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.

中文翻译：

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线性PDE和边界场等式中格林函数的精确关系：守恒律的一般化

许多物理方程的形式为\（\ mathbf {J}（\ mathbf {x}）= \ mathbf {L}（\ mathbf {x}）\ mathbf {E}（\ mathbf {x}）-\ mathbf {h }（\ mathbf {X}）\）与源\（\ mathbf {H}（\ mathbf {X}）\）和字段\（\ mathbf {E} \）和\（\ mathbf {Ĵ} \）满足微分约束，用\（\ mathcal E \）中的\（\ mathbf {E} \），\（\ mathcal J \中的\（\ mathbf {J} \）来表示，其中\（\ mathcal E \），\（\ mathcal J \）是正交空间。我们证明，如果\（\ mathbf {L}（\ mathbf {x}）\）取某些非线性流形中的值\（\ mathcal M \），并且矫顽力和有界条件成立，那么无限的物体格林函数（基本解）就满足精确的恒等式。该理论还将格林的各种问题的功能联系在一起。该分析基于复合材料的精确关系理论，但没有假设\（\ mathbf {L}（\ mathbf {x}）\）中变化的长度尺度，也没有更一般的方程式，例如，有损媒体，是允许的。对于物体\（\ Omega \），在其中\\\\ mathbf {L}（\ mathbf {x}）\ \ mathcal {M} \）中，给出响应的“ Dirichlet-to-Neumann映射”也精确关系。这些边界场等式概括了守恒定律的概念：\（\ Omega \）内部的场满足某些约束条件，这些约束条件在这些字段中留下了广泛的选择，但是它们给出了边界字段所满足的标识，并且还对身体内部的字段提供了约束。结果如下：如果具有无散度列的矩阵值字段\（\ mathbf {Q}（\ mathbf {x}）\）取集合\（\ mathcal B中\（\ Omega \）内的值\）（独立于\（\ mathbf {x} \））位于非线性流形上，我们在流形上找到条件，在\（\ mathcal B \）上找到条件，在边界通量\（\ mathbf {q}（\ mathbf {x}）= \ mathbf {n}（\ mathbf {x}} \ cdot \ mathbf {Q}（\ mathbf {x}）\）（其中\（\ mathbf {n}（ \ mathbf {x}）\）是向外法线\（\局部\欧米茄\） ）力\（\ mathbf {Q}（\ mathbf {X}）\）内\（\欧米茄\）采取值的子空间\（\ mathcal d \ ）。这会强制\（\ mathbf {q}（\ mathbf {x}} \）取\（\ mathbf {n}（\ mathbf {x}）\ cdot \ mathcal D \）中的值。我们发现\（\ Omega \）内还有其他无散度场，这些场又会生成其他边界场等式。因此，存在矢量势\（\ mathbf {w} \）及其梯度的部分零拉格朗日函数，函数\（F（\ mathbf {w}，\ nabla \ mathbf {w}}）\当\（\ nabla \ mathbf {w} \）被约束为\（\ omegab中的\（\ mathbf {x} \）以采用适当的非线性流形的某些集合\（\ mathcal A \）中的值，并且当\（\ mathbf {w} \）满足适当的边界条件时。还概述了对某些非线性最小化问题的扩展。