# Some orthogonal decompositions of Sobolev spaces and applications

Colloquium Mathematicae (2001)

- Volume: 89, Issue: 2, page 199-212
- ISSN: 0010-1354

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topH. Begehr, and Yu. Dubinskiĭ. "Some orthogonal decompositions of Sobolev spaces and applications." Colloquium Mathematicae 89.2 (2001): 199-212. <http://eudml.org/doc/284219>.

@article{H2001,

abstract = {Two kinds of orthogonal decompositions of the Sobolev space W̊₂¹ and hence also of $W₂^\{-1\}$ for bounded domains are given. They originate from a decomposition of W̊₂¹ into the orthogonal sum of the subspace of the $Δ^\{k\}$-solenoidal functions, k ≥ 1, and its explicitly given orthogonal complement. This decomposition is developed in the real as well as in the complex case. For the solenoidal subspace (k = 0) the decomposition appears in a little different form.
In the second kind decomposition the $Δ^\{k\}$-solenoidal function spaces are decomposed via subspaces of polyharmonic potentials. These decompositions can be used to solve boundary value problems of Stokes type and the Stokes problem itself in a new manner. Another kind of decomposition is given for the Sobolev spaces $W^\{m\}_\{p\}$. They are decomposed into the direct sum of a harmonic subspace and its direct complement which turns out to be $Δ(W^\{m+2\}_\{p\} ∩ W̊_\{p\}²)$. The functions involved are all vector-valued.},

author = {H. Begehr, Yu. Dubinskiĭ},

journal = {Colloquium Mathematicae},

keywords = {Sobolev spaces; orthogonal decomposition; solenoidal functions; polyharmonic functions; Stokes problem; bounded Lipschitz domain; subspaces of harmonic functions; variational problems; boundary value problems},

language = {eng},

number = {2},

pages = {199-212},

title = {Some orthogonal decompositions of Sobolev spaces and applications},

url = {http://eudml.org/doc/284219},

volume = {89},

year = {2001},

}

TY - JOUR

AU - H. Begehr

AU - Yu. Dubinskiĭ

TI - Some orthogonal decompositions of Sobolev spaces and applications

JO - Colloquium Mathematicae

PY - 2001

VL - 89

IS - 2

SP - 199

EP - 212

AB - Two kinds of orthogonal decompositions of the Sobolev space W̊₂¹ and hence also of $W₂^{-1}$ for bounded domains are given. They originate from a decomposition of W̊₂¹ into the orthogonal sum of the subspace of the $Δ^{k}$-solenoidal functions, k ≥ 1, and its explicitly given orthogonal complement. This decomposition is developed in the real as well as in the complex case. For the solenoidal subspace (k = 0) the decomposition appears in a little different form.
In the second kind decomposition the $Δ^{k}$-solenoidal function spaces are decomposed via subspaces of polyharmonic potentials. These decompositions can be used to solve boundary value problems of Stokes type and the Stokes problem itself in a new manner. Another kind of decomposition is given for the Sobolev spaces $W^{m}_{p}$. They are decomposed into the direct sum of a harmonic subspace and its direct complement which turns out to be $Δ(W^{m+2}_{p} ∩ W̊_{p}²)$. The functions involved are all vector-valued.

LA - eng

KW - Sobolev spaces; orthogonal decomposition; solenoidal functions; polyharmonic functions; Stokes problem; bounded Lipschitz domain; subspaces of harmonic functions; variational problems; boundary value problems

UR - http://eudml.org/doc/284219

ER -

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