Let $u_n $ be a sequence in $H^1 (\Omega,\mathbb{C})$ where $\Omega \subset \mathbb{R} ^N$ is bounded. Assume that we know that all the functions $u_n$ are smooth and $\Vert u_n \Vert _{C^m} \leq K(m,N)$ and moreover $\vert u_n \vert \leq K_1$ and $\vert \nabla u_n \vert\leq K_2$ where $K_i>0$ is a constant (not depending on $n$). I wonder if $\lbrace u_n\rbrace$ is strongly convergent in $H^1 (\Omega)$, up to a subsequence.

Since I have the uniform bounds $\vert u_n \vert \leq K_1$ and $\vert \nabla u_n \vert\leq K_2$, a good attempt is to use Dominated Convergence Theorem. But then I also need convergence pointwise of $u_n$ to some $u\in H^1(\Omega,\mathbb{C})$. If I show that $u_n$ converges pointwise to $u$ then, by DCT in $L^p$ spaces, $u_n \rightarrow u$ in $L^2$. Similarly, $\nabla u_n \rightarrow \nabla u$ in $L^2$. So, therefore $u_n \rightarrow u$ in $H^1$. I think that this reasoning is right... so I only have to prove that $u_n$ and $\nabla u_n$ converge pointwise. Any help to show that?

Thank you in advance.

  • $\begingroup$ What rules out an "oscillating" sequence like $u_n = (-1)^n$? Or do you mean to ask for convergence of a subsequence? $\endgroup$ Nov 1 '19 at 18:55
  • $\begingroup$ Good point, I mean up to a subsequence. I have edited the post. $\endgroup$ Nov 1 '19 at 18:56
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    $\begingroup$ Okay, so with your $C^m$ bounds, Arzela-Ascoli should give you that a subsequence converges uniformly on compact sets, along with its gradients. $\endgroup$ Nov 1 '19 at 18:57
  • $\begingroup$ So I will only have the pointwise a.e. convergence on compact subsets of $\Omega$? What about if $\Omega$ is compact? $\endgroup$ Nov 1 '19 at 19:00
  • $\begingroup$ Well, uniform convergence on compact sets certainly implies pointwise convergence (points are compact :-) ) So this gives you what you need, and more. $\endgroup$ Nov 1 '19 at 19:01

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