I am studying PDE now using mostly Evan's book. I have been thinking about the following problem and I don't know whether there is a solution or not.
Problem Formulation: Let $A$ be a subset of $R^n$. Assume further $A$ is both open and bounded. Let $a \in L^p(A)$ where $1<p<\infty$ and $A$ has $C^1$ boundary. Can we find a sufficient and necessary condition such that $a \in W_0^{1,p}$. I was motivated by inequalities http://www.cims.nyu.edu/~chou/notes/pde11 in the end of this material and by the MathSE post Proof of Sobolev Inequality Theroem.
Maybe this is well know fact, sorry if it is.