Two kinds of orthogonal decompositions of the Sobolev space W̊₂¹ and hence also of for bounded domains are given. They originate from a decomposition of W̊₂¹ into the orthogonal sum of the subspace of the -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 -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 . They are decomposed into the direct sum of a harmonic subspace and its direct complement which turns out to be . The functions involved are all vector-valued.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm89-2-5,
author = {H. Begehr and Yu. Dubinski\u\i },
title = {Some orthogonal decompositions of Sobolev spaces and applications},
journal = {Colloquium Mathematicae},
volume = {89},
year = {2001},
pages = {199-212},
zbl = {0988.46023},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm89-2-5}
}
H. Begehr; Yu. Dubinskiĭ. Some orthogonal decompositions of Sobolev spaces and applications. Colloquium Mathematicae, Tome 89 (2001) pp. 199-212. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm89-2-5/