论文标题
非局部性，非线性和对公制量梯度函数的dirichlet问题的解决方案的存在
Non-locality, non-linearity, and existence of solutions to the Dirichlet problem for least gradient functions in metric measure spaces
论文作者
论文摘要
我们研究了配备加倍度量和支持A（1,1） - p-Poincaré不平等的公制域中最小梯度函数的Dirichlet问题。在这种情况下，由Malý，Lahti，Shanmugalingam和Speight显示了解决连续边界数据的解决方案。我们扩展了这些结果，显示了边界数据的解决方案的存在，这些解决方案可以通过连续函数在上方和下方近似。我们还表明，对于l^1（\partialΩ）中的每一个$ f \，$在$ω$中具有最小梯度功能，其跟踪在$ f $的连续性点上与$ f $一致，因此我们获得了边界数据解决方案的存在，几乎到处都是连续的。这与Spradlin和Tamasan的结果相反，Spradlin和Tamasan在单位圆圈上构建了一个$ l^1 $功能，该单位圆圈中的单位磁盘中没有最低梯度解决方案在$ \ Mathbb {r}^2中。曲率条件。
We study the Dirichlet problem for least gradient functions for domains in metric spaces equipped with a doubling measure and supporting a (1,1)-Poincaré inequality when the boundary of the domain satisfies a positive mean curvature condition. In this setting, it was shown by Malý, Lahti, Shanmugalingam, and Speight that solutions exist for continuous boundary data. We extend these results, showing existence of solutions for boundary data that is approximable from above and below by continuous functions. We also show that for each $f\in L^1(\partialΩ),$ there is a least gradient function in $Ω$ whose trace agrees with $f$ at points of continuity of $f$, and so we obtain existence of solutions for boundary data which is continuous almost everywhere. This is in contrast to a result of Spradlin and Tamasan, who constructed an $L^1$-function on the unit circle which has no least gradient solution in the unit disk in $\mathbb{R}^2.$ Modifying the example of Spradlin and Tamasan, we show that the space of solvable $L^1$-functions on the unit circle is non-linear, even though the unit disk satisfies the positive mean curvature condition.
