I am studying the regularity of elliptic PDE with Robin boundary condition. One first prove the regularity with homogeneous Robin boundary condition and then extend it to non-homogeneous one. For the non-homogeneous Robin condition I need the answer of the following question:
Let $\Omega$ be an open, bounded set in $\mathbb{R^n}$ with $C^{k+2}$ boundary. Suppose $\sigma \ (> 0)$ and $\xi$ are two functions defined on $\partial \Omega$. What should be the regularities of $\sigma$ and $\xi$ such that there exists a function $\Phi \in H^{k+2}(\Omega)$ which satisfies \begin{equation*} \nabla \Phi \cdot \nu + \sigma \Phi = \xi \ \ \text{on $\partial \Omega$}, \end{equation*} where $\nu$ is the outward unit normal to the boundary?