一类欠定椭圆型PDE的光滑紧支撑解及其胶合应用
Smooth compactly supported solutions of some underdetermined elliptic PDE, with gluing applications
摘要：
给出了任意阶欠定椭圆型偏微分方程构造光滑紧支撑解的充分条件。特别是，我们证明了某些欠定线性椭圆算子$P$的核中的两个光滑元可以在一个选定的区域中胶合，以获得一个新的光滑解。这个新的解正好等于胶合区域外的起始元素。这一结果与通常的确定或超定椭圆算子的唯一延拓完全相反。作为推论，我们得到了$P$的核上的紧支撑解以及在所选的相对紧的开区域上消失的解。我们将结果应用于自然的几何和物理背景，如发散自由场或TT张量。
英文摘要：
We give sufficient conditions for some underdetermined elliptic PDE of any order to construct smooth compactly supported solutions. In particular we show that two smooth elements in the kernel of certain underdetermined linear elliptic operators $P$ can be glued in a chosen region in order to obtain a new smooth solution. This new solution is exactly equal to the starting elements outside the gluing region. This result completely contrasts with the usual unique continuation for determined or overdetermined elliptic operators. As a corollary we obtain compactly supported solutions in the kernel of $P$ and also solutions vanishing in a chosen relatively compact open region. We apply the result for natural geometric and physics contexts such as divergence free fields or TT-tensors.
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