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Some orthogonal decompositions of Sobolev spaces and applications

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Begehr H., Dubinskiĭ Y.

Colloquium Mathematicum

2001

tom 89

nr 2

199-212

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.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Begehr H.

1 Dubinskiĭ Y.

1 2001

Wyniki wyszukiwania

Some orthogonal decompositions of Sobolev spaces and applications