度量测度空间中最小梯度函数狄利克雷问题解的非局部性、非线性性及存在性
Non-locality, non-linearity, and existence of solutions to the Dirichlet problem for least gradient functions in metric measure spaces
摘要：
研究了度量空间中具有加倍测度且支持（1，1）-Poincaré不等式的区域的最小梯度函数的狄利克雷问题，当区域的边界满足正的中曲率条件时.在这种情况下，Malcolm、Lahti、Shanmugalingam和Speight证明了连续边界数据存在解。我们扩展这些结果，显示存在的解决方案的边界数据，是近似从上面和下面的连续函数。我们还证明了对于每个$f\in L^1(\partialΩ),$，在$Ω$中存在一个最小梯度函数，其迹在$f$的连续点处与$f$一致，从而得到了几乎处处连续的边界数据解的存在性.这与Spradlin和Tamasan的结果相反，他们在单位圆上构造了一个$L^1$ -函数，该函数在$\mathbb{R}^2.$中的单位圆上没有最小梯度解。修改Spradlin和Tamasan的例子，我们证明了单位圆上的可解$L^1$ -函数空间是非线性的，即使单位圆满足正的中曲率条件。
英文摘要：
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.
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