Sobolev embedding theorem

In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under slightly stronger conditions some Sobolev spaces are compactly embedded in others. They are named after Sergei Lvovich Sobolev.

Let $W k,p (R n)$ denote the Sobolev space consisting of all real-valued functions on $R n$ whose first $k$ weak derivatives are functions in $L p$ . Here $k$ is a non-negative integer and $1 \leq p < \infty$ . The first part of the Sobolev embedding theorem states that if $k > ℓ$ and $1 \leq p < q < \infty$ are two real numbers such that $(k - ℓ) p < n$ and:

then

and the embedding is continuous. In the special case of $k = 1$ and $ℓ = 0$ , Sobolev embedding gives

where $p *$ is the Sobolev conjugate of $p$ , given by

This special case of the Sobolev embedding is a direct consequence of the Gagliardo–Nirenberg–Sobolev inequality.

The second part of the Sobolev embedding theorem applies to embeddings in Hölder spaces $C r,α (R n)$ . If $(k - r - α)/ n = 1/ p$ with $α \in (0, 1]$ , then one has the embedding

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