Quantitative homogenization of Monge-Ampère equations in periodic media

Abstract

Let $u^{\epsilon }$ 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 $0<\alpha \le 1$, we prove \[ \|u^\varepsilon-u\|_{L^\infty(Ω)}\leq C\varepsilon^{m+α},\qquad \forall\,0<\varepsilon\leq1, \] 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 $O({\epsilon }^{\alpha })$ rate for locally weighted periodic Borel measures, with constants independent of the distribution of the microscopic measure.

arXiv comment22 pages