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.


  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!

  • 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
  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.

| cite | improve this answer | |
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.