Sobolev mapping

 ( Manifolds ) 

In mathematics, a Sobolev mapping is a mapping between 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 .

(2026). 9780444528339 ISBN 9780444528339

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Definition Given Riemannian manifolds $M$ and $N$, which is assumed by Nash's smooth embedding theorem without loss of generality to be isometrically embedded into $\backslash mathbb\{R\}^\backslash nu$ as

(2026). 9780821841907 . ISBN 9780821841907

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W^{s, p} (M, N)
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\{u \in W^{s, p} (M, \mathbb{R}^\nu) \, \vert \, u (x) \in N \text{ for almost every } x \in M\}. First-order ($s=1$) Sobolev mappings can also be defined in the context of .

(2026). 9780387856476 ISBN 9780387856476

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Approximation The strong approximation problem consists in determining whether smooth mappings from $M$ to $N$ are dense in $W^\{s,\; p\}\; (M,\; N)$ with respect to the norm topology. When $sp\; >\; \backslash dim\; M$, Morrey's inequality implies that Sobolev mappings are continuous and can thus be strongly approximated by smooth maps. When $sp\; =\; \backslash dim\; M$, Sobolev mappings have vanishing mean oscillation and can thus be approximated by smooth maps.

When , the question of density is related to obstruction theory: $C^\backslash infty\; (M,\; N)$ is dense in $W^\{1,\; p\}\; (M,\; N)$ if and only if every continuous mapping on a from a $\backslash lfloor\; p\backslash rfloor$–dimensional triangulation of $M$ into $N$ is the restriction of a continuous map from $M$ to $N$.

The problem of finding a sequence of weak approximation of maps in $W^\{1,\; p\}\; (M,\; N)$ is equivalent to the strong approximation when $p$ is not an integer. When $p$ is an integer, a necessary condition is that the restriction to a $\backslash lfloor\; p\; -\; 1\backslash rfloor$-dimensional triangulation of every continuous mapping from a $\backslash lfloor\; p\backslash rfloor$–dimensional triangulation of $M$ into $N$ coincides with the restriction a continuous map from $M$ to $N$. When $p\; =\; 2$, this condition is sufficient. For $W^\{1,\; 3\}\; (M,\; \backslash mathbb\{S\}^2)$ with $\backslash dim\; M\; \backslash ge\; 4$, this condition is not sufficient.

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Homotopy The homotopy problem consists in describing and classifying the path-connected components of the space $W^\{s,\; p\}(M,\; N)$ endowed with the norm topology. When and $\backslash dim\; M\; \backslash le\; sp$, then the path-connected components of $W^\{s,\; p\}\; (M,\; N)$ are essentially the same as the path-connected components of $C(M,\; N)$: two maps in $W^\{s,\; p\}\; (M,\; N)\; \backslash cap\; C\; (M,\; N)$ are connected by a path in $W^\{s,\; p\}\; (M,\; N)$ if and only if they are connected by a path in $C(M,\; N)$, any path-connected component of $W^\{s,\; p\}\; (M,\; N)$ and any path-connected component of $C\; (M,\; N)$ intersects $W^\{s,\; p\}\; (M,\; N)\; \backslash cap\; C\; (M,\; N)$ non trivially. When $\backslash dim\; M\; >\; p$, two maps in $W^\{1,\; p\}\; (M,\; N)$ are connected by a continuous path in $W^\{1,\; p\}\; (M,\; N)$ if and only if their restrictions to a generic $\backslash lfloor\; p\; -\; 1\backslash rfloor$-dimensional triangulation are homotopic.

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Extension of traces

The classical Trace operator states that any Sobolev map $u\; \backslash in\; W^\{1,\; p\}\; (M,\; N)$ has a trace $Tu\; \backslash in\; W^\{1\; -\; 1/p,\; p\}\; (\backslash partial\; M,\; N)$ and that when $N\; =\; \backslash mathbb\{R\}$, the trace operator is onto. The proof of the surjectivity being based on an averaging argument, the result does not readily extend to Sobolev mappings. The trace operator is known to be onto when $\backslash pi\_\{1\}\; (N)\; \backslash simeq\; \backslash dotsb\; \backslash pi\_\{\backslash lfloor\; p\; -\; 1\backslash rfloor\}(N)\; \backslash simeq\; \backslash \{0\backslash \}$ or when $p\backslash ge\; 3$, $\backslash pi\_\{1\}\; (N)$ is finite and $\backslash pi\_\{2\}\; (N)\; \backslash simeq\; \backslash dotsb\; \backslash pi\_\{\backslash lfloor\; p\; -\; 1\backslash rfloor\}(N)\; \backslash simeq\; \backslash \{0\backslash \}$. The surjectivity of the trace operator fails if $\backslash pi\_\{\backslash lfloor\; p\; -\; 1\backslash rfloor\}\; (N)\backslash not\; \backslash simeq\; \backslash \{0\backslash \}$ or if $\backslash pi\_\{\backslash ell\}\; (N)$ is infinite for some $\backslash ell\; \backslash in\; \backslash \{1,\; \backslash dotsc,\; \backslash lfloor\; p\; -\; 1\backslash rfloor\backslash \}$.

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Lifting

Given a covering map $\backslash pi\; :\; \backslash tilde\{N\}\; \backslash to\; N$, the lifting problem asks whether any map $u\; \backslash in\; W^\{s,\; p\}\; (M,\; N)$ can be written as $u\; =\; \backslash pi\; \backslash circ\; \backslash tilde\{u\}$ for some $\backslash tilde\{u\}\; \backslash in\; W^\{s,\; p\}\; (M,\; \backslash tilde\{N\})$, as it is the case for continuous or smooth $u$ and $\backslash tilde\{u\}$ when $M$ is simply-connected in the classical lifting theory. If the domain $M$ is simply connected, any map $u\; \backslash in\; W^\{s,\; p\}\; (M,\; N)$ can be written as $u\; =\; \backslash pi\; \backslash circ\; \backslash tilde\{u\}$ for some $\backslash tilde\{u\}\; \backslash in\; W^\{s,\; p\}\; (M,\; N)$ when $sp\; \backslash ge\; \backslash dim\; M$,

(2026). 9780821841907 ISBN 9780821841907

when $s\backslash ge\; 1$ and and when $N$ is compact, and . There is a topological obstruction to the lifting when and an analytical obstruction when .

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Further reading

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Categories: Manifolds, Maps Of Manifolds, Sobolev Spaces, Homotopy Theory

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