= Boundary compatibility in the self-adjoint Fredholm alternative
For real $c$ and $L=\Delta+c$ on a bounded regular domain, let $N$ be the zero-boundary kernel. The problem $Lu=f$, $u=\psi$ on the boundary is solvable exactly when $\int_\Omega fz+\int_{\partial\Omega}\psi\partial_\nu z=0$ for every $z\in N$. The sign follows from <Green second identity>. After subtracting a boundary extension of $\psi$, this is the usual orthogonality of the forcing to the <null space> of the <self-adjoint> Dirichlet realization. All solutions then differ by an element of $N$.
Back to article page
