Obstacle problems with double boundary condition for least gradient functions in metric measure spaces

TL;DR

This paper investigates obstacle problems with double boundary conditions for least gradient functions in metric measure spaces, establishing existence of solutions under certain regularity and boundary conditions, and extending classical Euclidean results.

Contribution

It extends the theory of obstacle problems with double boundary conditions to metric measure spaces, proving existence of solutions and identifying minimal solutions.

Findings

Existence of solutions for continuous obstacles and boundary data in uniform domains.

Existence of unique minimal solutions among possibly many solutions.

Generalization of Euclidean obstacle problem results to metric measure spaces.

Abstract

In the setting of a metric space equipped with a doubling measure supporting a $(1,1)$-Poincar\'e inequality, we study the problem of minimizing the BV-energy in a bounded domain $\Omega$ of functions bounded between two obstacle functions inside $\Omega$, and whose trace lies between two prescribed functions on the boundary of $\Omega.$ If the class of candidate functions is nonempty, we show that solutions exist for continuous obstacles and continuous boundary data when $\Omega$ is a uniform domain whose boundary is of positive mean curvature in the sense of Lahti, Mal\'y, 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.