| Abstract: |
| We extend the Boundary Control method beyond the hyperbolic setting to certain elliptic and parabolic inverse boundary value problems. The analytic framework is based on Schulze's edge calculus, with the parabolic case reduced to an elliptic problem. The reconstruction step is formulated through Boundary Control algebras generated by boundary-induced solutions. Under suitable separation assumptions, these algebras recover the underlying manifold, or in the parabolic case a compact spacetime strip, as a Gelfand spectrum. We then discuss applications to the Calder\`{o}n problem. In dimension \(n \ge 3\), this gives a real-analytic conductivity-metric result, while in dimension two it yields a Boundary Control approach based on holomorphic trace algebras and recovery up to the natural conformal gauge. We also outline an ongoing attempt to treat the higher-dimensional smooth anisotropic case within the same framework. |
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