Spectral invariants of the Stokes problem

TL;DR

This paper computes the initial heat invariants of the Stokes operator for smooth bounded domains, linking spectral data to geometric properties, and proves that the Stokes spectrum uniquely determines a ball in Euclidean space.

Contribution

It explicitly calculates the first two heat invariants for the Stokes operator and demonstrates the spectral uniqueness of the ball among smooth bounded domains.

Findings

First two heat invariants relate spectrum to volume and surface area.

The Stokes spectrum uniquely identifies a ball in Euclidean space.

Provides explicit formulas for heat invariants of the Stokes operator.

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.