= Global relation for a linear boundary value problem
{title2=$\mathcal G(\lambda)=0$}
= Global relation
{synonym}
= Global relations for a linear boundary value problem
{synonym}
A global relation is an identity connecting transforms or integrals of the boundary traces of a solution. For a formally self-adjoint operator $L=\Delta-k^2$, <Green second identity> gives
$$
\int_{\partial\Omega}(v\partial_nu-u\partial_nv)\,ds=0
$$
for every homogeneous adjoint solution $Lv=0$. Exponential adjoint families turn this into a spectral identity between the <Dirichlet boundary condition> and unknown <normal derivative>.
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