NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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CALDERÓN <strong>IN</strong>VERSE PROBLEM WITH PARTIAL DATA ON RIEMANN SURFACES COL<strong>IN</strong> GUILLARMOU and LEO TZOU Abstract On a fixed smooth compact Riemann surface with boundary (M0,g), we show that, for the Schrödinger operator g + V with potential V ∈ C 1,α (M0) for some α>0, the Dirichlet-to-Neumann map N |Ɣ measured on an open set Ɣ ⊂ ∂M0 determines uniquely the potential V . We also discuss briefly the corresponding consequences for potential scattering at zero frequency on Riemann surfaces with either asymptotically Euclidean or asymptotically hyperbolic ends. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2. Harmonic and holomorphic Morse functions on a Riemann surface . . . 86 3. Carleman estimate for harmonic weights with critical points . . . . . . . 95 4. Complex geometric optics on a Riemann surface . . . . . . . . . . . . . 99 5. Identifying the potential . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6. Inverse scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Appendix. Boundary determination . . . . . . . . . . . . . . . . . . . . . . 114 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 1. Introduction The problem of determining the potential in the Schrödinger operator by boundary measurement goes back to Calderón [8]andGelfand[12]. Mathematically, it amounts to asking if one can detect some data from boundary measurement in a domain (or manifold) with boundary. The typical model to have in mind is the Schrödinger operator P := g + V , where g is a metric and V is a potential; then we define the Cauchy data space by C := (u|∂,∂νu|∂); u ∈ H 1 (), u∈ ker P , DUKE MATHEMATICAL JOURNAL Vol. 158, No. 1, c○ 2011 DOI 10.1215/00127094-1276310 Received 17 September 2009. Revision received 22 July 2010. 2010 Mathematics Subject Classification. Primary 35R30; Secondary 58J32. Guillarmou’s work partially supported by National Science Foundation grant DMS-0635607. Tzou’s work partially supported by National Science Foundation grant DMS-0807502. 83
84 GUILLARMOU and TZOU where ∂ν is the interior pointing normal vector field to ∂ and where H 1 () = W 1,2 () is the space of L2 functions with one derivative in L2 . The first natural question is the following full data inverse problem: does the Cauchy data space determine uniquely the metric g and/or the potential V ?Ina sense, the most satisfying known results are those obtained when the domain ⊂ Rn is already known and g is the Euclidean metric, for which the identification of V has been proved in dimension n>2 by Sylvester and Uhlmann [32] and Novikov [30] (where reconstruction is also obtained), and very recently in dimension 2 by Bukgheim [6] when the domain is simply connected. In a recent work [16], we proved the identification of V on a Riemann surface with boundary from full data measurement. A related question is the conductivity problem which consists in taking V = 0 and replacing g by −divσ ∇, where σ is a positive definite symmetric tensor. An elementary observation shows that the problem of recovering a sufficiently smooth isotropic conductivity (i.e., σ = σ0Id for a function σ0) is contained in the abovementioned problem of recovering a potential V . For domain of R2 , Nachman [29] used the ¯∂ techniques to show that the Cauchy data space determines the conductivity. Recently a new approach developed by Astala and Päivärinta in [2] improved this result to assuming that the conductivity is only an L∞ scalar function on a simply connected domain. This was later generalized to L∞ anisotropic conductivities by Astala, Lassas, and Päivärinta in [3]. We notice that there still are rather few results in the direction of recovering the Riemannian manifold (,g) when V = 0, for instance the surface case by Lassas and Uhlmann [24] (seealso[4], [17]), the real analytic manifold case by Lassas, Taylor, and Uhlmann [23] (seealso[15] forthe Einstein case), the case of manifolds admitting limiting Carleman weights and in a same conformal class by Dos Santos Ferreira, Kenig, Salo, and Uhlmann [9]. The second natural, but harder, problem is the partial data inverse problem: if Ɣ1 and Ɣ2 are open subsets of ∂, does the partial Cauchy data space for P CƔ1,Ɣ2 := (u|Ɣ1,∂νu|Ɣ2); u ∈ H 1 (∂M0), Pu= 0, u= 0in∂ \ Ɣ1 determine the domain , the metric, the potential? For a fixed domain of R n ,the recovery of the potential if n>2 with partial data measurements was initiated by Bukhgeim and Uhlmann [7] and later improved by Kenig, Sjöstrand, and Uhlmann [21] to the case where Ɣ1 and Ɣ2 are respectively open subsets of the “front” and “back” ends of the domain. We refer the reader to the references for a more precise formulation of the problem. In dimension 2, the recent works of Imanuvilov, Uhlmann, and Yamamoto [19] solves the problem for fixed domains of R 2 in the case when Ɣ1 = Ɣ2 and when the potential are in C 2,α () for some α>0. In this article, we address the same question when the background domain is a fixed Riemann surface with boundary. We prove the following recovery result.
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SUR LA NON-DENSITÉ DES POINTS ENTI
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30 CASIM ABBAS Recall that the Reeb
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Page 53 and 54: 54 CASIM ABBAS formula (2.2) for th
Page 55 and 56: 56 CASIM ABBAS Restricting any of t
Page 57 and 58: 58 CASIM ABBAS where fτ (∞) = li
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Page 61 and 62: 62 CASIM ABBAS Recalling our origin
Page 63 and 64: 64 CASIM ABBAS THEOREM 3.7 Let µ,
Page 65 and 66: 66 CASIM ABBAS Since w2 − w1C 1,
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Page 73 and 74: 74 CASIM ABBAS W 1,p loc (C) since
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Page 79 and 80: 80 CASIM ABBAS Converting from coor
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NEAR OPTIMAL BOUNDS IN FREIMAN'S THEOREM

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