Exact relations for Green's functions in linear PDE and boundary field equalities: a generalization of conservation laws (Q2319967)

The authors derive Green's functions for various types of elliptic problems using their abstract forms. They reexpress the original system \[ \sum_{i=1}^d \frac \partial{\partial x_i}\left(\sum_{j=1}^d\sum_{\beta=1}^m L_{i\alpha j\beta}\frac {\partial u_\beta(\mathbf{x})}{\partial x_j} \right)=f_\alpha (\mathbf{x}),\ \alpha=1,\dots,m, \] for the potential \(\mathbf{u(x)}\) given by the source term \(\mathbf{f(x)}\) as \[ J_{i\alpha}(\mathbf{x})=\sum_{j=1}^d\sum_{\beta=1}^mL_{i\alpha j\beta}E_{j\beta}(\mathbf{x})-h_{i\alpha}(\mathbf{x}), E_{j\beta}(\mathbf{x})=\frac {\partial u_\beta(\mathbf{x})}{\partial x_j},\quad \sum_{i=1}^d \frac{\partial J_{i\alpha}(\mathbf{x})}{\partial x_i}=0, \] valid if the integral of \(\mathbf{f(x)}\) over \(\mathbb{R}^d\) is zero. After expressing it in the form \(\mathbf{J(x)}=\mathbf{L(x)E(x)}-\mathbf{h(x)}\) and analyzing the operator \(\mathbf{L(x)}\) they show that the infinite body Green's function (fundamental solution) satisfies exact identities. Various physical models are analyzed in the Appendix.