The Calderón problem with partial data on manifolds and applications. (English) Zbl 1335.35301

The Calderón inverse problem with partial data \[ \begin{aligned} -\Delta u+q u&=0 \quad \text{in } \Omega, \\ u|_{\Gamma_D}& =\varphi, \\ \partial_\nu u|_{\Gamma_N}&=\psi, \end{aligned} \] is considered in dimensions \(n \geq 3\), where \(u\in H_\Delta(\Omega)\) with \(H_\Delta (\Omega)=\{u\in L^2(\Omega): \Delta u \in L^2(\Omega)\}\) and \(\text{supp}(u|_{\partial\Omega})\subset \Gamma_D\), \(\varphi\in H^{-1/2}(\partial\Omega),\) \(\psi\in H^{-3/2}(\partial\Omega)\), and \(\Gamma_D, \Gamma_N\subset\Omega\). The goal here is to determine a potential \(q\in L^\infty(\Omega)\).
If the inaccessible part of the boundary satisfies a (conformal) flatness condition in one direction, it is shown that this problem reduces to the invertibility of a broken geodesic ray transform. In Euclidean space, sets satisfying the flatness condition include parts of cylindrical sets, conical sets, and surfaces of revolution. The authors prove local uniqueness for the Calderón problem with partial data in admissible geometries and global uniqueness under an additional concavity assumption. The proofs are based on improved Carleman estimates with boundary terms, complex geometric optics solutions involving reflected Gaussian beam quasimodes, and the invertibility of (broken) geodesic ray transforms.


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