Leray's inequality in general multi-connected domains in ℝ ⁿ

Consider the stationary Navier-Stokes equations in a bounded domain Ω ⊃ ℝ ⁿ whose boundary ∂Ω consists of L + 1 smooth (n - 1)-dimensional closed hypersurfaces Γ ₀, Γ ₁, . . ., Γ _L, where Γ ₁, . . ., Γ _L lie inside of Γ ₀ and outside of one another. The Leray inequality of the given boundary data β on ∂Ω plays an important role for the existence of solutions. It is known that if the flux γ _i ≡ ∫ _Γi β · νd S = 0 on Γ _i(ν: the unit outer normal to Γ _i) is zero for each i = 0, 1, . . ., L, then the Leray inequality holds. We prove that if there exists a sphere S in Ω separating ∂Ω in such a way that Γ ₁, . . ., Γ _k (1 ≦ k ≦ L) are contained inside of S and that the others Γ _k+1, . . ., Γ _L are outside of S, then the Leray inequality necessarily implies that γ ₁ + · · · + γ _k = 0. In particular, suppose that there are L spheres S ₁, . . ., S _L in Ω lying outside of one another such that Γ _i lies inside of S _i for all i = 1, . . ., L. Then the Leray inequality holds if and only if γ ₀ = γ ₁ = · · · = γ _L = 0.