Some orthogonal decompositions of Sobolev spaces and applications

H. Begehr¹ ; Yu. Dubinskiĭ²

¹ I. Math. Institut Freie Universität Berlin Arnimallee 3 D-14195 Berlin, Germany
² Moscow Power Engineering Institute Krasnokazarmennaja 14 Moscow 111250, Russia

Colloquium Mathematicum, Tome 89 (2001) no. 2, pp. 199-212

Résumé

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.

Détail

DOI : 10.4064/cm89-2-5

Keywords: kinds orthogonal decompositions sobolev space mathring hence bounded domains given originate decomposition mathring orthogonal sum subspace mit delta solenoidal functions its explicitly given orthogonal complement decomposition developed real complex solenoidal subspace decomposition appears little different form second kind decomposition mit delta solenoidal function spaces decomposed via subspaces polyharmonic potentials these decompositions solve boundary value problems stokes type stokes problem itself manner another kind decomposition given sobolev spaces decomposed direct sum harmonic subspace its direct complement which turns out mit delta cap mathring functions involved vector valued