I am studying partial differential equation optimization, and this is the elliptic model problem:
(1) $- \nabla \space \cdot (e^u\nabla y_i)=q_i\space\space\space x\in\Omega$
(2) $\nabla y_i \space \cdot \space n=0$
(3) $\int_\Omega y_i\space d\Omega = 0\space\space \space i=1,...,n_s$
Where $\Omega \subset \mathbb{R}^3$. My biggest problem is equation 3 which is said to ensure uniqueness. I know that $n$ is the normal vector on the boundary $\partial\Omega$ but I can only guess $n_s$ is the number of normal vectors, but I'm not totally sure. I know that this means that the integral is over the entire volume, but it is also equal to zero, so doesn't that make each y a point? Also, how does the $i$ subscript factor into this?