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In the book of Gilbar Trudinger (Elliptic partial differential equations of second order), the theorem 4.6 (page $60$) give the following estimate: $$|u|'_{2,\alpha;B_1}\leq C(|u|_{0;B_2}+R^2|f|'_{0,\alpha;B_2}),$$ where $u$ is a Poisson equation solution $(\Delta u=f)$, and $B_1$ and $B_2$ are two concentrics balls.

In the end of the page, he claims that this estimate implies in the equicontinuity on compact subdomains of the second derivatives of any bounded set of solutions of Poisson's equation.

I can see why the first derivatives are equicontinuity, because with this estimate applied in the second derivatives, implies in the boundness uniform of the second derivatives and this implies that the first derivatives of the bounded set of solutions of Poisson's equations is uniformly Lipschitz. But I can't to explain why the second derivatives are equicontinuity. Someone understood this consequence? Thank you.

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Functions that are uniformly bounded in a Holder space are equicontinuous, in this case, the second derivatives are uniformly bounded in $C^\alpha$.

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  • $\begingroup$ In fact, all derivatives are in a Holder space, this is correct? $\endgroup$ – José Carlos Feb 11 '13 at 2:58
  • $\begingroup$ Up to the 2nd derivatives, yes. $\endgroup$ – Ray Yang Feb 11 '13 at 9:33
  • $\begingroup$ This norm $|\cdot|_{2,\alpha,B}$ envolves every derivatives of order $0$, $1$ and $2$? $\endgroup$ – José Carlos Feb 11 '13 at 23:25
  • $\begingroup$ Yes, it does. It's usually defined as $\|u\|_{2,\alpha,B} = \|u\|_\infty + \|Du\|_\infty + \|D^2 u\|_\infty + [D^2 u]_{C^\alpha}$, although equivalent formulations are possible. $\endgroup$ – Ray Yang Feb 12 '13 at 15:38

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