# General Least Gradient Problems with Obstacle

## TL;DR

This paper investigates the existence, structure, and regularity of solutions to obstacle problems involving least gradient functionals, extending classical results by incorporating obstacles and establishing new regularity and uniqueness properties.

## Contribution

It introduces a comprehensive analysis of obstacle least gradient problems, including existence, structure, and regularity, with novel insights into the behavior of minimizers and associated vector fields.

## Key findings

- Minimizers also solve the unconstrained least gradient problem.
- Existence of a divergence condition vector field T that characterizes minimizers.
- Regularity results based on maximum principles for minimal surfaces.

## Abstract

We study existence, structure, uniqueness and regularity of solutions of the obstacle problem \begin{equation*} \inf_{u\in BV_f(\Omega)}\int_{\mathbb{R}^n}\phi(x,Du), \end{equation*} where $BV_f(\Omega)=\{u\in BV(\Omega): u\geq \psi \text{ in }\Omega\text{ and } u|_{\partial \Omega}=f|_{\partial \Omega}\}$, $f \in W^{1,1}_0(\mathbb{R}^n)$, $\psi$ is the obstacle, and $\phi(x,\xi)$ is a convex, continuous and homogeneous function of degree one with respect to the $\xi$ variable. We show that every minimizer of this problem is also a minimizer of the least gradient problem \[\inf_{u\in \mathcal{A}_f(\Omega)}\int_{\mathbb{R}^n}\phi(x,Du),\] where $\mathcal{A}_f(\Omega)=\{u\in BV(\Omega): u\geq \psi, \text{ and } u=f \text{ in }\Omega^c\}$. Moreover, there exists a vector field $T$ with $\nabla \cdot T \leq 0$ in $\Omega$ which determines the structure of all minimizers of these two problems, and $T$ is divergence free on $\{x\in \Omega: u(x)>\psi(x)\}$ for any minimizer $u$. We also present uniqueness and regularity results that are based on maximum principles for minimal surfaces. Since minimizers of the least gradient problems with obstacle do not hit small enough obstacles, the results presented in this paper extend several results in the literature about least gradient problems without obstacle.