Uniqueness in the Calderón Problem

Question. Does ${\Lambda }_{{\gamma }_1}={\Lambda }_{{\gamma }_2}$ imply ${\gamma }_1={\gamma }_2$ for isotropic conductivities?

Main results, in chronological order:

Year	Authors	Setting
1980	Calderón	injectivity of the linearisation at constant $\gamma$
1984–85	Kohn, Vogelius	[[Boundary Determination
1987	Sylvester, Uhlmann	$n\ge 3$, smooth $\gamma$ ($C^{\infty }$), via Complex Geometrical Optics Solutions
1988	Nachman; Novikov	$n\ge 3$, reconstruction procedures (see Reconstruction Algorithms with Guarantees)
1996	Nachman	$n=2$, $\gamma \in W^{2,p}$, $p>1$, constructive (D-bar Method)
1997	Brown, Uhlmann	$n=2$, $\gamma \in W^{1,p}$, $p>2$
2006	Astala, Päivärinta	$n=2$, $\gamma \in L^{\infty }$ (only bounded above and below)
2013	Haberman, Tataru	$n\ge 3$, $\gamma \in C^1$, or Lipschitz and close to constant
2015	Haberman	$n=3,4$, $\gamma \in W^{1,n}$
2016	Caro, Rogers	$n\ge 3$, $\gamma$ Lipschitz

The two-dimensional problem is special. There the conductivity equation can be rewritten as a Beltrami equation, and quasiconformal mapping techniques give uniqueness in $L^{\infty }$. In dimension $n\ge 3$, whether $L^{\infty }$ conductivities are uniquely determined is still open.

Uniqueness fails for Anisotropic Conductivities: they are determined at most up to a boundary-fixing diffeomorphism.

References

1. J. Sylvester, G. Uhlmann (1987). A Global Uniqueness Theorem for an Inverse Boundary Value Problem. Ann. of Math. 125(1), 153–169. doi:10.2307/1971291
2. A. I. Nachman (1996). Global Uniqueness for a Two-Dimensional Inverse Boundary Value Problem. Ann. of Math. 143(1), 71–96. doi:10.2307/2118653
3. R. M. Brown, G. Uhlmann (1997). Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions. Comm. PDE 22(5–6), 1009–1027. doi:10.1080/03605309708821292
4. K. Astala, L. Päivärinta (2006). Calderón’s inverse conductivity problem in the plane. Ann. of Math. 163(1), 265–299. doi:10.4007/annals.2006.163.265
5. B. Haberman, D. Tataru (2013). Uniqueness in Calderón’s problem with Lipschitz conductivities. Duke Math. J. 162(3), 497–516. doi:10.1215/00127094-2019591
6. B. Haberman (2015). Uniqueness in Calderón’s Problem for Conductivities with Unbounded Gradient. Comm. Math. Phys. 340, 639–659. doi:10.1007/s00220-015-2460-3
7. P. Caro, K. M. Rogers (2016). Global Uniqueness for the Calderón Problem with Lipschitz Conductivities. Forum Math. Pi 4, e2. doi:10.1017/fmp.2015.9
8. 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
9. G. Uhlmann (2009). Electrical impedance tomography and Calderón’s problem. Inverse Problems 25(12), 123011. doi:10.1088/0266-5611/25/12/123011