Spectral invariants of the Stokes problem

Abstract

For a given bounded domain $\Omega\subset {\Bbb R}^n$ with smooth boundary, we explicitly calculate the first two coefficients of the asymptotic expansion of the heat trace associated with the Stokes operator as $t\to 0^+$. These coefficients (i.e., heat invariants) provide precise information for the volume of the domain $\Omega$ and the surface area of the boundary $\partial \Omega$ in terms of the spectrum of the Stokes problem. As an application, we show that an $n$-dimensional ball is uniquely defined by its Stokes spectrum among all Euclidean bounded domains with smooth boundary.