Quantitative homogenization of Monge-Amp\`ere equations in periodic media

Abstract

Let $u^\varepsilon$ and $u$ be the convex solutions of \[ \det D^2u^\varepsilon=F(x,x/\varepsilon),\qquad \det D^2u=\overline F(x) \] on a bounded convex domain $\Omega$, with the same Dirichlet data. Here, $F$ is uniformly positive and periodic in its second variable, and $\overline F(x)=\int_{\mathbb T^n}F(x,y)\,dy$. For $m=0,1$ and $0<\alpha\leq1$, we prove \[ \|u^\varepsilon-u\|_{L^\infty(\Omega)}\leq C\varepsilon^{m+\alpha},\qquad \forall\,0<\varepsilon\leq1, \] provided that $u\in C^{m+2,\alpha}(\overline\Omega)$ and $F\in C_x^{m,\alpha}(\overline\Omega;C_y^{0,\gamma}(\mathbb T^n))$ for some $0<\gamma<1$. Both exponents are optimal in their respective regularity scales. We also establish the $O(\varepsilon^{\alpha})$ rate for locally weighted periodic Borel measures, with constants independent of the distribution of the microscopic measure.