In the book "[*Regularity Theory for elliptic PDE*][1]", written by Fernández-Real, page 67, $ \tilde{u}_{k} $ converge to $ \tilde{u} $ only in $ C^1 $ norm, but the result is that we can get a equation for $ \partial_{ij}\tilde{u} $. The writer gave the sentence "At this point, the passage to the limit requires a more involved argument, and can be done, for example, using the notion of viscosity solutions and their properties (see Definition 1.20 and Section 4.3)", but I cannot understand how to get the result of limits. Can you give me some hints?


  [1]: https://sma.epfl.ch/~xafernan/Documents/Fernandez-Real_Ros-Oton_Book.pdf