Uniqueness results for inverse Robin problems with bounded coefficient

In this paper we address the uniqueness issue in the classical Robin inverse problem on a Lipschitz domain $\mathrm{\Omega }\subset {\text{RR}}^n$, with $L^{\mathrm{\infty }}$ Robin coefficient, $L^2$ Neumann data and isotropic conductivity of class $W^{1,r}(\mathrm{\Omega })$, $r\text{textgreater}n$. We show that uniqueness of the Robin coefficient on a subpart of the boundary given Cauchy data on the complementary part, does hold in dimension $n=2$ but needs not hold in higher dimension. We also raise on open issue on harmonic gradients which is of interest in this context.