is equipped with the quotient norm kvkHr,p(Ω) = inf{kwkHr,p(Rn):w∈Hr,p(Rn), w|Ω=v}. We have the continuous inclusions Hr,p(Ω) ,→Ht,p(Ω) whenever r≥tby the definition of the quotient norm. We also define the spaces Hr,p F(Rn) = {u∈Hr,p(Rn) : spt(u)⊂F} e Hr,p(Ω) = closure of C∞ c(Ω) in Hr,p(Rn) Hr,p 0(Ω) = closure of C∞ c(Ω) in Hr,p(Ω) 5 where F⊂Rnis a closed set. Note that e Hr,p(Ω) ⊂Hr,p 0(Ω) since the restriction map |Ω:Hr,p(Rn)→Hr,p(Ω) is by definition continuous. One can also see that e Hr,p(Ω) ⊂Hr,p Ω(Rn). If Ω is a bounded C∞-domain and 1 <p<∞, then we have [48, Theorem 1 in section 4.3.2] e Hr,p(Ω) = Hr,p Ω(Rn), r ∈R Hr,p 0(Ω) = Hr,p(Ω), r ≤1 p. Some authors (especially in [8, 44]) use the notation Wr,p(Ω) for Bessel potential spaces. We have decided to use the notation Hr,p(Ω) so that these spaces are not confused with the Sobolev-Slobodeckij spaces which are in general different from the Bessel potential spaces [14, Remark 3.5]. The equation (1) we study is nonlocal. Instead of putting boundary conditions we impose exterior values for the equation. This can be done by saying that u=fin Ωeif u−f∈e Hs(Ω). Motivated by this we define the (abstract) trace space X=Hr(Rn)/e Hr(Ω), i.e. functions in Xare the same (have the same trace) if they agree in Ωe. If Ω is a Lipschitz domain, then we have X=Hr(Ωe) and X∗=H−r Ωe(Rn) [18, p.463]. 2.2. Properties of the fractional Laplacian. The fractional Laplacian admits two important properties which we need in our proofs. The first one is unique continuation property (UCP) which is used in proving the Runge approximation property. Lemma 2.1 (UCP).Let s∈R+\Z,r∈Rand u∈Hr(Rn). If (−∆)su|V= 0 and u|V= 0 for some nonempty open set V⊂Rn, then u= 0. Lemma 2.1 is proved in [12] for s > 1 by reducing the problem to the UCP result for s∈(0,1) in [18]. Note that such property is not true for local operators like the classical Laplacian (−∆). The second property we need is the Poincar´e inequality, which is used in showing that the forward problem for the perturbed fractional Schr¨odinger equation is well-posed. Lemma 2.2 (Poincar´e inequality).Let s∈R+\Z,K⊂Rncompact set and u∈Hs K(Rn). There exists a constant c=c(n, K, s)>0such that kukL2(Rn)≤c  (−∆)s/2u  L2(Rn). Many different proofs for lemma 2.2 are given in [12]. We note that in the literature the fractional Poincar´e inequality is typically considered only when s∈(0,1). Finally, we recall the fractional Leibniz rule, also known as the Kato-Ponce inequality. It is used to show the boundedness of the bilinear forms associated to the perturbed fractional Schr¨odinger equation in the case when the coefficients of the PDO have bounded