Solvability and Regularity for an Elliptic System Prescribing the Curl, Divergence, and Partial Trace of a Vector Field on Sobolev-Class Domains

C. H.Arthur Cheng, Steve Shkoller

Research output: Contribution to journalArticlepeer-review

14 Scopus citations

Abstract

We provide a self-contained proof of the solvability and regularity of a Hodge-type elliptic system, wherein the divergence and curl of a vector field u are prescribed in an open, bounded, Sobolev-class domain Ω ⊆ Rn, and either the normal component u· N or the tangential components of the vector field u× N are prescribed on the boundary ∂Ω. For k > n / 2 , we prove that u is in the Sobolev space Hk + 1(Ω) if Ω is an Hk + 1-domain, and the divergence, curl, and either the normal or tangential trace of u has sufficient regularity. The proof is based on a regularity theory for vector elliptic equations set on Sobolev-class domains and with Sobolev-class coefficients, and with a rather general set of Dirichlet and Neumann boundary conditions. The resulting regularity theory for the vector u is fundamental in the analysis of free-boundary and moving interface problems in fluid dynamics.

Original languageEnglish
Pages (from-to)375-422
Number of pages48
JournalJournal of Mathematical Fluid Mechanics
Volume19
Issue number3
DOIs
StatePublished - 1 Sep 2017

Keywords

  • elliptic estimates
  • Hodge theory
  • regularity
  • Sobolev-class coefficients
  • Sobolev-class domains

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