标题： 关于紧致Hessian流形上Monge-Ampère方程的最佳传输方法 显示英文标题

标题： An optimal transport approach to Monge-Ampère equations on compact Hessian manifolds

摘要： 本文我们研究紧致Hessian流形上的Monge-Ampère方程，或者等价地研究某些无界凸域$\Omega\subseteq \mathbb{R}^n$上的Monge-Ampère方程，并带有由仿射群作用给出的周期性约束。 在仿射群作用保持体积的情况下，即流形为特殊流形时，Cheng和Yau利用连续性方法建立了相应Monge-Ampère方程的可解性。 在一般情况下，我们构建了一个变分框架，涉及某些对偶流形以及经典Legendre变换的推广。我们给出了存在性和唯一性结果，并详细讨论了与最优传输及凸域的拟周期铺砖之间的联系。 显示英文摘要

摘要： In this paper we consider Monge-Amp\`ere equations on compact Hessian manifolds, or equivalently Monge-Amp\`ere equations on certain unbounded convex domains $\Omega\subseteq \mathbb{R}^n$, with a periodicity constraint given by the action of an affine group. In the case where the affine group action is volume-preserving, i.e., when the manifold is special, the solvability of the corresponding Monge-Amp\`ere equation was established using the continuity method by Cheng and Yau. In the general case we set up a variational framework involving certain dual manifolds and a generalization of the classical Legendre transform. We give existence and uniqueness results, elaborate on connections to optimal transport and quasi-periodic tilings of convex domains.