fractional derivatives. Lemma 2.3 (Kato-Ponce inequality).Let s≥0,1< r < ∞,1< q1≤ ∞ and 1< p2≤ ∞ such that 1 r=1 p1+1 q1=1 p2+1 q2. If f∈Lp2(Rn),Jsf∈Lp1(Rn),g∈Lq1(Rn)and Jsg∈Lq2(Rn), then Js(fg)∈Lr(Rn)and kJs(fg)kLr(Rn)≤C(kJsfkLp1(Rn)kgkLq1(Rn)+kfkLp2(Rn)kJsgkLq2(Rn)) where Jsis the Bessel potential of order sand C=C(s, n, r, p1, p2, q1, q2). The proof of lemma 2.3 can be found in [20] (see also [19, 22]). 2.3. Spaces of rough coefficients. Following [35, Ch. 3], we introduce the space of multipliers M(Hr→Ht) between pairs of Sobolev spaces. Here we are assuming that r, t ∈R. The coefficients of P(x, D) in theorem 1.1 will be picked from such spaces of multipliers. If f∈ D0(Rn) is a distribution, we say that f∈M(Hr→Ht) whenever the norm kfkr,t := sup{|hf, uvi| ;u, v ∈C∞ c(Rn),kukHr(Rn)=kvkH−t(Rn)= 1} 6 is finite. Here uv indicates the pointwise product of functions, while h·,·i is the duality pairing. If the distribution fhappens to be a function, the duality pairing can be defined as hf, uvi=ZRn f(x)u(x)v(x)dx. By M0(Hr→Ht) we indicate the closure of C∞ c(Rn) in M(Hr→Ht)⊂ D0(Rn). If f∈ M(Hr→Ht) and u, v ∈C∞ c(Rn) are both non-vanishing, we have the multiplier inequality (3) |hf, uvi| =*f, u kukHr(Rn) v kvkH−t(Rn)+kukHr(Rn)kvkH−t(Rn)≤ kfkr,t kukHr(Rn)kvkH−t(Rn). By the density of C∞ c(Rn)×C∞ c(Rn) in Hr(Rn)×H−t(Rn) with respect to the product norm k(u, v)k= max{kukHr(Rn),kvkH−t(Rn)}and estimate (2.3), there is a unique continuous extension of (u, v)7→ hf, uviacting on (u, v)∈Hr(Rn)×H−t(Rn). More precisely, each f∈M(Hr→Ht) gives rise to a linear multiplication map mf:Hr(Rn)→Ht(Rn) defined by hmf(u), vi:= lim i→∞hf, uiviifor all (u, v)∈Hr(Rn)×H−t(Rn), where (ui, vi)∈C∞ c(Rn)×C∞ c(Rn) is any Cauchy sequence in Hr(Rn)×H−t(Rn) converging to (u, v). The existence of the limit is granted by completeness and formula (2.3), which ensures that hf, uiviiis also a Cauchy sequence. In fact, we have |hf, umvmi−hf, unvni| ≤ |hf, um(vm−vn)i| +|hf, (um−un)vni| ≤ kfkr,t kumkHr(Rn)kvm−vnkH−t(Rn)+kum−unkHr(Rn)kvnkH−t(Rn) ≤ kfkr,t kumkHr(Rn)+kvnkH−t(Rn)k(um, vm)−(un, vn)k, where kumkHr(Rn)+kvnkH−t(Rn)is bounded by a constant independent of mand n. The independence of the limit on the particular sequence (ui, vi) can be showed by a similar estimate. We can analogously define the unique adjoint multiplication map m∗ f:H−t(Rn)→H−r(Rn) such that m∗ f(v), u:= lim i→∞hf, uiviifor all (u, v)∈Hr(Rn)×H−t(Rn). Since one sees that the adjoint of mfis m∗ f, the chosen notation is justified. For convenience, in the rest of the paper we will just write fu for both mf(u) and m∗ f(u). Remark 2.4. The spaces of rough coefficients we use are generalizations of the