  Many physics problems have $J(x)=L(x)E(x)+h(x)$, source $h(x)$, fields
$E$,$J$ satisfying differential constraints, symbolized by $E\in\cal
E$,$J\in\cal J$ where $\cal E$,$\cal J$ are orthogonal spaces. If $L(x)$ takes
values in certain nonlinear manifolds $\cal M$, and coercivity, boundedness
hold, then the Green's function satisfies exact identities. We also link
Green's functions of different problems. The analysis, based on the theory of
exact relations for composites, does not assume microscale variations in
$L(x)$, and allows for other equations, such as for waves in lossy media. For
bodies $\Omega$, in which $L(x)\in{\cal M}$, the Dirichlet-to-Neumann map
satisfies boundary field equalities. These generalize the notion of
conservation laws: the constraints on the fields inside $\Omega$ give
identities satisfied by the boundary fields, and provide extra constraints on
the interior fields. A consequence is this: if a matrix valued field $Q(x)$
with $\nabla\cdot Q=0$ takes values in a set $\cal B$ (independent of $x$) that
lies on a nonlinear manifold, we find conditions on the manifold, and on $\cal
B$, that with appropriate conditions on the boundary fluxes $q(x)=n(x)\cdot
Q(x)$ (where $n(x)$ is the outwards normal to $\partial\Omega$) force $Q(x)$
within $\Omega$ to take values in a subspace $\cal D$. This forces $q(x)$ to
take values in $n(x)\cdot\cal D$. We find there are additional divergence free
fields inside $\Omega$ that in turn generate additional boundary field
equalities. There exist partial Null-Lagrangians, functionals $F(w,\nabla w)$
of a vector potential $w$ and its gradient, that act as null-Lagrangians when
$\nabla w$ is constrained for $x\in\Omega$ to take values in certain sets $\cal
A$, of appropriate non-linear manifolds, and when $w$ satisfies appropriate
boundary conditions. The extension to certain non-linear minimization problems
is also sketched.
