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

Abstract

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.