ones considered in [44]. In fact, the space Z−s(Rn)used there coincides with our space M(Hs→H−s). In the next lemma we state some elementary properties of the spaces of multipliers. Other interesting properties may be found in [35]. Lemma 2.5. Let λ, µ ≥0and r, t ∈R. Then (i) M(Hr→Ht) = M(H−t→H−r), and the norms associated to the two spaces also coincide. (ii) M(Hr−λ→Ht+µ),→M(Hr→Ht)continuously. (iii) M(Hr→Ht) = {0}whenever r < t. Proof. (i) Let f∈ D0(Rn) be a distribution. Then by just using the definition we see that kfkr,t = sup{|hf, uvi| ;u, v ∈C∞ c(Rn),kukHr(Rn)=kvkH−t(Rn)= 1} = sup{|hf, vui| ;v, u ∈C∞ c(Rn),kvkH−t(Rn)=kukH−(−r)(Rn)= 1}=kfk−t,−r. (ii) Observe that the given definition of kfkr,t is equivalent to the following: kfkr,t = sup{|hf, uvi| ;u, v ∈C∞ c(Rn),kukHr(Rn)≤1,kvkH−t(Rn)≤1}. 7 Since λ, µ ≥0, we also have kukHr−λ(Rn)≤ kukHr(Rn),kvkH−(t+µ)(Rn)≤ kvkH−t(Rn). This implies kfkr,t ≤ kfkr−λ,t+µ, which in turn gives the wanted inclusion. (iii) If 0 ≤r < t, then this was considered in [35, Ch. 3]. The proof given there recalls the easier one for Sobolev spaces ([35, Sec. 2.1]), which is based on the explicit computation of derivatives of aptly chosen exponential functions. If r < t ≤0, then by point (i) we have M(Hr→Ht) = M(H−t→H−r). We need to show that M(H−t→H−r) = {0}whenever 0 ≤ −t < −r. This reduces the problem back to the case of non-negative Sobolev scales. If r≤0< t, then −r≥0. Now by point (ii), we have M(Hr→Ht)⊆M(Hr+(−r)→ Ht) = M(L2→Ht). It is therefore enough to show that this last space is trivial, which again immediately follows from the case of non-negative Sobolev scales. If r < 0≤t, then the problem can be reduced again to the earlier cases.  Remark 2.6. We also have M0(Hr−λ→Ht+µ)⊆M0(Hr→Ht)whenever λ, µ ≥0, since the inclusion in (ii) is continuous. Remark 2.7. In light of lemma 2.5 (ii) we are only interested in M(Hr→Ht)in the case r≥t, the case r < t being trivial. For our theorem 1.1, this translates into the condition m≤2s. We decided not to consider the limit case m= 2sin this work, as our machinery (in particular, the coercivity estimate (4.1)) breaks down in this case. However, it should be noted that since by assumption we have m∈Zand s6∈ Z, the equality m= 2scan only arise if mis odd, which forces s= 1/2 + kwith k∈Z. This case was excluded in [8, 18] as well. Propositions 1.2 and 1.3 relate our spaces of multipliers with some special Bessel potential spaces. This is interesting since in the coming section 3 we will consider the inverse problem for coefficients coming from such spaces. We now prove those propositions. Proof of proposition 1.2. Throughout the proof we assume that u, v ∈C∞ c(Rn) such that kukH−r(Rn)= kvkHt(Rn)= 1. In parts (i) and (ii) we can assume that r0< t since if r0≥t, then we have the continuous inclusion Hr0,∞(Ω) ,→Hr00,∞(Ω) where max{0, r} ≤ r00 < t (such r00 always exists since t > max{0, r}). (i) Let f∈e Hr0,∞(Ω). Now f=f1+f2where f1∈C∞ c(Ω) and kf2kHr0,∞(Rn)≤. Then |hf2, uvi| ≤ kf2vkHr0(Rn)kukH−r0(Rn)≤Ckf2kHr0,∞(Rn)kvkHr0(Rn)kukH−r(Rn) ≤C kvkHt(Rn)=C. Here we used the Kato-Ponce inequality (lemma 2.3)   Jr0(f2v)  L2(Rn)≤C(kf2kL∞(Rn)  Jr0v  L2(Rn)+  Jr0f2  L∞(Rn)kvkL2(Rn)) ≤Ckf2kHr0,∞(Rn)kvkHr0(Rn) and the assumption max{0, r} ≤ r0< t. Therefore kf−f1k−r,−t=kf2k−r,−t≤C which shows that f∈M0(H−r→H−t). (ii) Let f∈Hr0,∞ 0(Ω). Now f=f1+f2where f1∈C∞ c(Ω) and kf2kHr0,∞(Ω) ≤. By the definition of the quotient norm k·kHr0,∞(Ω) we can take F∈Hr0,∞(Rn) such that F|Ω=f2 and kFkHr0,∞(Rn)≤2kf2kHr0,∞(Ω). The assumptions imply the duality (H−r0(Ω))∗=Hr0 0(Ω) ⊂ Hr0(Ω). Using the Kato-Ponce inequality for the extension Fwe obtain as in the proof of part (i) that  Jr0(Fv)  L2(Rn)≤CkFkHr0,∞(Rn)kvkHr0(Rn)≤2Ckf2kHr0,∞(Ω) kvkHt(Rn)≤2C and hence |hf2, uvi| ≤ kf2vk(H−r0(Ω))∗kukH−r0(Ω) ≤ kf2vkHr0(Ω) kukH−r(Rn) 8 where u1,k, u∗ 2,k ∈e Hs(Ω) respectively solve (−∆)su1,k +X |α|≤m a1,αDαu1,k = 0 in Ω, u1,k −f1,k ∈e Hs(Ω) and (−∆)su∗ 2,k +X |α|≤m (−1)|α|Dα(a2,αu∗ 2,k) = 0 in Ω, u∗ 2,k −f2,k ∈e Hs(Ω) and r1,k, r2,k →0 in e Hs(Ω) as k→ ∞. By the assumption on the exterior DN maps and the Alessandrini identity from lemma 3.8 we have 0 = h(ΛP1−ΛP2)[f1,k],[f2,k]i=X |α|≤m h(a1,α −a2,α),(Dαu1,k)u∗ 2,ki.(18) On the other hand, the support conditions imply that X |α|≤m h(a1,α −a2,α),(Dαu1,k)u∗ 2,ki=X |α|≤m h(a1,α −a2,α),(Dα(u1,k −f1,k))(u∗ 2,k −f2,k)i =X |α|≤m h(a1,α −a2,α),(Dα(v1+r1,k))(v2+r2,k)i. Thus by taking the limit k→ ∞ and using lemma 3.2, we obtain (19) X |α|≤m h(a1,α −a2,α),(Dαv1)v2i= 0 for all v1, v2∈C∞ c(Ω) by formula (3.2). Step 2. Assume that we have a1,α|Ω=a2,α|Ωfor all αsuch that |α|< N for some N∈N. We show that the equality of the coefficients also holds for αfor which |α|=N, and this will prove the theorem by the principle of complete induction. To this end, consider v2∈C∞ c(Ω), and then take v1∈C∞ c(Ω) such that v1(x) = xα on supp(v2)bΩ. Recall that since α= (α1, α2, ..., αn)∈Nnis a multi-index and x= (x1, x2, ..., xn)∈Rn, the symbol xαis intended to mean xα1 1xα2 2... xαn n. With this choice of v1, v2, equation (3.2) becomes 0 = X |β|≤m h(a1,β −a2,β),(Dβv1)v2i=X N≤|β|≤m h(a1,β −a2,β),(Dβxα)v2i(20) =X N<|β|≤m h(a1,β −a2,β),(Dβxα)v2i+X |β|=N, β6=α h(a1,β −a2,β),(Dβxα)v2i +h(a1,α −a2,α),(Dαxα)v2i. If |β|> N =|α|, then there must exist k∈ {1,2, ..., n}such that βk> αk. This is true also if |β|=Nwith