TY - JOUR

T1 - Lr-Helmholtz-Weyl decomposition for three dimensional exterior domains

AU - Hieber, Matthias

AU - Kozono, Hideo

AU - Seyfert, Anton

AU - Shimizu, Senjo

AU - Yanagisawa, Taku

N1 - Funding Information:
The research of H. Kozono was partially supported by JSPS Grant-in-Aid for Scientific Research (S) 16H06339 . The research of S. Shimizu was partially supported by Fostering Joint Research Program (B)-18KK0072 .
Publisher Copyright:
© 2021 Elsevier Inc.

PY - 2021/10/15

Y1 - 2021/10/15

N2 - In this article the Helmholtz-Weyl decomposition in three dimensional exterior domains is established within the Lr-setting for 1r-vector field u, there exist h∈Xharr(Ω), w∈H˙1,r(Ω)3 with divw=0 and p∈H˙1,r(Ω) such that u may be decomposed uniquely as u=h+rotw+∇p. If for the given Lr-vector field u, its harmonic part h is chosen from Vharr(Ω), then a decomposition similar to the above one is established, too. However, its uniqueness holds in this case only for the case 1r-variational inequality allowing to construct w∈H˙1,r(Ω)3 and p∈H˙1,r(Ω) for given u∈Lr(Ω)3 as weak solutions to certain elliptic boundary value problems.

AB - In this article the Helmholtz-Weyl decomposition in three dimensional exterior domains is established within the Lr-setting for 1r-vector field u, there exist h∈Xharr(Ω), w∈H˙1,r(Ω)3 with divw=0 and p∈H˙1,r(Ω) such that u may be decomposed uniquely as u=h+rotw+∇p. If for the given Lr-vector field u, its harmonic part h is chosen from Vharr(Ω), then a decomposition similar to the above one is established, too. However, its uniqueness holds in this case only for the case 1r-variational inequality allowing to construct w∈H˙1,r(Ω)3 and p∈H˙1,r(Ω) for given u∈Lr(Ω)3 as weak solutions to certain elliptic boundary value problems.

KW - Exterior domains

KW - Harmonic vector fields

KW - Helmholtz-Weyl decomposition

KW - Vector and scalar potentials

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U2 - 10.1016/j.jfa.2021.109144

DO - 10.1016/j.jfa.2021.109144

M3 - Article

AN - SCOPUS:85108638216

VL - 281

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 8

M1 - 109144

ER -