Short-Time Heat Content Asymptotics via the Wave and Eikonal Equations

Abstract

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 ∫Sexp(tΔ)(f1S)dV=∫SfdV-tπ∫∂SfdA+o(t),t→0+,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\int _{S} \\exp (t\\Delta ) (f \\mathbb {1}_{S}) \\,\\mathrm {d}V= \\int _S f \\,\\mathrm {d}V- \\sqrt{\\frac{t}{\\pi }} \\int _{\\partial S} f \\,\\mathrm {d}A+ o(\\sqrt{t}),\\quad t \\rightarrow 0^+, \\end{aligned}$$\\end{document}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^mDelta ^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{-tDelta }) defined by an even Schwartz function k.