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 $$ u(x,y)=\begin{cases} 1-x & \text{if } x>0, \ |y|<x \\ 1+x & \text{if } x<0, \ |y|<-x \\ 1-y & \text{if } y>0, \ |x|<y \\ 1+y & \text{if } y<0, \ |x|<-y \end{cases} $$ I would like to know for which $p$ 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?
Your intuition is correct, but remember that you have to prove that the functions you give (which technically aren't functions, and only defined up to measure zero) are actually the weak derivatives of $u(x,y)$, i.e., calculate that $$\int_U v \varphi \, dx = - \int_U u \varphi_x \, dx$$ where $v$ is your candidate for $u_x$, etc, and $\varphi \in C_c^{\infty}(U)$.