TY - JOUR

T1 - Solvability and Regularity for an Elliptic System Prescribing the Curl, Divergence, and Partial Trace of a Vector Field on Sobolev-Class Domains

AU - Cheng, C. H.Arthur

AU - Shkoller, Steve

N1 - Publisher Copyright:
© 2016, Springer International Publishing.

PY - 2017/9/1

Y1 - 2017/9/1

N2 - 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.

AB - 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.

KW - Hodge theory

KW - Sobolev-class coefficients

KW - Sobolev-class domains

KW - elliptic estimates

KW - regularity

UR - http://www.scopus.com/inward/record.url?scp=85027561728&partnerID=8YFLogxK

U2 - 10.1007/s00021-016-0289-y

DO - 10.1007/s00021-016-0289-y

M3 - 期刊論文

AN - SCOPUS:85027561728

SN - 1422-6928

VL - 19

SP - 375

EP - 422

JO - Journal of Mathematical Fluid Mechanics

JF - Journal of Mathematical Fluid Mechanics

IS - 3

ER -