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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. ==Sobolev embedding theorem== Let denote the Sobolev space consisting of all real-valued functions on whose first weak derivatives are functions in . Here is a non-negative integer and . The first part of the Sobolev embedding theorem states that if and are two real numbers such that and: : then : and the embedding is continuous. In the special case of and , Sobolev embedding gives : where is the Sobolev conjugate of , 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 . If with , 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Sobolev inequality」の詳細全文を読む スポンサード リンク
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