Regularity of harmonic functions with robin data up to the boundary

Question

I want to prove that if $u$ is a solution of $\Delta u = 0$ in $\Omega$ with Robin boundary conditions $\frac{\partial u}{\partial n} = \lambda u$, where $\Omega \subset \mathbb{R}^n$ has analytic boundary, then $u$ is analytic up to the boundary (i.e. there exists analytic extension of $u$ to $U \supset \overline{\Omega}$). I know that showing that $u \in \mathcal{C}^\infty(U)$ would be enough.

Answer 1

I found the solution in the book by Lions and Magenes: Non-homogeneous boundary value problems and applications, vol 3, in the chapter about elliptic iterates.