Bootstrapping For Elliptic Operators in Holder Space

Question

Do we have a proposition in the following form:

L is an elliptic operator on $\Omega$. If we have $Lu = v$ with $v\in C^{k-2,\beta}(\Omega)$ and $u\in C^{k}(\Omega)$. We actually know that $u\in C^{k,\beta}(\Omega)$?

Any reference would be nice.

Answer 1

The rough idea is correct, but you need to be careful about some specifics, for instance the smoothness of the coefficients of your operator. You can check chapters 4 and 6 of Gilbarg and Trudinger’s “Elliptic Partial Differential Equations of Second Order“ and the references therein, or Caffarelli-Cabre “Fully Nonlinear Elliptic Equations”, Chapter 8, for the nonlinear case.