TITLE: Elliptic regularity for the Neumann problem
QUESTION [18 upvotes]: I'm trying to understand how to establish regularity for elliptic equations on bounded domains with Neumann data.
For simplicity, let's presume we are focusing on $-\Delta u = f$ in $\Omega$ and $\frac{\partial u}{\partial \nu} = 0$ on $\partial \Omega$. 
Interior regularity works the same as always.
When proving boundary regularity, for the dirichlet boundary case we first consider some ball $B(0,1) \cap \mathbb{R}_+^n$ and let $\xi = 1$ on $B(0,1/2)$, $\xi = 0$ on $\mathbb{R}^n - B(0,1)$ and then estimate all derivatives $\frac{\partial^2 u}{\partial x_i \partial x_j}$ except $\partial^2 u/\partial x_n^2$. Two main points are needed
1) $\xi$ vanishes on the curved part of $B(0,1) \cap \mathbb{R}_+^n$\
2) $u=0$ on $\{x_n=0\}$.
This allows us to let $-\partial_{x_i} (\xi \partial_{x_j} u)$ (with derivatvies replaced by difference quotients) be an admissible test function for our weak definitoin of a solution.
I presume the main difficulty in neumann boundary data is making your test function be admissible. In other words, we would need $\int v = 0$ since our existence was established on $H^1(\Omega)$ restricted to mean value zero functions. 
So in order to proceed, can we just subtract off a constant from our original $-\partial_{x_i}(\xi \partial_{x_j}u)$?
Is there some more natural way to establish regularity in this case? I do not want to take advantage of the fact that we have a green's function in this case however as I only chose the Laplace equation for simplicity.

REPLY [6 votes]: It's true that to prove existence for the Neumann problem you work in the space $X=\{v\in H^1(\Omega):\, \int_\Omega v\,dx=0\}$ and so a weak solution $u$ satisfies 
$$\int_\Omega\nabla u\cdot\nabla v\,dx=\int_\Omega fv\,dx$$
for all $v\in X$. However, if you take $w\in H^1(\Omega)$ and consider the function $v=w-\frac1{|\Omega|}\int_\Omega v\,dy$ you get 
$$\int_\Omega\nabla u\cdot\nabla w\,dx=\int_\Omega f\left(w-\frac1{|\Omega|}\int_\Omega v\,dy\right)\,dx=\int_\Omega fw\,dx-0$$
since $\int_\Omega f\,dx=0$. So You can actually work in the entire space $H^1(\Omega)$ rather than in the smaller space $X$.
Difference quotients work fine since the constraint is gone. If $\Omega=B^+(0,1)$ you can do difference quotients in the tangential directions $e_i$, $i=1,\cdots, n-1$ (so no need to assume $u=0$ on $\{x_n=0\}$) to prove that all second derivatives but $\partial^2 u/\partial x_n^2$ are in $L^2 (B^+(0,1/2))$ and then you use the equation (which is satisfied pointwise a.e. by interior regularity) to conclude that also $\partial^2 u/\partial x_n^2$ is  in $L^2 (B^+(0,1/2))$.