Dual of Sobolev space $W^{1,p}(U)$ for $U$ an arbitrary subset of $\mathbb R^n$

Question

this question may be shameful, but nevertheless I can't help myself.

Let $U \subset \mathbb R^n$ be arbitrary, in particular not the whole of the space itself. I wonder about the dual of the space $W^{1,p}(U)$, for $p < \infty$.

For $U = \mathbb R^n$, we have $(W^{1,p})' = (W^{1,p'})$ with $p' = \frac{p}{p-1}$. How about different $U$?

For example, in case $U = B_1(0)$ being the closed $1$-Ball, it seems the dual is not a function space. Just recall that the trace is well-defined, linear and continuous on $W^{1,p}(U)$ and, with $S_1$ the boundary of $B_1(0)$ and $w \in L^p(S_1)$, we are given are continuous linear functional by

$ W^{1,p}(B_1(0)) \longrightarrow \mathbb C \, , f \mapsto \int_{S_1} w \cdot tr f dx $.

In fact, I wouldn't be surprised if the above example were somehow prototypical, but I have no clue how to proceed from this point. I regard this relevant, as these spaces are ubiquitous in analysis.

Thank you!

Answer 1

I think you can find the answer yourself - I will just give you a hint: You are considering the wrong pairing between $W^{1,p}(U)$ and $W^{1,p'}(U)$. (The pairing you consider doesn't give you the desired isomorphism even in the $\mathbb R^n$ case.)

Answer 2

How about embedding $W^{1,p}$ in $L^p\times (L^p)^n$ and using Hahn-Banach and Riesz' representation theorem to get a nice characterization of elements in the dual