ncts

Let $\Omega$ be a domain with smooth domain in $\mathbb{R}^{n}$ with $n \ge 2$ . Let $\gamma(x)>0$ (with sufficient regularity) represent the conductivity of the medium $\Omega$ . Given any sufficiently regular $f$ (voltage), there exists a unique solution $u$ of the conductivity equation

$\nabla \cdot (\gamma(x) \nabla u) = 0 \quad \text{in}\,\,\Omega.$

This equation is derived by the conservation law. The Calder\'{o}n problem asked whether the electrical conductivity $\gamma(x)$ can be uniquely determined by making voltage $f$ and current measurements $\gamma \partial{\nu}u$ on its boundary $\partial \Omega$ . In this talk, starting from the famous Calder\'{o}n problem, we will give an introduction to the inverse problem. Moreover, we will exhibit some recent results of some inverse problems.