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Hi I would like to know if there is a definition of Sobolev space as a set of mappings $f:U\subset \mathbb R^n \rightarrow \mathbb R^m$ (instead to $\mathbb R$). In particular what should be $H^{1,2}(U)_{\mathbb R^m}$. Does it make sense to define it as the set of maps $f=(f_1,...,f_m)$ such that $f_i \in H^{1,2}(U)$? What would be a natural norm? Would it be a Hilbert space as in the case where $f$ goes to $\mathbb R$?

thank you

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up vote 2 down vote accepted

Yes, it's defined coordinate-wise. It's just the direct sum of $m$ copies of $H^{1,2}$. The direct sum of Hilbert spaces is automatically a Hilbert space: just add the inner products together.

Usually denoted $H^1(U;\mathbb R^m)$ or $W^{1,2}(U;\mathbb R^m)$. The notation $H^{1,2}$ appears redundant to me: $H$ already indicates a Hilbert space, so $p=2$ is understood.

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thank you, where can I read more about them? – inquisitor Aug 16 '12 at 22:49
@inquisitor There is not much to say about a general Sobolev map into $\mathbb R^m$: it's just an $m$-tuple of Sobolev functions. Things become interesting with the introduction of manifolds or of variational problems, and often both. These two surveys are not very old and are a good indication of what's going on in the field. – user31373 Aug 16 '12 at 23:04

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