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Nirenberg's Theorem and Singular Solutions from Traceless Symmetric Matrices
Nirenberg's theorem gives interior $C^{2,α}$ regularity for continuous viscosity solutions of fully nonlinear uniformly elliptic equations $F(D^2u)=0$ in dimension two. In dimensions five and higher, the work of Nadirashvili--Tkachev--Vlăduţ shows that such solutions need not be $C^2$. We construct
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