The Journal of Geometric Analysis ( IF 1.2 ) Pub Date : 2020-05-12 , DOI: 10.1007/s12220-020-00416-z Nathanael Schilling
In this short paper, we derive an alternative proof for some known (van den Berg & Gilkey 2015) short-time asymptotics of the heat content in a compact full-dimensional submanifolds S with smooth boundary. This includes formulae like
$$\begin{aligned} \int _{S} \exp (t\Delta ) (f \mathbb {1}_{S}) \,\mathrm {d}V= \int _S f \,\mathrm {d}V- \sqrt{\frac{t}{\pi }} \int _{\partial S} f \,\mathrm {d}A+ o(\sqrt{t}),\quad t \rightarrow 0^+, \end{aligned}$$and explicit expressions for similar expansions involving other powers of \(\sqrt{t}\). By the same method, we also obtain short-time asymptotics of \(\int _S \exp (t^m\Delta ^m)(f \mathbb {1}_S)\,\mathrm {d}V\), \(m \in \mathbb N\), and more generally for one-parameter families of operators \(t \mapsto k(\sqrt{-t\Delta })\) defined by an even Schwartz function k.
中文翻译：
通过波动和Eikonal方程的短时热含量渐近
在这篇简短的论文中，我们推导了一些已知的（van den Berg＆Gilkey，2015年）具有光滑边界的紧凑全维子流形S中的热含量的渐近渐近性的替代证明。这包括像
$$ \ begin {aligned} \ int _ {S} \ exp（t \ Delta）（f \ mathbb {1} _ {S}）\，\ mathrm {d} V = \ int _S f \，\ mathrm { d} V- \ sqrt {\ frac {t} {\ pi}} \ int _ {\ partial S} f \，\ mathrm {d} A + o（\ sqrt {t}），\ quad t \ rightarrow 0 ^ +，\ end {aligned} $$以及涉及\（\ sqrt {t} \）其他幂的类似扩展的显式表达式。通过相同的方法，我们还获得\（\ int _S \ exp（t ^ m \ Delta ^ m）（f \ mathbb {1} _S）\，\ mathrm {d} V \），\ （m \ in \ mathbb N \），更一般地说，是由偶数Schwartz函数k定义的一参数运算符\（t \ mapsto k（\ sqrt {-t \ Delta}）\）。
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