The heat equation with inhomogeneous Dirichlet boundary conditions
M. van den Berg, Peter Gilkey · Communications in Analysis and Geometry · 1999
We establish the existence of an asymptotic expansion for the heat content asymptotics with inhomogeneous Dirichlet boundary conditions and compute the first 5 coefficients in the asymptotic expansion. Introduction.Let M be a smooth compact Riemannian manifold of dimension m with smooth boundary e C oc (dM), let £() (t) be the total heat energy content of M where the initial temperature is 0 and where the boundary of M is kept at temperature ; see §1 for a more precise definition.Let $^ be the harmonic function with boundary value 0. It is well known that for large time the temperature profile of M approaches $^,'and that lim S{ )(t) = I )(£) has an asymptotic expansion.It is somewhat surprising in contrast to the large time behaviour that the coefficients in that expansion are locally computable.There exist locally defined geometric invariants B n on the boundary so that B n ((j)) = / 0*5 n . JdMThe coefficients B n ((j)) express the net heat flow into and out of the manifold over the boundary dM.The case = 1 has particular geometrical significance since the coefficients are then invariants of the boundary of M. ^PSRC grant K85391 (UK)279