While working on a much more involved problem for quite some time it became apparent that I could gain at least some insight from understanding a much simpler problem, but to my slight embarrassment I am still unable to solve it myself or find any good reference material, so I thought I would ask here.
Let $s \in \mathbb{R}^2$ such that $|s|$ is large enough (I guess $s \neq 0$ suffices) and consider the following equation
$$ -\Delta u = f\quad \text{in}\quad \Omega $$ $$ \frac{\partial u}{\partial n} = g\quad \text{on}\quad\partial\Omega$$,
where $\Omega := B_{\frac{|s|}{2}}(0) \subset \mathbb{R}^2$. Suppose further that we know that $|f(x)| \leq C_1|x|^{-2}|s|^{-1}$ in $\Omega$ for some generic constant and there is no blow-up at the origin. Finally, we also know that $|g(x)| \leq C_2 |x-s|^{-2}$ on $\partial\Omega$, which, given that $x\in\partial\Omega \implies |x-s| \geq \frac{|s|}{2}$ also implies that $|g(x)| \leq 4C_2 |s|^{-2}$ (not sure if this is of any help).
The general question is: assuming that a solution $u$ to this problem exists, what can we say about the rate of decay of $\nabla u$? And I mean it quite generally, so I am equally interested in what can be said if $f$, $g$ and $u$ are assumed to be smooth - any point-wise estimates in that case, such as $|\nabla u(x)| \leq C_3 |x|^{-k_1}|s|^{-k_2}$ for some positive $k_1,k_2$ (the least I need is to be able to establish that $k_1 + k_2 = 2$, e.g. $k_1 = k_2 = 1$, which does not seem like much, does it?).
Alternatively a natural way to proceed would be to look at weak formulation and then get estimates such as $\|\nabla u\|_{L^2}\leq C_4 |s|^{-k_3}$. Here $k_3 = 3/2$ is the least I would be satisfied with.
The reason why I refer to the problem as 'decay estimates' is that I want to know what is the rate of decay of the gradient of solution with respect to $|s|$.
Any help will be much appreciated!