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Concept
Sobolev inequality
Summary
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(Rn) denote the Sobolev space consisting of all real-valued functions on Rn whose first k weak derivatives are functions in Lp. Here k is a non-negative integer and 1 ≤ p < ∞. The first part of the Sobolev embedding theorem states that if k > l, p < n and 1 ≤ p < q < ∞ are two real numbers such that
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 result should be interpreted as saying that if a function in has one derivative in , then itself has improved local behavior, meaning that it belongs to the space where . (Note that , so that .) Thus, any local singularities in must be more mild than for a typical function in .
The second part of the Sobolev embedding theorem applies to embeddings in Hölder spaces C r,α(Rn). If n < pk and
with α ∈ (0, 1) then one has the embedding
This part of the Sobolev embedding is a direct consequence of Morrey's inequality. Intuitively, this inclusion expresses the fact that the existence of sufficiently many weak derivatives implies some continuity of the classical derivatives. If then for every .
In particular, as long as , the embedding criterion will hold with and some positive value of . That is, for a function on , if has derivatives in and , then will be continuous (and actually Hölder continuous with some positive exponent ).
The Sobolev embedding theorem holds for Sobolev spaces W k,p(M) on other suitable domains M.
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