= Neumann compatibility from a shifted compact resolvent
{c}
{title2=$(I-K)u=Kf,\quad K=(L+I)^{-1}$}
For a symmetric uniformly elliptic divergence-form operator with zero conormal flux and zero reaction, <Lax-Milgram theorem> constructs the shifted inverse $K:L^2\to H^1$. The compact <Rellich-Kondrachov theorem> embedding makes it compact on $L^2$, and symmetry makes it self-adjoint. If the homogeneous kernel consists of constants, then $\ker(I-K)$ is the constant line and $K1=1$. The <Fredholm alternative> makes solvability equivalent to $Kf$ being orthogonal to constants, or $\langle Kf,1\rangle=\langle f,K1\rangle=\int f=0$. The <weak solution> is unique up to an additive constant. For a disconnected domain the identical argument gives one compatibility condition per connected component.
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