We consider the inverse problem of determining the coefficients of a general second order elliptic operator in two dimensions by measuring the corresponding Cauchy data on an arbitrary open subset of the boundary. We show that one can determine the coefficients of the operator up to natural limitations such as conformal invariance, gauge transformations and diffeomorphism invariance. We use the main result to prove that the curl of the magnetic field and the electric potential are uniquely determined by measuring partial Cauchy data associated to the magnetic Schrödinger equation on an arbitrary open subset of the boundary. We also show that any two of the three coefficients of a second order elliptic operator whose principal part is the Laplacian, are uniquely determined by their partial Cauchy data.
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Oleg Yu. Imanuvilov, Gunther Uhlmann, Masahiro Yamamoto, Partial Cauchy Data for General Second Order Elliptic Operators in Two Dimensions. Publ. Res. Inst. Math. Sci. 48 (2012), no. 4, pp. 971–1055