Questions on Nonlinear Elliptic Theory by Schauder

Question

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:

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!

Answer 1

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.