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 $sigma (geq 0)$ 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.