Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
add comment

1 Answer

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.

share|improve this answer
    
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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.