Heat invariants of the perturbed polyharmonic Steklov problem
Genqian Liu · arXiv (Cornell University) · 2014
For a given bounded domain $Ω$ with smooth boundary in a smooth Riemannian manifold $(\mathcal{M},g)$, we establish a procedure to get all the coefficients of the asymptotic expansion of the trace of the heat kernel associated with the perturbed polyharmonic Dirichlet-to-Neumann operator $Λ_m$ ($m\ge 1$) as $t\to 0^+$. We also explicitly calculate the first four coefficients of this asymptotic expansion. These coefficients (i.e., heat invariants) provide precise information for the area and curvatures of the boundary $\partial Ω$ in terms of the spectrum of the perturbed polyharmonic Steklov problem. In particular, when $m=1$ and $q\equiv 0$ our work recovers the previous corresponding results in \cite{PS} and \cite{Liu3}.