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Published online by Cambridge University Press: 30 May 2025
The global uniqueness for inverse boundary value problems of elliptic equations at fixed frequency in dimension n = 2 is quite particular and remained open for many years. Now these problems are well understood, with a variety of results appearing in the last 10 or 15 years, essentially all using the complex structure ℝ2 ⋍ℂ and ∂-techniques. This is therefore a good time to write a short survey on the subject. Although we tried to cover as much as we can, we do not pretend to be exhaustive and we apologize in advance for any forgotten reference, which is not a decision made on purpose but rather a sign of our ignorance. We have decided to give more details about the proofs of recent results based on Bukhgeim’s idea [2008], for there is already a survey by Uhlmann [2003] on the subject about older results. The results of Astala, Lassas, and Päivärinta using quasiconformal methods are the subject of a separate survey in this volume [Astala et al. 2013]. Finally, we do not discuss questions about stability and reconstruction, nor inverse scattering results
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