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I'm studying an anrticle on a non linear Schrodinger equation posed on $\Theta=(x\in R^2:|x|<1)$. I read this: "we will only consider initial data of Sobolev regularity $s<1/2$ and thus we will not need to specify the boundary condition on $R\times\partial\Theta$." Can someone explain me why?

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up vote 2 down vote accepted

Well, this does not really answer your question, but formally, you cannot define the trace for functions having regularity $s<1/2$.

Practically, it means that if you take fractional power of the Dirichlet-Laplace operator of order less than $1/4$, then boundary conditions disappear from the domain.

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Thank you for your answer Andras. I still have some doubt but i'm trying to figure out the question. –  hispa87 Oct 11 '12 at 23:54
    
I find these introductory remarks sometimes annoying. They are well-meant and help fellow experts placing the stuff, but if you are not experienced enough, you only understand them wen you read the paper. Go on reading, there will be some enlightening result later on. –  András Bátkai Oct 12 '12 at 7:12
    
I agree with you Andras. I'm mainly intersted in the measure theoric aspects (construction of a Gibbs measure and its properties), and i was trying to clarify quickly some analytical questions, but i will read carefully the full article and its references. –  hispa87 Oct 12 '12 at 19:22
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