2.4 Savin’s small perturbation theorem [00B2]

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2.4 Savin’s small perturbation theorem

Savin [40] proved that for a large class of second order elliptic equations satisfying certain structural conditions, any viscosity solution $C^{0}$ -close to a given smooth solution has interior $C^{2,\gamma}$ -bound. In particular this applies to complex MA equation. Combined with the Schauder estimate,

Theorem 2.8.

Fix $k\geq 2$ and $0<\gamma<1$ . On the unit ball, let $v$ be a given smooth solution to the complex Monge-Ampère equation $(dd^{c}v)^{n}=1$ . Then there are constants $0<\kappa\ll 1$ and $C$ depending on $n,k,\gamma,\left\lVert v\right\rVert_{C^{k,\gamma}}$ , such that if

(dd^{c}(u+v))^{n}=1+f,\quad\left\lVert f\right\rVert_{C^{k-2,\gamma}}<\kappa,

and $\left\lVert u\right\rVert_{C^{0}}<\kappa$ , then $\left\lVert u\right\rVert_{C^{k,\gamma}(B_{1/2})}\leq C\kappa$ .