Connections between a Conjecture of Schiffer’s and Incompressible Fluid Mechanics

We demonstrate connections that exists between a conjecture of Schiffer’s (which is equivalent to a positive answer to the Pompeiu problem), stationary solutions to the Euler equations, and the convergence of solutions to the Navier-Stokes equations to that of the Euler equations in the limit as viscosity vanishes. We say that a domain Ω ⊆ R, d ≥ 2, has the Pompeiu property if, given that the integral of a continuous function f : R → R is zero for all translations and rotations of Ω, it follows that f is identically zero. The Pompeiu problem is to determine whether balls are the only simply connected domains with Lipschitz boundary not having the Pompeiu property. From now on we assume that Ω is a nonempty simply connected domain with Lipschitz boundary Γ, having outward unit normal n (defined almost everywhere on Γ). Williams showed in [5] that an affirmative answer to the Pompeiu problem is equivalent to Conjecture 1 of Schiffer. Conjecture 1 (Schiffer’s conjecture). Let α and λ be nonzero real numbers. Then there exists a non-identically vanishing solution ω to the overdetermined equation