Nonhomogeneous boundary value problems for stationary navier-stokes equations in a multiply connected bounded domain

We consider the stationary Navier-Stokes equations on a multiply connected bounded domain Ω in R{double-struck}ⁿ for n = 2; 3 under nonhomogeneous boundary conditions. We present a new sufficient condition for the existence of weak solutions. This condition is a variational estimate described in terms of the harmonic part of solenoidal extensions of the given boundary data; we prove it by using the Helmholtz-Weyl decomposition of vector fields over Ω satisfying adequate boundary conditions. We also study the validity of Leray's inequality for various assumptions about the symmetry of Ω.