# real analysis – On regularity of boundary data for Robin boundary condition

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.