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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, 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, 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.
English: A DeVore-Triebel diagram describing the Sobolev embedding based on Morrey's inequality. A point (k, 1/p) represents the Sobolev space W^{k,p}. The space W^{3,p} (blue dot) embeds into the space C^{1,α} (red dot), which is the intersection of a line with slope n (space dimension) with the y-axis.
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.
A more concrete representation of the image of can be given using Sobolev-Slobodeckij spaces which generalize the concept of Hölder continuous functions to the -setting. Since ∂ Ω {\textstyle \partial \Omega } is a (n-1) -dimensional Lipschitz manifold embedded into R n {\textstyle \mathbb {R} ^{n}} an explicit characterization of these ...
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 notion of a Morse function can be generalized to consider functions that have nondegenerate manifolds of critical points. A Morse–Bott function is a smooth function on a manifold whose critical set is a closed submanifold and whose Hessian is non-degenerate in the normal direction. (Equivalently, the kernel of the Hessian at a critical ...