I am a bit confused by weak convergence in Sobolev spaces. I am making this post to hopefully clarify some of my doubts.
Recall that in a Banach space, we say a sequence $x_n\in X$ converges weakly to $x \in X$ if for every linear functional $f \in X^{*}$ we have: $$ \langle f , x_n \rangle \to \langle f , x\rangle \quad n \to \infty $$ where the above convergence is in the sense of real numbers. Now in $L^p$ spaces, with $p \neq \infty$, we may identify the dual of $L^p$ with $L^q$, where $1/p + 1/q = 1$, and write the action of any linear functional as integration against some $L^q$ function.
In Sobolev spaces, (let us solely work with $W^{1,p}(\Omega)$), the situation seems to be more delicate. In Haim Brezis's book, there is discussion about the space $W^{1,p}_0(\Omega)$ the Sobolev spaces with $0$ trace. In this case, the book identifies the objection $W^{-1,q}(\Omega)$ as the dual of $W^{1,p}_0$, but not $W^{1,p}$. It also provides a characterization of weak convergence, that is, every functional $F \in W^{-1,q}(\Omega) = W^{1,p}_0(\Omega)^*$ can be identified with: $$ \langle F , u \rangle = \int_\Omega f_0 u(x) \mathrm{d}x +\sum_{k = 1}^{n}\int_\Omega f_k(x)\partial_ku(x) $$ where $f_0, \cdots , f_n$ is a collection of $L^q$ functions, so these integrals make sense.
My question is: are there any representations for these functionals when the space is just $W^{1,p}(\Omega)$ and not $W^{1,p}_0(\Omega)$? More generally, is there an "easy" characterization of weak convergence, say in terms of weak-$L^p$ convergence of $u$, or its derivatives?
One has a standard isometric embedding $W^{1,p}(\Omega) \hookrightarrow L^p(\Omega)^{n+1}$ defined by $u \mapsto (u, \partial_1 u, \dots, \partial_n u)$. This will tell you that $u_n \rightharpoonup u$ in $W^{1,p}(\Omega)$ if and only if $u_n \rightharpoonup u$ and $\partial_i u_n \rightharpoonup \partial_i u$ in $L^p(\Omega)$ for each $i$.
Firstly, if $u_n \rightharpoonup u$ in $W^{1,p}(\Omega)$ and $f, g_i \in L^q(\Omega)$, then it is clear that $\phi_0(v) = \int_\Omega v f d \mu$ and $\phi_i(v) = \int_\Omega (\partial_i v) g_i$ are bounded linear functionals on $W^{1,p}(\Omega)$ by Holder's inequality and so $u_n \rightharpoonup u$ and $\partial_i u_n \to \partial_i u$ in $L^p(\Omega)$.
Conversely, note that via the above embedding, if $\phi \in W^{1,p}(\Omega)^*$, we can consider $\phi$ to be a functional on a closed subspace of $L^p(\Omega)^{n+1}$. Then, by Hahn-Banach $\phi$, extends to an element of $$(L^p(\Omega)^{n+1})^* \simeq (L^p(\Omega)^*)^{n+1} \simeq L^q(\Omega)^{n+1}.$$ Hence, to see that $u_n$ converges to $u$ weakly in $W^{1,p}(\Omega)$ it suffices to check that $$\int_\Omega u_n f d\mu \to \int_\Omega u f d \mu \qquad \text{and} \qquad \int_\Omega \partial_i u_n g_i d \mu \to \int_\Omega \partial_i u g_i d\mu$$ for all $f, g_i \in L^q(\Omega)$.