# Discrete Sobolev space of $R^n$ valued maps

Can some one tell me the reference or any idea how to take the Discrete Sobolev space work defined for a scalar valued map to the space of maps which are vector valued.Let's say
$f:\Omega \rightarrow R^n$ where $\Omega \subset R^d$. A Sobolev space of vector valued maps are defined. Though it looks pretty straightforward to lift the discrete Sobolev space of scalar valued maps to vector valued analogue but does there any work exists ?

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What is the discrete Sobolev space? –  timur Sep 19 '12 at 0:38
1) Could you be more precise about what you mean by a "discrete Sobolev space"? 2) Have you tried to do this yourself and run into any difficulties? If so, what? –  Deane Yang Sep 19 '12 at 0:38
Sorry for being too short.I am attaching a link.But what if f is a vector-valued map ?? facpub.stjohns.edu/ostrovsm/sobolev.pdf –  user26265 Sep 19 '12 at 9:54
The usual thing to do, if $F$ is vector-valued is to use the scalar valued function $f = |F|$, where $|\cdot|$ is the usual Euclidean norm. This usually produces results good enough for most applications. Have you tried this yet? If so, please explain what difficulties you've run into. In the meantime, I'm voting to close. –  Deane Yang Sep 19 '12 at 12:00
sincerely, i don't think deane's suggestion is very efficient. to begin with, when you integrate ||F|| by parts to check weak differentiability you get something which is not very well-behaved. a vector-valued function with coordinates that vanish in all but one components is probably more appropriate, if you need a test function. –  Delio Mugnolo Sep 19 '12 at 22:54

## 1 Answer

well, there are objects called bochner-lebesgue spaces which are nothing but lebesgue spaces of vector-valued functions. if the target space is finite dimensional, the theory is absolutely trivial, in the general case it is much more involved but still very well understood (see for instance this book; but the rule of thumb is that everything still is very easy if the target space is a separable, reflexive banach space). because this theory works well for a very large class of measures, including atomic ones, there is no problem carrying everything over to the case of sobolev spaces on graphs/point sets.

(i was reflecting on similar problems recently while writing this, which you might want to take a look at).

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