DISCONTINUOUS EIKONAL EQUATIONS IN METRIC MEASURE SPACES
DISCONTINUOUS EIKONAL EQUATIONS IN METRIC MEASURE SPACES
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DOI：
10.1090/tran/9294
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发表时间：
2024-09-25
影响因子：
1.3
通讯作者：
Zhou，Xiaodan
中科院分区：
文献类型：
--
作者：
Liu，Qing;Shanmugalingam，Nageswari;Zhou，Xiaodan
本文研究了度量测度空间中的程函方程，其中非齐次项允许在有限域上不连续、无界且仅可积。对于连续程函方程，已知Monge解的概念等价于粘性解的标准定义。推广蒙奇解决方案的概念，在我们的设置，我们建立了相关的Dirichlet边值问题的唯一性和存在性结果。在我们的方法中的关键是采用一种新的度量，基于最优控制的解释，它集成了不连续项，并将程函方程转换为标准的连续形式。在空间和非齐次项的正则性假设下，我们还讨论了唯一解关于原度量的Hölder连续性。引用
In this paper, we study the eikonal equation in metric measure spaces, where the inhomogeneous term is allowed to be discontinuous, unbounded and merely-integrable in the domain with a finite. For continuous eikonal equations, it is known that the notion of Monge solutions is equivalent to the standard definition of viscosity solutions. Generalizing the notion of Monge solutions in our setting, we establish uniqueness and existence results for the associated Dirichlet boundary problem. The key in our approach is to adopt a new metric, based on the optimal control interpretation, which integrates the discontinuous term and converts the eikonal equation to a standard continuous form. We also discuss the Hölder continuity of the unique solution with respect to the original metric under regularity assumptions on the space and the inhomogeneous term. References