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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 .
Sobolev's original proof of the Sobolev embedding theorem relied on the following, sometimes known as the Hardy–Littlewood–Sobolev fractional integration theorem. An equivalent statement is known as the Sobolev lemma in (Aubin 1982, Chapter 2). A proof is in (Stein 1970, Chapter V, §1.3). Let 0 < α < n and 1 < p < q < ∞.
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, 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.
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 ...
The Sobolev spaces , for < are defined as the closure of the set of compactly supported test functions with respect to the , ()-norm. The following alternative characterization holds: The following alternative characterization holds:
Dual to scalar-valued functions – maps – are maps , which correspond to curves or paths in a manifold. One can also define these where the domain is an interval [ a , b ] , {\displaystyle [a,b],} especially the unit interval [ 0 , 1 ] , {\displaystyle [0,1],} or where the domain is a circle (equivalently, a periodic path) S 1 , which yields ...
This category includes maps between manifolds, smooth or otherwise, particularly in geometric topology. Pages in category "Maps of manifolds" The following 14 pages are in this category, out of 14 total.