Stokes equations and elliptic systems with nonstandard boundary conditions
Chérif Amrouche, Nour El Houda Seloula · Comptes Rendus Mathématique · 2011
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 × n = h × n or u × n = g × n and π = π 0 on the boundary Γ . 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 ⋅ n and u × n on Γ .