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

## Abstract

Many physical equations have the form \(\mathbf{J }(\mathbf{x })=\mathbf{L }(\mathbf{x })\mathbf{E }(\mathbf{x })-\mathbf{h }(\mathbf{x })\) with source \(\mathbf{h }(\mathbf{x })\) and fields \(\mathbf{E }\) and \(\mathbf{J }\) satisfying differential constraints, symbolized by \(\mathbf{E }\in \mathcal E\), \(\mathbf{J }\in \mathcal J\) where \(\mathcal E\), \(\mathcal J\) are orthogonal spaces. We show that if \(\mathbf{L }(\mathbf{x })\) takes values in certain nonlinear manifolds \(\mathcal M\), and coercivity and boundedness conditions hold, then the infinite body Green’s function (fundamental solution) satisfies exact identities. The theory also links Green’s functions of different problems. The analysis is based on the theory of exact relations for composites, but without assumptions about the length scales of variations in \(\mathbf{L }(\mathbf{x })\), and more general equations, such as for waves in lossy media, are allowed. For bodies \(\Omega \), inside which \(\mathbf{L }(\mathbf{x })\in \mathcal{M}\), the “Dirichlet-to-Neumann map” giving the response also satisfies exact relations. These boundary field equalities generalize the notion of conservation laws: the field inside \(\Omega \) satisfies certain constraints that leave a wide choice in these fields, but which give identities satisfied by the boundary fields, and moreover provide constraints on the fields inside the body. A consequence is the following: if a matrix-valued field \(\mathbf{Q }(\mathbf{x })\) with divergence-free columns takes values within \(\Omega \) in a set \(\mathcal B\) (independent of \(\mathbf{x }\)) that lies on a nonlinear manifold, we find conditions on the manifold, and on \(\mathcal B\), that with appropriate conditions on the boundary fluxes \(\mathbf{q }(\mathbf{x })=\mathbf{n }(\mathbf{x })\cdot \mathbf{Q }(\mathbf{x })\) (where \(\mathbf{n }(\mathbf{x })\) is the outward normal to \(\partial \Omega \)) force \(\mathbf{Q }(\mathbf{x })\) within \(\Omega \) to take values in a subspace \(\mathcal D\). This forces \(\mathbf{q }(\mathbf{x })\) to take values in \(\mathbf{n }(\mathbf{x })\cdot \mathcal D\). We find there are additional divergence-free fields inside \(\Omega \) that in turn generate additional boundary field equalities. Consequently, there exist partial null Lagrangians, functionals \(F(\mathbf{w },\nabla \mathbf{w })\) of a vector potential \(\mathbf{w }\) and its gradient that act as null Lagrangians when \(\nabla \mathbf{w }\) is constrained for \(\mathbf{x }\in \Omega \) to take values in certain sets \(\mathcal A\), of appropriate nonlinear manifolds, and when \(\mathbf{w }\) satisfies appropriate boundary conditions. The extension to certain nonlinear minimization problems is also sketched.

## Keywords

Green’s functions Exact relations Inverse problems Boundary field equalities Inhomogeneous media## Mathematics Subject Classification 2000

35J08 35J25 65N21## Notes

### Acknowlegements

GWM is grateful to the National Science Foundation for support through the Research Grants DMS-1211359 and DMS-1814854. Both authors thank the Institute for Mathematics and its Applications at the University of Minnesota for hosting their visit there during the Fall 2016 where this work was initiated as part of the program on Mathematics and Optics. Yury Grabovsky and the referees are thanked for their comments which led to significant improvements of the manuscript.

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