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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.
[1] [2] In the following year, both authors improved their results and published them independently. [3] [4] [5] Nonetheless, a complete proof of the inequality went missing in the literature for a long time. Indeed, to some extent, both original works of Gagliardo and Nirenberg do not contain a full and rigorous argument proving the result.
When Ω is a ball, the above inequality is called a (p,p)-Poincaré inequality; for more general domains Ω, the above is more familiarly known as a Sobolev inequality. The necessity to subtract the average value can be seen by considering constant functions for which the derivative is zero while, without subtracting the average, we can have ...
In mathematics, a Sobolev mapping is a mapping between manifolds which has smoothness in some sense. Sobolev mappings appear naturally in manifold-constrained problems in the calculus of variations and partial differential equations , including the theory of harmonic maps .
Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sobolev spaces are named after the Russian mathematician Sergei Sobolev.
In mathematical analysis, Trudinger's theorem or the Trudinger inequality (also sometimes called the Moser–Trudinger inequality) is a result of functional analysis on Sobolev spaces. It is named after Neil Trudinger (and Jürgen Moser). It provides an inequality between a certain Sobolev space norm and an Orlicz space norm of a
equipped with the previous norm is a Banach space (a general definition of , (′) for non-integer > can be found in the article for Sobolev-Slobodeckij spaces). For the (n-1)-dimensional Lipschitz manifold define /, by locally straightening and proceeding as in the definition of /, (′).
Since an embedding is compact if and only if the inclusion (identity) operator is a compact operator, the Rellich–Kondrachov theorem implies that any uniformly bounded sequence in W 1,p (Ω; R) has a subsequence that converges in L q (Ω; R).