Regularity for viscosity solutions of fully nonlinear equations \(F(D^ 2u)=0\) (Q1918506)

The authors consider fully nonlinear equations (1) \(F(D^2u)=0\). The real-valued function \(F\) is defined on the space \(S\) of real symmetric \(n\times n\) matrices, normed by \(|M|= \max|\lambda_i|\) \((\lambda_i\) eigenvalue of \(M)\), and it satisfies \[ \lambda|N|\leq F(M+N)- F(M)\leq\mu |N|\quad \text{for } M,\;N\in S, \] \(N\) positive semidefinite, \(\lambda>0\), i.e., (1) is uniformly elliptic. The paper presents new and simpler proofs of the following two theorems, which are basically known: If \(u\) is a continuous viscosity solution of (1) in the unit ball \(B_1\), then \(u\) belongs to \(C^{1,\alpha}\), and the corresponding \(C^{1,\alpha}\)-norm in \(B_{1/2}\) is bounded by \(C(\max|u|+ |F(0) |)\), where the maximum is taken over \(\overline B_1\). If \(F\) is, in addition, concave, then \(u\in C^{2,\alpha}\), and the same estimate for the \(C^{2,\alpha}\)-norm holds. Here, \(\alpha \in (0,1)\) and \(C\) are universal constants. The proof uses the Krylov-Safonov Harnack inequality and related estimates.