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

TL;DR

This paper extends the existence results for solutions to the Dirichlet problem for least gradient functions in metric spaces, showing solutions for broader boundary data classes and analyzing the structure of solvable boundary functions.

Contribution

It generalizes previous results by establishing existence of solutions for boundary data approximable by continuous functions and characterizes the solvability of L^1 boundary data in metric spaces.

Findings

Solutions exist for boundary data approximable from above and below by continuous functions.

For each L^1 boundary function, a least gradient solution exists at points of continuity.

The space of solvable L^1 boundary functions is non-linear, even in spaces with positive mean curvature.

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)-Poincar\'e inequality when the boundary of the domain satisfies a positive mean curvature condition. In this setting, it was shown by Mal\'y, 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 $f\in L^1(\partial\Omega),$ there is a least gradient function in $\Omega$ 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.