# implications of convergence in sobolev spaces

If we are given that $O \subset \Omega$ is open and bounded and $a: \Omega \times \mathbb{R} \times \mathbb{R}^{n} \rightarrow \mathbb{R}$. We have a sequence $\{u_{m}\}$ satisfying $$u_{m} \rightharpoonup u\text{ } \text{ in }\text{ } W^{1,\infty}_{0}(O)$$ and

$$\limsup\limits_{m \rightarrow \infty}\int_{O} \langle a(x,u_{m}, \nabla u_{m}); \nabla(u_{m}-u)\rangle dx \leq 0$$ where $\langle \text{ } \rangle$ is the scalar product.

How does this imply the following two conditions to $u_{m} \rightharpoonup u$ in $W^{1,\infty}(O)$ and $\Vert \nabla u_{m} \Vert_{L^{\infty}(O)} \leq \sigma$ where $\sigma > 0$.

Please view this pdf link Research Paper Question Refers To. The above question refers to how we can use Lemma 1 at the start of the proof of Lemma 2.

Let me know if my question is still unclear. Thanks.

• What is $p$ and what are the conditions imposed in $a$? Also, is $u_m$ some particular sequence? With this generality, the result you want is not true. – Tomás Mar 8 '14 at 20:38
• Hi Tomas please view changes to question. I have added a hyperlink to pdf in question. This is the document that I am studying. To answer your question $a$ is taken as a Caratheordory function with $|a(x,s,\xi)| \leq k(x) = \beta(|s|^{p-1} + |\xi|^{p-1})$ for almost every $x \in \Omega$, for every $(x, \xi) \in \mathbb{R} \times \mathbb{R}^{n}$ and for some $k \in L^{p^{'}}(\Omega)$, $\beta \geq 0$, $p > 1$. – Lucio D Mar 9 '14 at 7:38
• Is there any possibility of taking this discussion to chat to discuss some points regarding this paper? – Lucio D Mar 9 '14 at 12:35
• Do you have Skype? My native language is not English, but I think I can talk something. – Tomás Mar 9 '14 at 16:36
• Hi Tomas, I unfortunately don't have access to a pc with Skype at the moment. I make use of campus computers(which is why my responses might me delayed at times). Have you had a chance to look at the paper? My email is dlucio50@yahoo.com, I can check this more regularly than the forum. If possible maybe we could discuss some points on the forum chat or facebook chat? Thanks for you responses, much appreciated. – Lucio D Mar 9 '14 at 20:54