4.4 Savin’s small perturbation theorem [004F]

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4.4 Savin’s small perturbation theorem

Savin [65] 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 4.8.

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 $0<\epsilon\ll 1$ 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}}<\epsilon,

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

Savin’s theorem has fully nonlinear nature, because the perturbative machinery only applies once the solution has a priori $C^{2}$ bound. His proof has two main parts: first he shows a Harnack inequality by a nontrivial application of Aleksandrov-Bakelman-Pucci estimates, and then uses a compactness argument to prove $C^{2,\gamma}$ estimate, similar to De Giorgi’s almost flatness theorem for minimal surfaces [21].