A decomposition technique for integrable functions with applications to the divergence problem

Let Ω ⊂ R n be a bounded domain that can be written as Ω = ⋃ t Ω t , where { Ω t } t ∈ Γ is a countable collection of domains with certain properties. In this work, we develop a technique to decompose a function f ∈ L 1 ( Ω ) , with vanishing mean value, into the sum of a collection of functions { f t − f ˜ t } t ∈ Γ subordinated to { Ω t } t ∈ Γ such that supp ( f t − f ˜ t ) ⊂ Ω t and ∫ f t − f ˜ t = 0 . As an application, we use this decomposition to prove the existence of a solution in weighted Sobolev spaces of the divergence problem div u = f and the well-posedness of the Stokes equations on Hölder-α domains and some other domains with an external cusp arbitrarily narrow. We also consider arbitrary bounded domains. The weights used in each case depend on the type of domain.