Inverse conductivity problem on Riemann surfaces. (English) Zbl 1151.35101
Summary: An electrical potential \(U\) on a bordered Riemann surface \(X\) with conductivity function \(\sigma >0\) satisfies equation \(d(\sigma d^c U)=0\). The problem of effective reconstruction of \(\sigma \) from electrical currents measurements (Dirichlet-to-Neumann mapping) on the boundary: \(U|_{bX} \rightarrowtail \sigma d^c U|_{bX}\) is studied. We extend to the case of Riemann surfaces the reconstruction scheme given, firstly, by R. G. Novikov [Funct. Anal. Appl. 22, No. 4, 263–272 (1988); translation from Funkts. Anal. Prilozh. 22, No. 4, 11–22 (1988; Zbl 0689.35098)] for simply connected \(X\). We apply for this new kernels for \(\bar{\partial}\) on the affine algebraic Riemann surfaces constructed by G. Khenkin [Cauchy-Pompeiu type formulas for \(d\)-bar on affine algebraic Riemann surfaces and some applications, preprint 2008, arXiv:0804.3761].
MSC:
| 35R30 | Inverse problems for PDEs |
| 81U40 | Inverse scattering problems in quantum theory |
| 35R05 | PDEs with low regular coefficients and/or low regular data |
| 35Q40 | PDEs in connection with quantum mechanics |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
Keywords:
Riemann surface; electrical current; inverse conductivity problem; \(\bar{ \partial}\) -method; homotopy formulasCitations:
Zbl 0689.35098References:
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