On the regularity of solutions to fully nonlinear elliptic equations via the Liouville property (Q2782621)

The paper under review is concerned with the fully nonlinear elliptic equation (1) \( F (D^2 u)=0,\) where \(F\) is assumed to satisfy the following conditions: \(F \in C^1\), \(F(0)=0,\) \(\lambda \|N\|\leq F(M+N)-F(M) \leq \Lambda \|N\|\) \(\forall M,N \in {\mathcal S },N\geq 0\) (\({\mathcal S}\) = the space of real \(n\times n\) symmetric matrices, \(\Lambda \geq \lambda > 0\)). A continuous function \(u\) in a domain \(\Omega \subset \mathbb R^n\) is called a viscosity solution to (1) if for \(x_0 \in \Omega\) and \(\varphi \in C^2\), \(u-\varphi \) attains a local maximum (resp. minimum) at \(x_0\) , then \(F(D^2 \varphi (x_0)) \geq 0\) (resp. \(\leq 0 \)). Equation (1) is said to satisfy the Liouville property if \(u \in C^{1,1}_{\text{loc}} (\mathbb R^n)\) is an entire viscosity of (1) with bounded \(D^2 u \) in \(\mathbb R^n\), then \(u\) must be a polynomial of degree \(\leq 2\). Under the above assumptions upon \(F\), the author proves the following result: Let \(u \in C^{1,1} (B_1 (0))\) be a viscosity solution of (1) in \(B_1(0)\). Let (1) satisfy the Liouville property. Then \(u \in C^{2,\alpha}(B_{1/2} (0))\) and \(\|D^2 u \|_{C^{\alpha}(B_{1/2}(0))} \leq C \) for any \(0< \alpha< 1\), where \(C\) depends on \(n, \lambda , \Lambda , \alpha , \|u \|_{C^{1,1}( B_1(0))}\), \(F\) and the moduls of continuity of \(DF\).