Uniqueness theorems for inverse boundary value problems in quasilinear anisotropic media
This study investigates the question of whether one can uniquely determine a scalar quasilinear conductivity in an anisotropic medium by making voltage and current measurements at the boundary whose quasilinear conductivity coefficient has smooth and limited regularity. In our study, the first result is to explore the outcomes in the situation of smooth quasilinear (non-analytic) coefficient matrices with dimension $n \geq 3 $. This research allows us to find a new way to investigate the unsolved aspects of quasilinear anisotropic inverse boundary value problems and obtain a new uniqueness theorem. We consider a special case where $A(x; t) = \gamma (x; t)A(x)$, where A(x) is known and one needs to recover the unknown scalar function (x; t). The most important technique applies to prove the uniqueness result is to use a combination of Ferrand's theorems about conformal diffeomorphisms in order to show that the quasilinear Dirichlet to Neumann map, for $\gamma_1(x; t) and \gamma_2(x; t)$ with $\Lambda _{\gamma _1}(x;t) =\Lambda_{\gamma _2}(x;t),$ determines $\gamma_1(x,t)$ and $\gamma_2(x,t)$. The second result is exploring the uniqueness theorem where quasilinear conductivity (coefficient metric) has less regularity $C^{2,\alpha}, 0 < \alpha < 1$. In this case, since the assumptions of the celebrated Ferrand's result on the action of conformal diffeomorphism on a manifold is in $C^1$, there is no need to use the theorems. However, we use Ferrand's original result in actions of conformal diffeomorphism on a compact manifold with C1 regularity to prove the second result.
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