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September 9th, 2013, 11:50 AM   #1
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Equivalence of integrals

The following question is simpler than it initially looks, its about the equivalence of two integrals which are slightly different, but I thought it best to give some detail about where they come from in case I am missing something that might influence the question.

Let me know if any detail is required. By the way this is based on notes from the book Functional Anlaysis, Sobolev Spaces by Brezis, section 'Extension Operators'.

Given , write with and set .

We define





Q can be thought of as a cylinder in way ("in a way" since its N dimensional).

Let and .
Consider the following extension of to all of :



Consider the following integral:

change of variable gives us:



can we then state that:


?
If so why would this be the case?

Thanks a lot, let me know if any extra detail is required.
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September 16th, 2013, 12:36 PM   #2
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Re: Equivalence of integrals

The summary of what u're saying is that , in which case you would have to include a minus sign on your last equation line.
Now I have a question for you - how much do you know or have acquired interest for partial differential equations with Neumann boundary condition?
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September 18th, 2013, 09:15 AM   #3
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Re: Equivalence of integrals

I have a question. ? ? Qc? (Qv) so ? and itís partial derivatives are continuous then they are Borl measurable .In this case the integrand is measurable but is that integrable ?
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September 18th, 2013, 10:31 AM   #4
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Re: Equivalence of integrals

Yes, because is in so the function is both and compactly supported.
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September 19th, 2013, 02:45 AM   #5
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Re: Equivalence of integrals

This might be a silly question, but is your reason why the integrals in the first post should be equivalent if simply because and therefore ?
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