Some orthogonal decompositions of Sobolev spaces and applications

Volume 89 / 2001

H. Begehr, Yu. Dubinskiĭ Colloquium Mathematicum 89 (2001), 199-212 MSC: Primary 46E35, 46E30; Secondary 35Q30, 31B30, 32F99. DOI: 10.4064/cm89-2-5


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.


  • H. BegehrI. Math. Institut
    Freie Universität Berlin
    Arnimallee 3
    D-14195 Berlin, Germany
  • Yu. DubinskiĭMoscow Power Engineering Institute
    Krasnokazarmennaja 14
    Moscow 111250, Russia

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