I am reading article with title " Integral equation methods in Inverse Obstacle scattering with a Generalized Impedance Boundary Condition" written by Rainer Kress. The problem is fomulated in the follwing. Let $D$ be simply connected domain in $\mathbb{R}^2$ with boundary $\partial D$ of Hölder class $C^{4,\alpha}$. $u$ satisfies the Helmholtz equation \begin{align} \Delta u+k^2 u=0 \end{align} with positive wave number $k$ and the generalized impedance boundary condition \begin{align} \frac{\partial u}{\partial \nu}+ik(\lambda u-\frac{d}{ds}\mu\frac{du}{ds})= 0 \,\text{on } \partial D \end{align} where $d/ds$ is tangential derivative and $\mu \in C^{1}(\partial D)$ and $\lambda \in C^{1}(\partial D)$.
The inverse problem is to determine $\lambda,\mu$ and $\partial D$.
Why do we choose $\lambda,\mu \in C^1(\partial D)$ and $\partial D $ in the class of $C^4$ for this problem? If we chose $\lambda, \mu \in H^1(\partial D)$, What would happen? Thank you for your help.