Boundary Determination

The first uniqueness result for the Calderón Problem concerns the boundary itself. Kohn and Vogelius (1984) showed that for smooth conductivities, ${\Lambda }_{\gamma }$ determines $\gamma {|}_{\partial \Omega }$ and all its normal derivatives ${\partial }_{\nu }^k\gamma {|}_{\partial \Omega }$.

Idea. Use boundary voltages that oscillate rapidly and are concentrated near a boundary point $x_0$. The corresponding solutions decay quickly away from the boundary, so $⟨{\Lambda }_{\gamma }f,f⟩$ only sees $\gamma$ in a thin layer near $x_0$. In the limit this gives $\gamma (x_0)$; higher-order asymptotics give the normal derivatives. In the language of microlocal analysis, ${\Lambda }_{\gamma }$ is a pseudodifferential operator of order one whose full symbol determines the Taylor series of $\gamma$ at the boundary (Sylvester–Uhlmann 1988).

Consequences.

• Real-analytic conductivities are uniquely determined, because their Taylor series at the boundary determines them. Kohn and Vogelius (1985) extended this to piecewise analytic conductivities.
• The same localisation explains why high-frequency Current Patterns only carry information about the region near the boundary.

References

1. R. V. Kohn, M. Vogelius (1984). Determining conductivity by boundary measurements. Comm. Pure Appl. Math. 37(3), 289–298. doi:10.1002/cpa.3160370302
2. R. V. Kohn, M. Vogelius (1985). Determining conductivity by boundary measurements II. Interior results. Comm. Pure Appl. Math. 38(5), 643–667. doi:10.1002/cpa.3160380513
3. J. Sylvester, G. Uhlmann (1988). Inverse boundary value problems at the boundary—continuous dependence. Comm. Pure Appl. Math. 41(2), 197–219. doi:10.1002/cpa.3160410205