Can someone look at the 14th line of page 9 in this article and give a hint that how with the embedding theorem we could find $u\in W^{2,q}(\Omega)$ and how $u_n$ coverge strongly to $u$ in $W^{2,q}(\Omega)$ .
And also can you say that , why in 4th line of page 9 of the same article , we could consider $u_n \in L^{\infty}(\Omega). $
And also how we can proof $\nabla v_n \to \nabla v (a.e.)$ in 10th line of page 9 of same article.
Thanks
Similar to what the authors do in the previous paragraph, by standard regularity you get that $\{u_n\}$ is a bounded sequence in $W^{3,1}$. Now you can apply Rellich-Kondrachov theorem to the sequence $\{u_n\}$ to obtain what you need.