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首页> 外文期刊>Journal of the Mathematical Society of Japan >Kiselman's principle, the Dirichlet problem for the Monge-Ampere equation, and rooftop obstacle problems
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Kiselman's principle, the Dirichlet problem for the Monge-Ampere equation, and rooftop obstacle problems

机译：Kiselman原理，Monge-Ampere方程的Dirichlet问题以及屋顶障碍物问题

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摘要

First, we obtain a new formula for Bremermann type upper envelopes, that arise frequently in convex analysis and pluripotential theory, in terms of the Legendre transform of the convex- or plurisubharmonic-envelope of the boundary data. This yields a new relation between solutions of the Dirichlet problem for the homogeneous real and complex Monge-Ampere equations and Kiselman's minimum principle. More generally, it establishes partial regularity for a Bremermann envelope whether or not it solves the Monge-Ampere equation. Second, we prove the second order regularity of the solution of the free-boundary problem for the Laplace equation with a rooftop obstacle, based on a new a priori estimate on the size of balls that lie above the non-contact set. As an application, we prove that convex- and plurisubharmonic-envelopes of rooftop obstacles have bounded second derivatives.
机译：首先，我们针对边界数据的凸或多亚次谐波包络的Legendre变换，获得了在凸分析和多能理论中经常出现的布雷默曼型上包络的新公式。这就在齐次实和复Monge-Ampere方程的Dirichlet问题的解与Kiselman的最小原理之间产生了新的关系。更一般而言，它为Bremermann包络建立了部分规则性，无论它是否解决Monge-Ampere方程。其次，基于对位于非接触集合上方的球的大小的新的先验估计，我们证明了具有屋顶障碍的Laplace方程的自由边界问题解的二阶正则性。作为一个应用，我们证明了屋顶障碍物的凸次谐波和多次谐波谐波信封已经界定了二阶导数。

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