Solutions of the divergence operator on John domains

AbstractIf Ω⊂Rn is a bounded domain, the existence of solutions u∈W01,p(Ω) of divu=f for f∈Lp(Ω) with vanishing mean value and 1<p<∞, is a basic result in the analysis of the Stokes equations. It is known that the result holds when Ω is a Lipschitz domain and that it is not valid for domains with external cusps.In this paper we prove that the result holds for John domains. Our proof is constructive: the solution u is given by an explicit integral operator acting on f. To prove that u∈W01,p(Ω) we make use of the Calderón–Zygmund singular integral operator theory and the Hardy–Littlewood maximal function.For domains satisfying the separation property introduced in [S. Buckley, P. Koskela, Sobolev–Poincaré implies John, Math. Res. Lett. 2 (5) (1995) 577–593], and 1<p