Calderon inverse Problem with partial data on Riemann Surfaces

By C. Guillarmou, Leo Tzou. 2009

103 citations recorded by the source corpus.

On a fixed smooth compact Riemann surface with boundary $(M_0,g)$, we show that for the Schr\"odinger operator $\Delta +V$ with potential $V\in C^{1,\alpha}(M_0)$ for some $\alpha>0$, the Dirichlet-to-Neumann map $N|_{\Gamma}$ measured on an open set $\Gamma\subset \partial M_0$ determines uniquely the potential $V$. We also discuss briefly the corresponding consequences for potential scattering at 0 frequency on Riemann surfaces with asymptotically Euclidean or asymptotically hyperbolic ends.

Subfields: Biomedical Engineering, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2f970a8e4e87dddb74ca73e2cdcb2bd7122f9ae3

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In Biomedical Engineering

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Electrical impedance tomography and Calderón's problem (2009, background): “In two dimensions, when the metric g is known, it is proven in [50] that one can uniquely determine the conductivity β. Guillarmou and Tzou [46] have shown that a potential is uniquely determined for the Schrödinger equation g − q, with g being the Laplace–Beltrami operator associated with the metr”
Inverse problems: seeing the unseen (2014, background): “The case of Riemannian surfaces was considered in [70].”
Partial Cauchy data for general second order elliptic operators in two dimensions (2010, result): “The latter result was extended to Riemann surfaces in [13].”
Inverse problem for the Riemannian wave equation with Dirichlet data and Neumann data on disjoint sets (2012, methodology): “For Riemannian surfaces the determination of the potential in a Riemann-Schrödinger equation with local data corresponding to S = R 6 = ∂M has been solved in [16].”
The Calderón problem for variable coefficients nonlocal elliptic operators (2017, background): “In the two-dimensional setting, if Λg,q1 = Λg,q2 on ∂M , then q1 = q2 in M whenever q1, q2 are continuous potential on M , see [22].”