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I recently started to study about Elliptic theory and below is a brief introduction my professor made:

Let $\;u:\mathbb R^n \to \mathbb R\;$ and $\;f:\mathbb R \to \mathbb R\;$ two functions which satisfy the following:

  • $\;\Delta u=f(u(x))\;$
  • Boundary conditions over $\;\Omega\;$=open,bounded subset of $\;\mathbb R^n\;$

If $\;f\in C^\alpha\;$ then $\;u\in C^{2+\alpha}\;$. In addition $\;{\vert u \vert}_{C^{2,\alpha}(\Omega)} \le K {\vert f \vert}_{C^\alpha(\Omega)}\;$ where $\;K\;$ is a positive constant.

He also mentioned that these estimates are due to Schauder and explained to me the "bootstrap" argument.

Questions:

  1. I would like to study this result in more details but I can't find this Theorem anywhere. Are there any suggestions of books that might be helpful here?
  2. I was wondering if I could use the above in this system of equations: $\;\Delta u_i=f_{u_i}(u(x))\;$ where $\;f_{u_i}=\frac {\partial f}{\partial u_i}(u)\;\;\;\forall 1\le i \le m\;$ and $\;u:\mathbb R^n \to \mathbb R^m\;$. The fact that I have $\;f_{u_i}(u(x))\;$ instead of $\;f_{u_i}(u_i(x))\;$ confuses me a lot.

EDIT: After the suggestion in the answer below I searched on Gilbarg & Trudinger 's book and I came across with this Theorem:

enter image description here

It seems to me it's quite close to the introduction my professor made. Although the extra term $\;{\vert u \vert}_{0;B_2}\;$ confuses me a lot. I tried to read about the norms and the notation of this chapter but I'm having a really hard time getting my head around them.

I would appreciate if somebody could enlighten me about these. Is Theorem 4.6 the right one?

Any help would be valuable. Thanks in advance!

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  • 1
    $\begingroup$ There is a wikipedia page about this with a few books at the end. $\endgroup$ – Gribouillis Sep 13 '17 at 17:00
  • $\begingroup$ I think $f$ has to satisfy some additional conditions. $\endgroup$ – Yuhang Sep 13 '17 at 18:00
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  1. Gilbarg and Trudinger (Elliptic Partial Differential Equations of Second Order) is a very popular reference for this kind of PDE, with chapter 6 being devoted to Schauder estimates.
  2. Assuming $f$ is has bounded derivatives of all orders, you can use the given Schauder estimate to bootstrap a $C^{0,\alpha}$ estimate on your $u$ to $C^\infty$. The easiest step is the first: if we assume $u \in C^{\alpha}$ (that is, each component $u_i \in C^\alpha$) and $f \in C^{1,1}$, then (exercise) the composition $f_{u_i}(u(x))$ is $C^\alpha$; so the PDE $$\Delta u_i = f_{u_i}(u(x))$$ along with the Schauder estimate tells us that each $u_i$ (and thus $u$ as a whole) is $C^{2,\alpha}.$ To continue the bootstrap, differentiate the PDE to get the system $$\Delta \partial_j u_i = \sum_k f_{u_i u_k}(u(x)) \partial_j u_k$$ for the first derivatives of $u$, which we can treat similarly: if we assume $f \in C^{2,1}$ and $u \in C^{1,\alpha}$ then the RHS is $C^\alpha$, and thus Schauder tells us $\partial u \in C^{2,\alpha}$; i.e. $u \in C^{3,\alpha}$. But we already know $u\in C^{2,\alpha}$ from the first step; so the only extra information we need to get this additional regularity is the condition on $f$. Continuing in this fashion by induction, we see that if $f\in C^{k,1}$ we have the implication $u \in C^\alpha \implies u \in C^{k+1,\alpha}$.

Note that the dependence of $f$ on all the different components of $u$ isn't a difficulty here. I expect it will, however, make getting the initial $C^\alpha$ estimate harder, which the Schauder approach cannot help you with for a nonlinear equation.

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  • $\begingroup$ First of all thanks a lot for your explicit answer. However coming back to it a few days later, I have some questions to make: 1) Why is necessary for $\;f\;$ to be bounded? 2)After you differentiate the PDE again, it's not that clear to me why this sum follows 3)How should I proceed in order to show $\;f_{u_i}(u(x))\;$ is $\;C^\alpha\;$? Could you give me some hints? $\endgroup$ – kaithkolesidou Sep 18 '17 at 15:51
  • $\begingroup$ 1. I suppose we don't really need this; I just wrote $f \in C^{1,1}$ rather than $\partial f \in C^{0,1}$ out of habit. 2. This is just the multivariable chain rule applied to the composition $f_{u_i}(u(x)).$ 3. You mean with my assumptions on $f$ and $u$? This is just the fact that the composition of a $C^{0,\alpha}$ and a $C^{0,\beta}$ function is a $C^{0,\alpha\beta}$ function, which you should be able to prove from the definition of Hölder continuity. $\endgroup$ – Anthony Carapetis Sep 19 '17 at 1:04
  • $\begingroup$ Oh, when you say $\;u\in C^\alpha \;$ you mean it satisfies the Holder condition: $\;\vert u(x_1)-u(x_2) \vert \le M \vert {x_1 - x_2 \vert}^{\alpha}\;\forall x_1,x_2 \in \mathbb R^n\;,\;M \gt 0\;$. I thought $\;u\in C^\alpha\;$ means $\;u\;$ has continuous derivatives up to $\;\alpha \;$ order.I misunderstood I guess. Ok then, it's easy now. I got it! May I ask you one more question? Theorem 6.2 is the one I need by Gilbarg&Trudinger 's book? The notation there is a bit confusing to me....However you 've been really helpful. Thanks a lot! $\endgroup$ – kaithkolesidou Sep 19 '17 at 9:12
  • $\begingroup$ I just edited my post. I think I found what I needed in Theorem 4.6 but I have -again- some questions. I was wondering if you could check it out. :) $\endgroup$ – kaithkolesidou Sep 19 '17 at 10:33

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