TY - JOUR
T1 - Lr-variational inequality for vector fields and the helmholtz-weyl decomposition in bounded domains
AU - Kozono, Hideo
AU - Yanagisawa, Taku
PY - 2009
Y1 - 2009
N2 - We show that every Lr-vector field on Ω can be uniquely decomposed into two spaces with scalar and vector potentials, and the harmonic vector space via operators rot and div, where Ω is a bounded domain in ℝ3 with the smooth boundary ∂Ω. Our decomposition consists of two kinds of boundary conditions such as u-v ∂Ω = 0 and u × ∂Ω = 0, where v denotes the unit outward normal to ∂Ω. Our results may be regarded as an extension of the well-known de Rham-Hodge-Kodaira decomposition of C∞-forms on compact Riemannian manifolds into Lr-vector fields on Ω. As an application, the generalized Biot-Savart law for the incompressible fluids in Ω is obtained. Furthermore, various bounds of u in Lr for higher derivatives are given by means of rot u and div u.
AB - We show that every Lr-vector field on Ω can be uniquely decomposed into two spaces with scalar and vector potentials, and the harmonic vector space via operators rot and div, where Ω is a bounded domain in ℝ3 with the smooth boundary ∂Ω. Our decomposition consists of two kinds of boundary conditions such as u-v ∂Ω = 0 and u × ∂Ω = 0, where v denotes the unit outward normal to ∂Ω. Our results may be regarded as an extension of the well-known de Rham-Hodge-Kodaira decomposition of C∞-forms on compact Riemannian manifolds into Lr-vector fields on Ω. As an application, the generalized Biot-Savart law for the incompressible fluids in Ω is obtained. Furthermore, various bounds of u in Lr for higher derivatives are given by means of rot u and div u.
KW - Betti number
KW - Div-curl lemma
KW - Harmonic vector fields
KW - L-vector fields
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U2 - 10.1512/iumj.2009.58.3605
DO - 10.1512/iumj.2009.58.3605
M3 - Article
AN - SCOPUS:70349619910
VL - 58
SP - 1853
EP - 1920
JO - Indiana University Mathematics Journal
JF - Indiana University Mathematics Journal
SN - 0022-2518
IS - 4
ER -