论文标题
部分可观测时空混沌系统的无模型预测
Obstacle problems with double boundary condition for least gradient functions in metric measure spaces
论文作者
论文摘要
在设置一个设置，配备了两倍尺寸的指标空间，以支持$（1,1）$ - 庞加莱不平等，我们研究在有限的域$ω$中最小化BV-能量的问题，该功能的功能的两个障碍函数之间的两个障碍函数之间的$ω$之间的两个障碍物在$ω$之间的范围内，并且在$ω$的范围内均可出现在$ω$上。当$ω$是一个统一的域时，连续的障碍物和连续边界数据在Lahti，Malý，Shanmugalingam和Speight（2019）的意义上，其边界是正平均曲率的。尽管这种解决方案通常不是唯一的，但我们显示了独特的最小解决方案的存在。我们的存在结果概括了Ziemer和Zumbrun（1999）的结果，他们在欧几里得环境中研究了这个问题，并具有单个障碍物和单个边界条件。
In the setting of a metric space equipped with a doubling measure supporting a $(1,1)$-Poincaré inequality, we study the problem of minimizing the BV-energy in a bounded domain $Ω$ of functions bounded between two obstacle functions inside $Ω$, and whose trace lies between two prescribed functions on the boundary of $Ω.$ If the class of candidate functions is nonempty, we show that solutions exist for continuous obstacles and continuous boundary data when $Ω$ is a uniform domain whose boundary is of positive mean curvature in the sense of Lahti, Malý, Shanmugalingam, and Speight (2019). While such solutions are not unique in general, we show the existence of unique minimal solutions. Our existence results generalize those of Ziemer and Zumbrun (1999), who studied this problem in the Euclidean setting with a single obstacle and single boundary condition.
