A Characterization of Harmonic $$L^r$$ L r -Vector Fields in Two-Dimensional Exterior Domains
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A Characterization of Harmonic Lr -Vector Fields in Two-Dimensional Exterior Domains Matthias Hieber1 · Hideo Kozono2,3 · Anton Seyfert1 · Senjo Shimizu4 Taku Yanagisawa5
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Received: 26 April 2019 © Mathematica Josephina, Inc. 2019
Abstract Consider the space of harmonic vector fields h in L r () for 1 < r < ∞ in the twodimensional exterior domain with the smooth boundary ∂ subject to the boundary conditions h · ν = 0 or h ∧ ν = 0, where ν denotes the unit outward normal to ∂. r () and V r (), respectively, it is shown that, in spite of Denoting these spaces by X har har the lack of compactness of , both of these spaces are finite dimensional and that their dimension of both spaces coincides with L for 2 < r < ∞ and L − 1 for 1 < r ≤ 2. . Here L is the number of disjoint simple closed curves consisting of the boundary ∂. Keywords Helmholtz–Weyl decomposition · Exterior domains · Harmonic vector fields · Betti number
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Senjo Shimizu [email protected] Matthias Hieber [email protected] Hideo Kozono [email protected] Anton Seyfert [email protected] Taku Yanagisawa [email protected]
1
Fachbereich Mathematik, Technische Universität Darmstadt, Schlossgartenstr. 7, 64289 Darmstadt, Germany
2
Department of Mathematics, Waseda University, Tokyo 169-8555, Japan
3
Research Alliance Center of Mathematical Sciences, Tohoku University, Sendai 980-8578, Japan
4
Graduate School of Human and Environmental Studies, Kyoto University, Kyoto 606-8501, Japan
5
Department of Mathematics, Nara Women’s University, Nara 630-8506, Japan
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M. Hieber et al.
Mathematics Subject Classification 35J57 · 35Q35
1 Introduction The Helmholtz–Weyl decomposition has a long history in physics and mathematics and is one of the fundamental theorems in the fluid mechanics. Roughly speaking, it describes a vector field in terms of its divergence-free and rotation-free components. Such a description simplifies for example the analysis of vector fields since central properties like incompressibility or vorticity can be investigated on the components directly. Regarding such as decomposition from a geometric point of view leads us to the de Rham–Hodge–Kodaira decomposition on compact Riemannian manifolds with or without boundaries, see, e.g., the monographs by Morrey [14] and Abraham et al. [1]. Whereas the classical theory of the Helmholtz–Weyl is mainly concerned with the L 2 -setting, it is nowadays well understood that the existence of such a decomposition within the L r -framework for 1 < r < ∞ is of essential importance in many problems of analysis, and in particular in the analysis of the Navier–Stokes equations. Let D ⊂ R2 be a bounded domain with the smooth boundary ∂ D and let 1 < r < ∞. It was then shown by the second and the fifth authors in [13] that the space L r (D) may be decomposed in two ways as direct sums of the form L (D) = r
X har (D) ⊕ rot Vσr (D) ⊕ ∇ H 1,r (D), Vhar (D) ⊕ rot X σr (D) ⊕ ∇ H01,r (D),
(1.1)
where the above spaces are defined by X har (D) :=
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