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In this paper we show uniqueness of the conductivity for the quasilinear Calderón's inverse problem. The nonlinear conductivity depends, in a nonlinear fashion, of the potential itself and its gradient. Under some structural assumptions on the direct problem, a real-valued conductivity allowing a small analytic continuation to the complex plane induce a unique Dirichlet-to-Neumann (DN) map. The method of proof considers some complex-valued, linear test functions based on a point of the boundary of the domain, and a linearization of the DN map placed at these particular set of solutions.
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Gunther Uhlmann, Claudio Muñoz, The Calderón problem for quasilinear elliptic equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 37 (2020), no. 5, pp. 1143–1166DOI 10.1016/J.ANIHPC.2020.03.004