β6=α. In both cases we can compute Dβ(xα) = (∂β1 x1xα1 1) (∂β2 x2xα2 2)... (∂βn xnxαn n) = 0 because ∂βk xkxαk k= 0. Therefore formula (3.2) becomes 0 = h(a1,α −a2,α),(Dαxα)v2i=α!ha1,α −a2,α, v2i which by the arbitrariety of v2∈C∞ c(Ω) implies a1,α|Ω=a2,α|Ωalso for αfor which |α|=N. Step 3. We have proved that a1,α|Ω=a2,α|Ωfor all αof order |α| ≤ m. Since this entails P1|Ω=P2|Ω, the proof is complete.  15 4. Main theorem for bounded coefficients We shall now study the case when the coefficients of PDOs are from the bounded spaces Hrα,∞(Ω). It should be noted, however, that most of the considerations of the previous section still apply identically. 4.1. Well-posedness of the forward problem. We shall define the bilinear forms for the problems (3.1) and (3.1) respectively by (3.1) and (3.1), just as in the case of singular coefficients. These will turn out to be bounded in Hs(Rn)×Hs(Rn) as well, but the proof we give of this fact is a fortiori different. Since now we assume that aα∈Hrα,∞(Ω) ⊂L∞(Ω) for rα≥0, the duality pairing haα,(Dαv)wibecomes an integral over Ω. Lemma 4.1 (Boundedness of the bilinear forms).Let Ω⊂Rnbe a bounded Lipschitz domain and s∈R+\Z,m∈Nsuch that 2s > m. Let aα∈Hrα,∞(Ω), with rαdefined as in (1.4). Then BPand B∗ Pextend as bounded bilinear forms on Hs(Rn)×Hs(Rn). Remark 4.2. Since s∈R+\Zand |α| ≤ m < 2s, we also have that max(0,|α| − s)≤rα< s for δ > 0small (see formula (1.4)). Proof of lemma 4.1. We only prove the boundedness of BP, as for B∗ Pone can proceed in the same way. If v, w ∈C∞ c(Rn), then |haα(x)Dαv, wi| =ZΩ aαw(Dαv)dx≤ kaαwk(H−rα(Ω))∗kDαvkH−rα(Ω). Since Ω is a Lipschitz domain and rα≥0, rα6∈ 1 2,3 2,5 2..., we have (H−rα(Ω))∗=Hrα 0(Ω) ⊂ Hrα(Ω). Therefore |haα(x)Dαv, wi| ≤ CkaαwkHrα(Ω)kDαvkH−rα(Ω) ≤CkAαwkHrα(Rn)kDαvkH−rα(Ω) (21) ≤CkJrα(Aαw)kL2(Rn)kvkH|α|−rα(Ω) where J= (Id −∆)1/2is the Bessel potential and Aαis an extension of aαfrom Ω to Rnsuch that Aα|Ω=aαand kAαkHrα,∞(Rn)≤2kaαkHrα,∞(Ω). Since rα≥0, we may estimate the last term of (4.1) by the Kato-Ponce inequality given in lemma 2.3 kJrα(Aαw)kL2(Rn)≤CkAαkL∞(Rn)kJrαwkL2(Rn)+kJrαAαkL∞(Rn)kwkL2(Rn) ≤CkAαkHrα,∞(Rn)kwkHrα(Rn)≤CkaαkHrα,∞(Ω)kwkHrα(Rn). Substituting this into (4.1) gives |haα(x)Dαv, wi| ≤ CkaαkHrα,∞(Ω)kwkHrα(Rn)kvkH|α|−rα(Ω) (22) ≤CkaαkHrα,∞(Ω)kwkHs(Rn)kvkHs(Rn) given that both rα< s and |α| − rα≤shold by remark 4.2. Eventually we obtain |BP(v, w)| ≤ |h(−∆)s/2v, (−∆)s/2wi| +X |α|≤m |haαDαv, wi| ≤ kwkHs(Rn)kvkHs(Rn)+X |α|≤m CkaαkHrα,∞(Ω)kwkHs(Rn)kvkHs(Rn) ≤CkwkHs(Rn)kvkHs(Rn). Next we shall prove existence and uniqueness of solutions for the problems (3.1) and (3.1). The reasoning is similar to the one for the proof of lemma 