Solutions of the divergence operator on John domains

By Gabriel Acosta, Ricardo G. Durán, M. Muschietti. 2006

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If 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 Calderon–Zygmund singular integral operator theory and the Hardy–Littlewood maximal function. For domains satisfying the separation property introduced in [S. Buckley, P. Koskela, Sobolev–Poincare implies John, Math. Res. Lett. 2 (5) (1995) 577–593], and 1<p<n, we also prove a converse result, thus characterizing in this case the domains for which a continuous right inverse of the divergence exists. In particular, our result applies to simply connected planar domains because they satisfy the separation property.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dd70ca03eed8263c532472eb73b89d5622cc1f28

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In Applied Mathematics

In Computational Theory and Mathematics

Eduard Feireisl

In Mathematical Physics

Citations in the corpus

Homogenization of the evolutionary Navier–Stokes system (2016, background): “[1] concerning the explicit construction of the so-called Bogovskii operator on John’s domains (see also Diening et al.”