I have a question which looks like some sort of inverse problem.
Let $B$ denote the unit ball centered at the origin in $R^N$ (take $N \ge 2$).
Given any $h:\partial B \rightarrow (0,\infty)$ (smooth) we would like to find some $ f \ge 0$ in $B$ (sufficiently regular, say $ f \in L^q(B) $ for some $q>N$) such that $v$ satisfies the following:
$\Delta v(x)=f(x)$ in $B$ with $ v=0$ on $ \partial B$ and $ x \cdot \nabla v(x)=h(x)$ on $ |x|=1$.
In the case of $h=C>0$ (constant) one sees they can just explicitly write out a solution. In the case of $h(x)=1 +\epsilon g(x)$ ($g$ fixed $ \epsilon $ small they can also do it).
Any comments would be greatly appreciated.