Problem from Evans' PDE book, chapter 5, problem 5 I'm taking my first theoretical math course in a year and am bashing my head against a rock with this problem. "The sets $U,V$ are open, with $V \subset \subset U $ (compactly contained). Show that there exists a smooth function $\zeta$ such that $\zeta \equiv 1$ on V, $\zeta = 0$ near $\delta U$. (Hint: Take $V \subset \subset W \subset \subset U $ and mollify $\chi_{W}$.) Now, I've figured that $\chi_{W}$ has to be 1 on $W$ and 0 elsewhere, and that the clue is to mollify $\chi_{W}$ so that the function spends the space between $\delta V$ and $\delta U$ going to zero. How to formalise this into mathematics and "show" it... I'm at a fault. Does anyone have any suggestions?

The reason why you need to find a space $W$ satisfying $V\subset\subset W \subset\subset U$ is that you need to smoothly extend the function from set $V$, which takes value 1 to the set close to the boundary of $U$, which takes value $0$. So consider the classical solution of Laplace equation with scale and shift terms included to match the structure.