论文标题
分数凸度
Fractional convexity
论文作者
论文摘要
我们介绍了一个分数凸度的概念，该概念自然地扩展了欧几里得空间中凸的通常的概念，以至于分数设置。有了这个分数凸度的概念，我们研究了外部基准域内的分数凸封（在外面基准下方的域内最大的分数凸函数），并表明分数凸构的元素在非局部方程中的iftions $ nibention uniondions $ nifions unsions unions unions unions $ nifions uniondions $ nibension unconions $ 1-拉普拉斯。对于这个方程式，我们证明存在，独特性和比较原理（在粘度解决方案的框架中）。此外，我们发现凸封的方程溶液与分数Monge-Ampere方程的溶液有关。
We introduce a notion of fractional convexity that extends naturally the usual notion of convexity in the Euclidean space to a fractional setting. With this notion of fractional convexity, we study the fractional convex envelope inside a domain of an exterior datum (the largest possible fractional convex function inside the domain that is below the datum outside) and show that the fractional convex envelope is characterized as a viscosity solution to a non-local equation that is given by the infimum among all possible directions of the $1-$dimensional fractional Laplacian. For this equation we prove existence, uniqueness and a comparison principle (in the framework of viscosity solutions). In addition, we find that solutions to the equation for the convex envelope are related to solutions to the fractional Monge-Ampere equation.
