论文标题
通过波和Eikonal方程的短时热含量渐近级
Short-time heat content asymptotics via the wave and eikonal equations
论文作者
论文摘要
在这篇简短的论文中，我们为一些已知的[van den Berg＆Gilkey 2015]的替代证明是对紧凑的全维次曼菲尔德$ s $具有平稳边界的热含量的短期渐近级。这包括\ begin {qore*} \ int_ {s} \ exp（tδ）\ left（f \ mathbb 1_s \ right）\，\ mathrm {d} x = \ int_s f \ \， s} f \，\ mathrm {d} a + o（\ sqrt t），\ quad t \ rightarrow 0 \，.。 \ end {equation*}和（部分新的）显式表达式，用于涉及$ \ sqrt t $的其他功率的类似扩展。通过同样的方法，我们还获得了$ \ int_s \ exp（t^mδ^m）\ left（f \ mathbb 1_s \ right）\，\ mathrm {d} x $，$ m \ in \ mathbb n $的短期渐近学（f \ mathbb 1_s \ right）通过均匀的schwartz函数$ k $。
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 compact full-dimensional submanifolds $S$ with smooth boundary. This includes formulae like \begin{equation*} \int_{S} \exp(tΔ)\left( f \mathbb 1_S\right)\, \mathrm{d}x = \int_S f \,\mathrm{d}x - \sqrt{\frac{t}π} \int_{\partial S} f \,\mathrm{d}A + o(\sqrt t),\quad t \rightarrow 0\,. \end{equation*} and (partially new) explicit expressions for similar expansions involving other powers of $\sqrt t$. By the same method, we also obtain short-time asymptotics of $\int_S \exp(t^mΔ^m)\left(f \mathbb 1_S\right)\, \mathrm{d}x$, $m \in \mathbb N$, and more generally for one-parameter families of operators $t \mapsto k(\sqrt{-tΔ})$ defined by an even Schwartz function $k$.
