Preliminaries

Abstract

Let Ω ⊂ ℝ^d, d = 2, 3 be a bounded domain. In this contents, we refer to L ²(Ω),H ^k(Ω),H ₀ ^k (Ω),k an integer, as the standard Lebesgue and Sobolev spaces, see [9] for further details. These spaces are endowed with the standard scalar products and their induced norms ‖· ‖ _k . Further, H ^−k(Ω) is the space that is dual to H ^k (Ω) ∩ H ₀ ¹ (Ω). We make frequent use of the notation (·, ·) ≡ (·, ·) _L ². L ₀ ² is the subspace of L ²(Ω) consisting of functions with vanishing spatial average, which is isomorphic to L ²(Ω)/ℝ. Finally, we employ the notation H ^k/ℝ := H ^k ∩ L ²/ℝ.