Non-locality, non-linearity, and existence of solutions to the Dirichlet problem for least gradient functions in metric measure spaces

Abstract

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 (2019) that solutions exist for continuous boundary data. We extend these results, showing existence of solutions for boundary data that are approximable from above and below by continuous functions. We also show that for each $f\in L^1(\partial \Omega )$, there is a least gradient function in $\Omega$ whose trace agrees with $f$ at all points of continuity of $f$, and so we obtain existence of solutions for boundary data which are continuous almost everywhere. This is in contrast to a result of Spradlin and Tamasan (2014), 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.