Spectral invariants of the Stokes problem

2014/10/16 by Liu, Genqian
#35P20 #35Q30 #58J35 #76D07 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1410.4279

Abstract

For a given bounded domain Ω⊂ \Bbb Rⁿ 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→ 0⁺. These coefficients (i.e., heat invariants) provide precise information for the volume of the domain Ω and the surface area of the boundary ∂ Ω 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.

Related

• One can know the area and total curvatures of the boundary by hearing the resonances of a Stokes flow
• Stokes Resolvent Estimates in Spaces of Bounded Functions
• Generic properties of the spectrum of the Stokes system with Dirichlet boundary condition in R³
• Behaviour of the Stokes operators under domain perturbation
• Heat invariants of the Steklov problem