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Calderón inverse problem with partial data on Riemann surfaces. (English) Zbl 1222.35212

Let \((M_0,g)\) be a compact Riemann surface with boundary, \(\Delta_g\) its nonnegative Laplacian, and \(\partial_\nu\) the interior pointing normal field. Let \(\Gamma\subset\partial M_0\) be a nonempty open subset of the boundary. The Cauchy partial data space of a Schrödinger operator with potential \(V\), \(\Delta_g+V\), is defined as follows:
\[ {\mathcal C}_V^\Gamma= \{(u|_\Gamma,\partial_\nu u|\Gamma);\;u\in H^1(M_0),\;(\Delta_g+V)u=0,\;u|_{\Gamma_0}=0\}, \]
where \(\Gamma_0=\partial M_0\setminus\Gamma\). The main result of the paper is Theorem 1.1 which says that two potentials \(V_1,V_2\in C^{1+\alpha}(M_0)\) are equal if \({\mathcal C}_{V_1}^\Gamma={\mathcal C}_{V_2}^\Gamma\). By a standard observation, the identifiability for the isotropic conductivity equation follows as a corollary. The authors also discuss consequences for potential scattering at zero frequency on Riemann surfaces with either asymptotically Euclidean or asymptotically hyperbolic ends.
The method to identify the potential follows O. Y. Imanuvilov and G. Uhlmann and M. Yamamoto [J. Am. Math. Soc. 23, 655–691 (2010; Zbl 1201.35183)]. In particular, the authors use the idea of A. L. Bukhgeim [J. Inverse Ill-Posed Probl. 16, 19–33 (2008; Zbl 1142.30018)] to reconstruct \(V(p)\) using the method of stationary phase in \(p\). The main contribution of the paper is a geometric construction of the holomorphic phase functions which enter into the Carleman estimates and the stationary phase expansion. The paper is written very well.

MSC:

35R30 Inverse problems for PDEs
58J32 Boundary value problems on manifolds
35J10 Schrödinger operator, Schrödinger equation
35R01 PDEs on manifolds
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References:

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