Let $\Omega$ be the unit disk in $\mathbb{R}^2$ and consider the inverse problem $$ - \Delta u = f, $$ where we want to find $f$ when we are given $\partial_\nu u = g$ on $\partial \Omega$ and we know that $f$ is harmonic.
A hint for this problem says introduce a new function $v = \Delta u$. So in this case the equation becomes $$ -v = f $$ and taking the Laplacian of both sides we get $$ -\Delta v = \Delta f = 0. $$ For the boundary condition I have $$ v = \Delta u \Longleftrightarrow \int_\Omega v = \int_\Omega \Delta u = \int_{\partial \Omega} \partial_\nu u = \int_{\partial \Omega} g. $$ So we have the following problem for $v$: $$ - \Delta v = 0, \\ \int_\Omega v = \int_{\partial \Omega} g. $$ Is this correct? If so, how do we solve this last expression for $v$?