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

Let Ω⊂RnΩ⊂Rn be a bounded domain that can be written as Ω=⋃tΩtΩ=⋃tΩt, where {Ωt}t∈Γ{Ωt}t∈Γ is a countable collection of domains with certain properties. In this work, we develop a technique to decompose a function f∈L1(Ω)f∈L1(Ω), with vanishing mean value, into the sum of a collection of functions {ft−f˜t}t∈Γ subordinated to {Ωt}t∈Γ{Ωt}t∈Γ such that supp(ft−f˜t)⊂Ωt and ∫ft−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 divu=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.