2026/08/05 by Ling Wang
Mathematics · #math.AP
41 pages
arxiv created 2026/08/05 · arxiv updated 2026/08/06
We study entire solutions and periodic correctors for the coupled-drift Monge-Ampère equation det D2u = exp\-a⋅ Du+b⋅ x+V(x)-c0\, D2u>0. For \(V≡0\), 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 \(a≠0\) and \(a⋅ b≤0\); and non-quadratic entire solutions exist when \(a=0\) and \(b≠0\), or when \(a⋅ b>0\). The main new ingredient is a scalar maximum-principle argument valid in every dimension \(n≥2\), which proves that the null case \(a≠0\), \(a⋅ b=0\) admits no entire smooth strictly convex solution. For periodic \(V\), we establish the existence and uniqueness for the drifted cell problem det(A+D2ψ) = exp\-a⋅ Dψ+V-cA\, A+D2ψ>0 \quadon \mathbb Tn. 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.