具有边界衰减估计的de Rham复形的支撑保持同伦
A support preserving homotopy for the de Rham complex with boundary decay estimates
AI总结：
本文构造了一个具有边界衰减估计的de Rham复形的支撑保持同伦，提供了在欧几里得空间和广义相对论中散度算子的右逆。
摘要：
我们研究了具有边界的紧致流形上的相对微分形式的de Rham复形。这种复形的链同伦高度非唯一，不同的同伦可以有不同的分析性质，特别是在边界附近。我们构造了一个具有 desirable 支持传播性质的链同伦，并且满足相对于加权Sobolev范数的估计，其中权重测量边界处的衰减。这些估计在考虑de Rham微分在边界扩张下的齐次性质时是最佳的，并且通过证明该同伦是一个b-伪微分算子而获得。作为推论，我们得到了在欧几里得空间上保持支撑在原点周围大球体上的散度算子的右逆，且满足测量无穷远处衰减的估计。此类保持支撑的右逆之前由Bogovskii构造，但其映射性质在衰减方面并不最优。作为进一步的推论，在三维情况下，我们得到了对称无迹矩阵上散度算子的右逆，因此也得到了关于平坦空间的广义相对论线性化约束算子的右逆。
英文摘要：
We study the de Rham complex of relative differential forms on compact manifolds with boundary. Chain homotopies for this complex are highly non-unique, and different homotopies can have different analytic properties, particularly near the boundary. We construct a chain homotopy that has desirable support propagation properties, and that satisfies estimates relative to weighted Sobolev norms, where the weights measure decay at the boundary. The estimates are optimal given the homogeneity properties of the de Rham differential under boundary dilation, and are obtained by showing that the homotopy is a b-pseudodifferential operator. As a corollary we obtain a right inverse of the divergence operator on Euclidean space that preserves support on large balls around the origin, and satisfies estimates that measure decay at infinity. Such a support preserving right inverse was constructed before by Bogovskii, but its mapping properties are not optimal with respect to decay. As a further corollary, in three dimensions we obtain a right inverse of the divergence operator on symmetric traceless matrices, and therefore of the linearized constraint operator of general relativity about flat space.
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