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

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be a bounded domain that can be written as $\Omega ={\bigcup }_t{\Omega }_t$, where $\{{\Omega }_t{\}}_{t\in \Gamma }$ is a countable collection of domains with certain properties. In this work, we develop a technique to decompose a function $f\in L^1(\Omega )$, with vanishing mean value, into the sum of a collection of functions $\{f_t-{\tilde{f}}_t{\}}_{t\in \Gamma }$ subordinated to $\{{\Omega }_t{\}}_{t\in \Gamma }$ such that $Supp(f_t-{\tilde{f}}_t)\subset {\Omega }_t$ and $\int f_t-{\tilde{f}}_t=0$. 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-$\alpha$ 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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