Heat content asymptotics for higher order Laplacians
Let \(M\) be a compact Riemannian manifold with smooth boundary and let \(P\) be a partial differential operator of even order with scalar positive leading symbol. \(P\) is acting on smooth sections of a vector bundle \(V\). Furthermore, assume that an elliptic boundary condition, \({\mathcal B}\), of Lopatinski-Shapiro type is given. Then the heat semigroup of \(P_{\mathcal B}\), \(e^{-t P_{\mathcal B}}\), is well-defined. Finally let \(f_1\in C^\infty(V), f_2\in C^\infty(V^*)\) be smooth sections of \(V\) resp. the dual bundle \(V^*\). Then the author considers the heat content: \[ \beta(f_1,f_2,P,{\mathcal B})(t)=\int_M \langle e^{-t P_{\mathcal B}} f_1,f_2\rangle dx. \] It follows from standard elliptic theory that \(\beta(t)\) has an asymptotic expansion as \(t\to 0+\) of the form \(\beta(t)\sim_{t\to 0+} \sum_{n=0}^\infty \beta_n t^{n/\text{ord P}}\). The coefficients \(\beta_n=\beta_n(f_1,f_2,P,{\mathcal B})\) depend on the sections \(f_1,f_2\), the operator and the boundary condition. The heat content asymptotics is somewhat related to the heat trace asymptotics, however it is a relatively new subject introduced by \textit{M. van den Berg} and \textit{E. B. Davies} [Math. Z. 202, 463-482 (1989; Zbl 0687.35045)]. The explicit calculation of (some of) the coefficients \(\beta_n\) has been the subject of extensive study. So far these explicit calculations are limited to second order operators, e.g. generalized Laplacians. In the current paper under review the author considers higher order Laplacians, i.e. operators of the form \(\Delta^v+\) lower order terms, where the lower order terms have to satisfy certain further conditions. The author calculates \(\beta_0,\beta_1,\beta_2,\beta_3\) for arbitrary higher order Laplacians and, in addition, \(\beta_4\) for \(4\)th order Laplacians with Dirichlet or Neumann type boundary conditions. The main technique used is invariance theory.
scientific article; zbMATH DE number 1308121