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I have a question on Sobolev space. This is one of exercises in Evans PDE textbook. Let $U=\{(x,y) | |x|<1, |y<1|\} \subset \mathbb{R}^2$. Define a function $u(x,y)$ by $$ u(x,y)=\begin{cases} 1-x & \text{if } x>0, \ |y|0, \ |x|$p$ the function $u$ in $W^{1,p}(U)$.

It seems to me that weak derivatives are given by $u_x(x,y)=-1,1,0,0$ and $u_y(x,y)=0,0,-1,1$ in each region respectively, but then this question is too trivial. Could someone point out any mistake I made?

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    How did you find that?2012-11-04

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