3.4, but the details of the computations are quite different. Lemma 4.3 (Well-posedness).Let Ω⊂Rnbe a bounded Lipschitz domain and s∈R+\Z, m∈Nsuch that 2s>m. Let aα∈Hrα,∞(Ω), with rαdefined as in (1.4). There exist a real number µ > 0and a countable set Σ⊂(−µ, ∞)of eigenvalues λ1≤λ2≤... → ∞ such that if 16 λ∈R\Σ, for any f∈Hs(Rn)and F∈(e Hs(Ω))∗there exists a unique u∈Hs(Rn)such that u−f∈e Hs(Ω) and BP(u, v)−λhu, vi=F(v)for all v∈e Hs(Ω). One has the estimate kukHs(Rn)≤CkfkHs(Rn)+kFk( e Hs(Ω))∗. The function uis also the unique u∈Hs(Rn)satisfying rΩ (−∆)s+X |α|≤m aα(x)Dα−λ u=F in the sense of distributions in Ωand u−f∈e Hs(Ω). Moreover, if (3.1) holds then 0/∈Σ. Proof. Again it is enough to find unique ˜u∈e Hs(Ω) such that BP(˜u, v)−λh˜u, vi=˜ F(v), where ˜ F:= F−BP(f, ·) + λhf, ·i. Consider v, w ∈C∞ c(Ω) and rα6= 0. Since 0 < rα< s, the interpolation inequality kwkHrα(Rn)≤Ckwk1−rα/s L2(Rn)kwkrα/s Hs(Rn) holds. Using this and formula (4.1) we get, for a constant C=C(Ω, n, s, rα) which may change from line to line, |haα(x)Dαv, wi| ≤ CkaαkHrα,∞(Ω)kvkHs(Rn)kwkHrα(Rn) (23) ≤CkaαkHrα,∞(Ω)kvkHs(Rn)kwk1−rα/s L2(Rn)kwkrα/s Hs(Rn) ≤ kaαkHrα,∞(Ω)kvkHs(Rn)Crα/(rα−s)kwkL2(Rn)+kwkHs(Rn). In the last step of (4.1) we used formula (3.1) with q=s rα , p =s s−rα , b =kwkrα/s Hs(Rn), a =Ckwk1−rα/s L2(Rn), η =. If instead rα= 0, just by formula (4.1) we already have |haα(x)Dαv, wi| ≤ CkaαkL∞(Ω)kvkHs(Rn)kwkL2(Rn). Moreover, the two estimates above also hold for v, w ∈e Hs(Ω) by the density of C∞ c(Ω) in e Hs(Ω). Now we use formula (3.1) again, but this time we choose q=p= 2, b =kvkHs(Rn), a =kvkL2(Rn), η =s/(s−rα). This leads to |haα(x)Dαv, vi| ≤ kaαkHrα,∞(Ω)kvkHs(Rn)Crα/(rα−s)kvkL2(Rn)+kvkHs(Rn) =kaαkHrα,∞(Ω) Crα/(rα−s)kvkL2(Rn)kvkHs(Rn)+kvk2 Hs(Rn) ≤ kaαkHrα,∞(Ω) Crα+s rα−skvk2 L2(Rn)+(C+ 1)kvk2 Hs(Rn) ≤CkaαkHrα,∞(Ω) rα+s rα−skvk2 L2(Rn)+kvk2 Hs(Rn) ≤C0kaαkHrα,∞(Ω) M+s M−skvk2 L2(Rn)+kvk2 Hs(Rn) where C=C(Ω, n, s, rα) and C0=C0(Ω, n, s) are constants changing from line to line and M∈[0, s) is defined by M:= max|α|≤mrα. Eventually BP(v, v)≥ k(−∆)s/2vk2 L2(Rn)−X |α|≤m |haα(x)Dαv, vi|(24) ≥ k(−∆)s/2vk2 L2(Rn)−C0M+s M−skvk2 L2(Rn)+kvk2 Hs(Rn)X |α|≤m kaαkHrα,∞(Ω) 17 =k(−∆)s/2vk2 L2(Rn)−C0C00 M+s M−skvk2 L2(Rn)+kvk2 Hs(Rn) where C00 := P|α|≤mkaαkHrα,∞(Ω) is a constant independent of and v. By the higher order Poincar´e inequality (lemma 2.2) (4.1) turns into BP(v, v)≥ck(−∆)s/2vk2 L2(Rn)+kvk2 L2(Rn)−C0C00 M+s M−skvk2 L2(Rn)+kvk2 Hs(Rn) ≥ckvk2 Hs(Rn)−C0C00 M+s M−skvk2 L2(Rn)+kvk2 Hs(Rn) for some constant c=c(Ω, n, s) changing from line to line. For small enough (notice that M−s < 0), this eventually gives the coercivity estimate (25) BP(v, v)≥c0kvk2 Hs(Rn)−µkvk2 L2(Rn) for some constants c0, µ > 0 independent of v. The proof is now concluded as in lemma 3.4.  