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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.
In mathematics, and in particular in mathematical analysis, the Gagliardo–Nirenberg interpolation inequality is a result in the theory of Sobolev spaces that relates the -norms of different weak derivatives of a function through an interpolation inequality.
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 .
Whether a space supports a Poincaré inequality has turned out to have deep connections to the geometry and analysis of the space. For example, Cheeger has shown that a doubling space satisfying a Poincaré inequality admits a notion of differentiation. [3] Such spaces include sub-Riemannian manifolds and Laakso spaces.
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of L p-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete , i.e. a Banach space .
The trace operator can be defined for functions in the Sobolev spaces , with <, see the section below for possible extensions of the trace to other spaces. Let Ω ⊂ R n {\textstyle \Omega \subset \mathbb {R} ^{n}} for n ∈ N {\textstyle n\in \mathbb {N} } be a bounded domain with Lipschitz boundary.
This is a documentation subpage for Template:Adams Fournier Sobolev Spaces. It may contain usage information, categories and other content that is not part of the original template page. Calling
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