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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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 DirichletLaplace operator of order less than $1/4$, then boundary conditions disappear from the domain. 

