[Sniffnoy]

In what way did Schauder's work turn out to be false? It simply doesn't apply to the situations discussed here.

[storus]

There was this sentence in the article: "...he realized that nonuniformly elliptic PDEs that seem well behaved can have irregular solutions even when they satisfy the condition Schauder had identified"

[Sniffnoy]

Yes, nonuniformly elliptic PDEs. Schauder's theorem applies to uniformly elliptic PDEs.