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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 .
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.
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.
The theory of interpolation of vector spaces began by an observation of Józef Marcinkiewicz, later generalized and now known as the Riesz-Thorin theorem.In simple terms, if a linear function is continuous on a certain space L p and also on a certain space L q, then it is also continuous on the space L r, for any intermediate r between p and q.
There are several mapping spaces between manifolds which can be viewed as Hilbert spaces by only considering maps of suitable Sobolev class. For example we can consider the space L M {\displaystyle \operatorname {L} M} of all H 1 {\displaystyle H^{1}} maps from the unit circle S 1 {\displaystyle \mathbf {S} ^{1}} into a manifold M ...
with Sobolev-Slobodeckij spaces , for non-integer > defined on through transformation to the planar case , (′) for ′, whose definition is elaborated in the article on Sobolev-Slobodeckij spaces. The operator T m {\textstyle T_{m}} extends the classical normal traces in the sense that
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