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Fourier approximation and embeddings of Sobolev spaces

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Edmunds D. E., Moscatelli V. B.

Instytut Matematyczny Polskiej Akademii Nauk

EN

CONTENTS Introduction............................................................................................................ 5 1. Preliminaries............................................................................................................. 8 2. Embedding into $W^{m,p}(Ω)$ into $L^S(Ω)$ (n>1).......................................... 10 3. The case n = 1.......................................................................................................... 28 4. Embedding $W^{m,p}(Ω)$ into $L^φ(Ω)$............................................................ 29 5. Embedding $W^{m,p}_0(Ω)$ into $L^S(Ω)$ and $L^φ(Ω)$............................. 35 6. Applications to the type of the embedding.......................................................... 35 7. Unfortunate technicalities....................................................................................... 37 References.................................................................................................................... 46

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Entropy numbers of embeddings of Sobolev spaces in Zygmund spaces

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Edmunds D. E., Netrusov Y.

Studia Mathematica

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1998

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tom 128

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nr 1

71-102

EN

Let id be the natural embedding of the Sobolev space $W_p^l(Ω)$ in the Zygmund space $L_q(log L)_a(Ω)$, where $Ω = (0,1)^n$, 1 < p < ∞, l ∈ ℕ, 1/p = 1/q + l/n and a < 0, a ≠ -l/n. We consider the entropy numbers $e_k(id)$ of this embedding and show that $e_k(id) ≍ k^{-η}$, where η = min(-a,l/n). Extensions to more general spaces are given. The results are applied to give information about the behaviour of the eigenvalues of certain operators of elliptic type.

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Some remarks on variants of the Navier-Stokes equations

56%

Dyer R. H., Edmunds D. E.

Colloquium Mathematicum

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1970-1971

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tom 22

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nr 2

311-316

Ograniczanie wyników

1 Colloquium Mathematicum

1 Studia Mathematica

3 Edmunds D. E.

1 Dyer R. H.

1 Moscatelli V. B.

1 Netrusov Y.

1 1998

1 1977

1 1971

Fourier approximation and embeddings of Sobolev spaces

Entropy numbers of embeddings of Sobolev spaces in Zygmund spaces

Some remarks on variants of the Navier-Stokes equations