论文标题
最佳传输和高斯曲率方程
Optimal transport and the Gauss curvature equation
论文作者
论文摘要
在此简短说明中，我们考虑了在欧几里得空间中域上的函数图的高斯曲率和图像的规定问题。高斯图的图像的处方将其变成第二个边界值问题。我们的主要观察结果是，可以将此问题作为最佳运输问题，其中目标是$ \ Mathbb {s}^n $的下半球的子集。结果，我们在曲率上的轻度假设下获得了解决方案的存在和规律性，以及由于urbas而导致的梯度爆炸结果的定量版本，事实证明，该结果属于最佳运输框架。
In this short note, we consider the problem of prescribing the Gauss curvature and image of the Gauss map for the graph of a function over a domain in Euclidean space. The prescription of the image of the Gauss map turns this into a second boundary value problem. Our main observation is that this problem can be posed as an optimal transport problem where the target is a subset of the lower hemisphere of $\mathbb{S}^n$. As a result we obtain existence and regularity of solutions under mild assumptions on the curvature, as well as a quantitative version of a gradient blowup result due to Urbas, which turns out to fall within the optimal transport framework.
