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Many physics problems have $J(x)=L(x)E(x)+h(x)$, source $h(x)$, fields$E$,$J$ satisfying differential constraints, symbolized by $E\in\calE$,$J\in\cal J$ where $\cal E$,$\cal J$ are orthogonal spaces. If $L(x)$ takesvalues in certain nonlinear manifolds $\cal M$, and coercivity, boundednesshold, then the Green's function satisfies exact identities. We also linkGreen's functions of different problems. The analysis, based on the theory ofexact relations for composites, does not assume microscale variations in$L(x)$, and allows for other equations, such as for waves in lossy media. Forbodies $\Omega$, in which $L(x)\in{\cal M}$, the Dirichlet-to-Neumann mapsatisfies boundary field equalities. These generalize the notion ofconservation laws: the constraints on the fields inside $\Omega$ giveidentities satisfied by the boundary fields, and provide extra constraints onthe 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 $x$) thatlies on a nonlinear manifold, we find conditions on the manifold, and on $\calB$, that with appropriate conditions on the boundary fluxes $q(x)=n(x)\cdotQ(x)$ (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 $q(x)$ totake values in $n(x)\cdot\cal D$. We find there are additional divergence freefields inside $\Omega$ that in turn generate additional boundary fieldequalities. 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 $\calA$, of appropriate non-linear manifolds, and when $w$ satisfies appropriateboundary conditions. The extension to certain non-linear minimization problemsis also sketched.

Exact relations for Green's functions in linear PDE and boundary field equalities: a generalization of conservation laws