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Let $\Omega \subset \mathbb{R}^N (N \geq 2)$ an open and bounded set with smooth boundary. Let $(u_n)$ a sequence of real functions defined in $\Omega$. Suppose that $u_n \in C^{2}(\Omega) \cap C(\overline{\Omega})$ with $|\Delta u_n(x)| \leq C$ for all $x \in \Omega, $ and for all $n \in \mathbb{N}$ where the constant $C$ does not depend on $n \in \mathbb{N}.$ My question is the sequence $u_n$ is equicontinuous? I am trying to obtain this by using the mean value theorem, but i am getting anywhere.

Someone could help me to prove or disprove the statemente?

thanks in advance

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Take $\Omega = B(0,1) \subset \mathbb R^2$ and take $u_n = n (x^2 - y^2)$. Then $\Delta u_n = 0$ for each $n$. But surely this family of functions is not equicontinuous?

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  • $\begingroup$ Even simpler: take $u_n\equiv n$. Maybe OP forgot to mention boundary conditions? $\endgroup$ – Bowditch Feb 28 '17 at 18:57
  • $\begingroup$ @Bowditch, $u_n = n$ is equicontinuous, actually! $\endgroup$ – Kenny Wong Feb 28 '17 at 19:02
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    $\begingroup$ Indeed, you are right -- I was thinking of the hypotheses of Arzela-Ascoli! $\endgroup$ – Bowditch Feb 28 '17 at 19:59

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