I am trying to implement a numerical scheme for a PDE, and it takes as input dirichlet boundary conditions, i.e. $u=0$ on $\partial\Omega$, as well as the hessian of $u$ on the boundary also.
I am handling a non standard boundary condition, called the second boundary condition, where we prescribe $\Omega,\Omega^*\subset\Bbb R^2$, convex, and our requirement is that $\nabla u(\Omega)=\Omega^*$.
It can be shown that this is equivalent to $\nabla u(\partial\Omega)=\partial \Omega^*$.
But say we can find a map $-F:\partial\Omega\to\partial\Omega^*$, of closed form, so we can state:
$\nabla u(x)=-F(x)$ on $\partial\Omega$,
Then taking the divergence of both sides we get:
$-\Delta u=f(x)$, on $\partial\Omega$, where $f(x)=\nabla\cdot F$.
So now we have poissons equation, but my issue is that the PDE is defined on a closed domain, namley $\partial \Omega$.
Is this a pathological problem? since if not, we can find $u$ (with a free constant), and by induce dirichlet boundary conditions.
Thank you for your help in advance.