论文标题
奇异的Abreu方程和线性化的Monge-ampère方程
Singular Abreu equations and linearized Monge-Ampère equations with drifts
论文作者
论文摘要
我们研究了在受凸限制的凸函数近似中产生的奇异abreu方程的可溶性。先前的工作确定了其第二个边界值问题在二维中或在较小的条件或径向对称条件下的较高维度。在这里，我们通过将奇异的abreu方程转换为带有漂移的线性化monge-ampère方程来解决较高的维情况。我们建立了线性化的Monge-Ampère方程的全局Hölder估计值，并在合适的假设下漂移，然后将其用于较高维度的单数ABREU方程的第二个边界值问题的规律性和解决性。许多具有一般右侧的案件也将被讨论。
We study the solvability of singular Abreu equations which arise in the approximation of convex functionals subject to a convexity constraint. Previous works established the solvability of their second boundary value problems either in two dimensions, or in higher dimensions under either a smallness condition or a radial symmetry condition. Here, we solve the higher dimensional case by transforming singular Abreu equations into linearized Monge-Ampère equations with drifts. We establish global Hölder estimates for the linearized Monge-Ampère equation with drifts under suitable hypotheses, and then use them to the regularity and solvability of the second boundary value problem for singular Abreu equations in higher dimensions. Many cases with general right-hand side will also be discussed.
