Fredholm alternative for an elliptic Dirichlet problem

For a uniformly elliptic operator $L$ on a bounded regular domain, exactly one of the following equivalent descriptions applies to its homogeneous Dirichlet problem. If $\ker L^{\ast }=0$, every forcing has a unique solution. Otherwise $Lu=f$ is solvable exactly when $f$ is orthogonal to $\ker L^{\ast }$, and any two solutions differ by an element of $\ker L$. Moreover $\dim \ker L=\dim \ker L^{\ast }$.

The zero-boundary problem $\Delta u+cu=f$ is uniquely solvable if $c$ has a sufficiently small positive part depending only on the bounded regular domain. Write $c=c_{-}+c_{+}$ with $c_{-}\le 0$. The supremum norm barrier for an elliptic Dirichlet problem gives $\Vert u{\Vert }_{\infty }\le C_{\Omega }\Vert c_{+}u{\Vert }_{\infty }$ for a homogeneous solution. If $C_{\Omega }\Vert c_{+}{\Vert }_{\infty }<1$, its kernel is zero and the Fredholm alternative for an elliptic Dirichlet problem gives existence for every forcing and boundary datum.

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 $N$.