Stokes equations and elliptic systems with nonstandard boundary conditions

In a three-dimensional bounded possibly multiply-connected domain of class C-1.1, we consider the stationary Stokes equations with nonstandard boundary conditions of the form u . n = g and curl u x n = h x n or u x n = g x n and pi = pi(0) on the boundary Gamma. We prove the existence and uniqueness of weak, strong and very weak solutions corresponding to each boundary condition in L-P theory. Our proofs are based on obtaining Inf-Sup conditions that play a fundamental role. And finally, we give two Helmholtz decompositions that consist of two kinds of boundary conditions such as u nand u x n on Gamma. (C) 2011 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.