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

Nathanael Schilling

doi:10.1007/s12220-020-00416-z

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.

Highlights

Let (M, g) be a complete, boundaryless,1 oriented Riemannian manifold with Laplace–Beltrami operator, and volume dV
The heat semi-group Tt := exp(t ) acting on L2(M, dV ) is well defined ( is essentially self-adjoint on Cc∞(M) [2]) and its behaviour as t → 0+ has been extensively investigated in the literature
Our aim is to show that slightly weaker results can be obtained by considerably lower technical effort

Summary

Introduction

Let (M, g) be a complete, boundaryless, oriented Riemannian manifold with Laplace–Beltrami operator , and volume dV. On a codimension-1 submanifold of M, we write d A for the induced surface (hyper)-area form. The heat semi-group Tt := exp(t ) acting on L2(M, dV ) is well defined ( is essentially self-adjoint on Cc∞(M) [2]) and its behaviour as t → 0+ has been extensively investigated in the literature. For a set S ⊂ M, the heat content of the form. S, f (t) := S Tt ( f 1S) dV , f ∈ C∞(M), has recently received much attention; see, for instance, [7,11,12] and the references therein. Let us briefly recall some known results. On Rn, sets S of finite perimeter P(S) are characterized by [7, Thm. 3.3 ]

Objectives

Conclusion