Let $u:\mathbb R^n \to \mathbb R^m$ be a solution of $\Delta u=({\nabla}_u f(u))^T (*)$ for some non-negative function $f:\mathbb R^m \to \mathbb R$. I'm only interested in $n=1,n=2$.

I try to understand the regularity of this solution under the following cases:

  1. $u\in W^{1,2}_{loc}(\mathbb R)$ is a weak solution of $(*)$ and $f$ is only continuous.
  2. $u\in W^{1,2}_{loc}(\mathbb R)$ is a classical solution of $(*)$ and $f\in C^1$.
  3. $u\in W^{1,2}_{loc}(\mathbb R^2)$ is a classical solution of $(*)$ and $f\in C^3$.

I believe that the answers for 2. adn 3. are that $u\in C^2$ and $u\in C^4$ respectively. For the first one I believe that $u\in C^1$ but I'm a bit unsure.

I'm looking for a general elliptic regularity Theorem that I might miss, which I could apply to the above cases. Is there any Theorem that guarantees the implication $f\in C^k \Rightarrow u\in C^{k+1}$?

I would really appreciate if somebody could take the time to enlighten me. I've already searched on Gilbarg&Trudinger's book but I'm completely lost.

Thanks in advance!


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