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

• Published in: Journal of mathematical analysis and applications Vol. 418; no. 1; pp. 79 - 99
• Main Author: López García, Fernando
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.10.2014
• Subjects: Bad domains; Cuspidal domains; Decomposition; Divergence problem; Hölder-α domains; Stokes equations; Weighted Sobolev spaces; Bad domains; Hölder-α domains; Decomposition; Divergence problem; Cuspidal domains; Weighted Sobolev spaces; Stokes equations
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2014.03.080

Let Ω⊂Rn be a bounded domain that can be written as Ω=⋃tΩt, where {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), with vanishing mean value, into the sum of a collection of functions {ft−f˜t}t∈Γ subordinated to {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.