## Some orthogonal decompositions of Sobolev spaces and applications

### Volume 89 / 2001

#### Abstract

Two kinds of orthogonal decompositions of the Sobolev space
$\mathring W{}_2^1$ and hence also
of $W^{-1}_{2}$ for bounded domains are given. They originate
from a decomposition of $\mathring W{}_2^1$ into the orthogonal sum of the subspace of the
${\mit \Delta }^{k}$-solenoidal functions, $k \ge 1$, 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 ${\mit \Delta }^{k}$-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 ${\mit \Delta }(W^{m+2}_{p}\cap \mathring W{}_p^2)$.
The functions involved are all
vector-valued.