Assuming as in Section 3 that both (3.1) and (3.1) hold, by means of the above lemma 4.3 we can define the DN-maps ΛP,Λ∗ Pjust as in lemma 3.6. Definition 4.4. Let Ω⊂Rnbe a bounded open set. Let s∈R+\Zand m∈Nsuch that 2s>m, and let aα∈Hrα,∞(Ω), with rαdefined as in (1.4). The exterior DN maps ΛPand Λ∗ Pare ΛP:X→X∗defined by hΛP[f],[g]i:= BP(uf, g) and Λ∗ P:X→X∗defined by hΛ∗ P[f],[g]i:= B∗ P(u∗ f, g) where uf, u∗ fare the unique solutions to the equations (−∆)su+X |α|≤m aαDαu= 0 in Ω, u −f∈e Hs(Ω) and (−∆)su∗+X |α|≤m (−1)|α|Dα(aαu∗)=0 in Ω, u∗−f∈e Hs(Ω) with f, g ∈Hs(Rn). 4.2. Proof of injectivity. We also arrive at the same Alessandrini identity and Runge approximation property which we get in lemmas 3.8 and 3.9. Lemma 4.5 (Alessandrini identity).Let Ω⊂Rnbe a bounded Lipschitz domain and s∈R+\Z, m∈Nsuch that 2s > m. Let aα∈Hrα,∞(Ω), with rαdefined as in (1.4). For any f1, f2∈ Hs(Rn), let u1, u∗ 2∈Hs(Rn)respectively solve (−∆)su1+X |α|≤m a1,α(x)Dαu1= 0 in Ω, u1−f1∈e Hs(Ω) and (−∆)su∗ 2+X |α|≤m (−1)|α|Dα(a2,α(x)u∗ 2) = 0 in Ω, u∗ 2−f2∈e Hs(Ω). Then we have the integral identity h(ΛP1−ΛP2)[f1],[f2]i=X |α|≤m h(a1,α −a2,α)Dαu1, u∗ 2i. Lemma 4.6 (Runge approximation property).Let Ω, W ⊂Rnrespectively be a bounded Lipschitz domain and a non-empty open set such that W∩Ω = ∅. Let s∈R+\Z,m∈Nsuch that 18 2s > m. Let aα∈Hrα,∞(Ω), with rαdefined as in (1.4). Moreover, let R:= {uf−f:f∈ C∞ c(W)} ⊂ e Hs(Ω), where ufsolves (−∆)suf+X |α|≤m aα(x)Dαuf= 0 in Ω, uf−f∈e Hs(Ω) and R∗:= {u∗ f−f:f∈C∞ c(W)} ⊂ e Hs(Ω), where u∗ fsolves (−∆)su∗ f+X |α|≤m (−1)|α|Dα(aα(x)u∗ f) = 0 in Ω, u∗ f−f∈e Hs(Ω). Then Rand R∗are dense in e Hs(Ω). With this at hand, we can prove the main theorem for bounded coefficients. 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World Scientific, Third edition, 2014. 20 Department of Mathematics and Statistics, University of Jyv¨ askyl¨ a, Jyv¨ askyl¨ a, Finland Current address: Institut fur Angewandte Mathematik, Ruprecht-Karls-Universit¨at Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany Email address:[email protected] Department of Mathematics and Statistics, University of Jyv¨ askyl¨ a, Jyv¨ askyl¨ a, Finland Email address:[email protected] Seminar for Applied Mathematics, Department of Mathematics, ETH Zurich, Z¨ urich, Switzerland Current address: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, UK Email address:[email protected] Department of Mathematics, University of Washington, Seattle, USA / Jockey Club Institute for Advanced Study, HKUST, Hong Kong Email address:[email protected] 21