Simultaneous identification of diffusion and absorption coefficients in a quasilinear elliptic problem

In this work we consider the identifiability of two coefficients $a(u)$ and $c(x)$ in a quasilinear elliptic partial differential equation from observation of the Dirichlet-to-Neumann map. We use a linearization procedure due to Isakov [On uniqueness in inverse problems for semilinear parabolic equations. Archive for Rational Mechanics and Analysis, 1993] and special singular solutions to first determine $a(0)$ and $c(x)$ for $x \in \Omega$. Based on this partial result, we are then able to determine $a(u)$ for $u \in \mathbb{R}$ by an adjoint approach.