Some orthogonal decompositions of Sobolev spaces and applications

• Autorzy: H. Begehr , Yu. Dubinskiĭ
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 31B30: Biharmonic and polyharmonic equations and functions
• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• 46E35: Sobolev spaces and other spaces of ``smooth'' functions, embedding theorems, trace theorems
• 35Q30: Navier-Stokes equations
• 32F99: None of the above, but in this section

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2001
• Tom: 89
• Numer: 2
• Strony: 199-212

Daty

wydano

2001

Twórcy

autor

H. Begehr

• I. Math. Institut, Freie Universität Berlin, Arnimallee 3, D-14195 Berlin, Germany

autor

Yu. Dubinskiĭ

• Moscow Power Engineering Institute, Krasnokazarmennaja 14, Moscow 111250, Russia

Identyfikatory

DOI

10.4064/cm89-2-5