论文标题
Korevaar-Schoen在强大的空间上的能量
Korevaar-Schoen's energy on strongly rectifiable spaces
论文作者
论文摘要
我们扩展了Korevaar-Schoen的公制sobolev映射理论，以涵盖源空间为RCD空间的情况。在这种情况下，似乎没有版本的“副引理”的版本：要获得近似能量的极限的存在，又要获得极限能量的较低的半分化。 - 这样的空间是“强烈纠正”的事实，这是本质上一阶的概念（与二阶相反的类似于测量的诱导性能相反）。这一事实在与Kirchheim的度量不同定理结合使用方面特别有用，因为它允许获得近似的度量可不同结果，从而迅速为能量密度提供了表示形式 - 第一作者开发的差分演算，这要归功于我们在这里证明的能量的表示公式，以从抽象差异的闭合中获得所需的下半持续性。 当目标空间是CAT（0）时，我们还可以将能量密度识别为差分的Hilbert-Schmidt Norm，这与平滑情况一致。
We extend Korevaar-Schoen's theory of metric valued Sobolev maps to cover the case of the source space being an RCD space. In this situation it appears that no version of the `subpartition lemma' holds: to obtain both existence of the limit of the approximated energies and the lower semicontinuity of the limit energy we shall rely on: - the fact that such spaces are `strongly rectifiable' a notion which is first-order in nature (as opposed to measure-contraction-like properties, which are of second order). This fact is particularly useful in combination with Kirchheim's metric differentiability theorem, as it allows to obtain an approximate metric differentiability result which in turn quickly provides a representation for the energy density, - the differential calculus developed by the first author which allows, thanks to a representation formula for the energy that we prove here, to obtain the desired lower semicontinuity from the closure of the abstract differential. When the target space is CAT(0) we can also identify the energy density as the Hilbert-Schmidt norm of the differential, in line with the smooth situation.
