arXiv 1712.035972017-12math.AP

线性PDE方程和边界场方程中格林函数的精确关系：守恒律的推广

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

Graeme W. Milton, Daniel Onofrei

备注：: 43 Pages, 3 figures

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摘要：

许多物理问题具有满足微分约束的$J(x)=L(x)E(x)+h(x)$、源$h(x)$、域$E$、$J$，用$E\in\cal E$、$J\in\cal J$表示，其中$\cal E$、$\cal J$是正交空间。如果$L(x)$在某些非线性流形$\cal M$中取值，并且有界性，有界性成立，那么格林函数满足精确恒等式。我们还联系格林函数的不同问题。分析，基于复合材料的精确关系的理论，不假设$L(x)$的微尺度变化，并允许其他方程，如在有损介质中的波。对于物体$Ω$，其中$L(x)\in{\cal M}$，Dirichlet到Neumann映射满足边界场等式。这些推广了守恒律的概念：$Ω$内部场的约束给出了边界场所满足的恒等式，并提供了内部场的额外约束。结果是这样的：如果具有$\nabla\cdot Q=0$的矩阵值域$Q(x)$在位于非线性流形上的集合$\cal B$（独立于$x$）中取值，则我们在流形上和$\cal B$上找到条件，该条件在边界通量$q(x)=n(x)\cdot Q(x)$（其中$n(x)$是$\partialΩ$的向外法线）上具有适当的条件，从而迫使$Ω$内的$Q(x)$在子空间$\cal D$中取值。这将强制$q(x)$采用$n(x)\cdot\cal D$中的值。我们发现有额外的发散自由领域内$Ω$，反过来又产生额外的边界场等式。存在部分零拉格朗日函数，矢量势$w$及其梯度的泛函$F(w,\nabla w)$，当$\nabla w$被约束为$x\inΩ$在适当的非线性流形的某些集合$\cal A$中取值时，以及当$w$满足适当的边界条件时，其充当零拉格朗日函数。某些非线性最小化问题的延伸也勾画。

英文摘要：

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 $Ω$, 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 $Ω$ 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 $\partialΩ$) force $Q(x)$ within $Ω$ 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 $Ω$ 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 $x\inΩ$ 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.

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