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

2013/08/20 by Fernando López García, García, Fernando López · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1308.4346

3 figures

arxiv created 2013/08/20 · arxiv updated 2013/08/21

Abstract

Let Ω⊂ ℝⁿ be a bounded domain that can be written as Ω=\bigcup_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∈ L¹(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 \di\uu=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.

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