The heat equation with inhomogeneous Dirichlet boundary conditions
type: Item
instance of: scholarly article
title: The heat equation with inhomogeneous Dirichlet boundary conditions
zbMATH Open document ID: 0940.58017
author: Peter B. Gilkey
author: M. van den Berg
published in: Communications in Analysis and Geometry
publication date: 2000-07-17T00:00:00Z
review text: The authors prove that there exists an asymptotic expansion for the heat content asymptotics with inhomogeneous Dirichlet boundary conditions. More precisely, let \(M\) be a smooth compact Riemannian manifold of dimension \(m\) with smooth boundary \(\partial M,\) \(\varphi \in C^\infty(\partial M)\) and \(\varepsilon (\varphi) (t)\) be the total heat energy content of \(M.\) Let us suppose that the initial temperature is \(0\) and the boundary of \(M\) is kept at temperature \(\varphi.\) The authors show that as \(t \to 0,\) \(\varepsilon (\varphi)(t)\) is equivalent to \(\sum_{n\geq 1} \beta_n(\varphi)t^{{n}\over{2}};\) then there exist locally defined geometric invariants \(B_n\) on the boundary so that \(\beta_n(\varphi)= \int_{\partial M}\varphi B_n.\) Moreover, it is proved that in the above asymptotic expansion the first five coefficients are computable.
reviewed by: Maria Alessandra Ragusa
Mathematics Subject Classification ID: 58J37
Mathematics Subject Classification ID: 58J32
Mathematics Subject Classification ID: 35S15
Mathematics Subject Classification ID: 35S35
Mathematics Subject Classification ID: 58J35
zbMATH DE Number: 1309814
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: heat content asymptotics
zbMATH Keywords: inhomogeneous Dirichlet boundary conditions
MaRDI profile type: Publication
full work available at URL: cag.1999.v7.n2.a3
OpenAlex ID: W1589123119
DOI: 10.4310/CAG.1999.V7.N2.A3
description: scientific article; zbMATH DE number 1309814