AI 中文总结
该研究针对耦合漂移Monge-Ampère方程，通过标量最大值原理论证得到全空间可解性分类，证明漂移单元问题解的存在唯一性，并推导渐近二次整体解的性质。
AI 中文摘要
我们研究耦合漂移Monge-Ampère方程\\( \det D^2u = \exp\{-a\cdot Du+b\cdot x+V(x)-c_0\}, \quad D^2u>0 \\)的整体解与周期校正子。当\\(V\equiv0\\)时，我们得到全空间可解性区域的精确分类：当\\(a=b=0\\)时，所有整体光滑严格凸解均为二次型；当\\(a\neq0\\)且\\(a\cdot b\le0\\)时，不存在此类解；当\\(a=0\\)且\\(b\neq0\\)或\\(a\cdot b>0\\)时，存在非二次型整体解。主要新要素是适用于所有维数\\(n\ge2\\)的标量最大值原理论证，证明了\\(a\neq0\\)、\\(a\cdot b=0\\)的零情况不存在整体光滑严格凸解。对于周期\\(V\\)，我们建立了漂移单元问题\\( \det(A+D^2\psi) = \exp\{-a\cdot D\psi+V-c_A\}, \quad A+D^2\psi>0 \\)在\\(\mathbb T^n\\)上的存在性与唯一性，还证明任何渐近二次整体解必满足\\(b=Aa\\)，若其余项有界，则该解为对应的二次-周期校正子（相差一个加法常数）。
英文摘要
In this paper, we study entire solutions and periodic correctors for the coupled-drift Monge-Ampère equation \[ \det D^2u = \exp\{-a\cdot Du+b\cdot x+V(x)-c_0\}, \quad D^2u>0. \] For $V\equiv0$, we obtain a sharp classification of the whole-space solvability regimes: all entire smooth strictly convex solutions are quadratic when $a=b=0$; no such solution exists when $a\neq0$ and $a\cdot b\le0$; and non-quadratic entire solutions exist when $a=0$ and $b\neq0$, or when $a\cdot b>0$. For the null case $a\neq0$, $a\cdot b=0$, we give a scalar maximum-principle argument in every dimension $n\ge2$. For periodic $V$, we prove existence and uniqueness of the normalized pair $(ψ_A,c_A)$ solving the drifted cell problem \[ \det(A+D^2ψ) = \exp\{-a\cdot Dψ+V-c_A\}, \quad A+D^2ψ>0 \quad\text{on }\mathbb T^n. \] We also prove that any asymptotically quadratic entire solution must satisfy $b=Aa$. If its remainder is bounded, then the solution is the corresponding quadratic-periodic corrector up to an additive constant.
Comments38 pages; simplified some arguments and corrected several typos
