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Limiting behaviour of intrinsic seminorms in fractional order Sobolev spaces

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Arcangéli R., Torrens J.

Studia Mathematica

2013

tom 214

nr 2

101-120

We collect and extend results on the limit of $σ^{1-k}(1-σ)^{k}|v|_{l+σ,p,Ω}^{p}$ as σ → 0⁺ or σ → 1¯, where Ω is ℝⁿ or a smooth bounded domain, k ∈ {0,1}, l ∈ ℕ, p ∈ [1,∞), and $|·|_{l+σ,p,Ω}$ is the intrinsic seminorm of order l+σ in the Sobolev space $W^{l+σ,p}(Ω)$. In general, the above limit is equal to $c[v]^{p}$, where c and [·] are, respectively, a~constant and a seminorm that we explicitly provide. The particular case p = 2 for Ω = ℝⁿ is also examined and the results are then proved by using the Fourier transform.

Ograniczanie wyników

1 Studia Mathematica

1 Arcangéli R.

1 Torrens J.

1 2013

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Limiting behaviour of intrinsic seminorms in fractional order Sobolev spaces