The heat equation with inhomogeneous Dirichlet boundary conditions
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.
inhomogeneous Dirichlet boundary conditions
scientific article; zbMATH DE number 1309814