# $L^p$ regularity of least gradient functions

## TL;DR

This paper establishes optimal $L^p$ regularity for solutions to the anisotropic least gradient problem, demonstrating their boundedness and precise boundary behavior in various settings.

## Contribution

It proves the optimal $L^{rac{Np}{N-1}}$ regularity for solutions and provides explicit boundary blow-up bounds in specific cases.

## Key findings

- Solutions are in $L^{rac{Np}{N-1}}()$ for boundary data in $L^p()$.
- Solutions are locally bounded with explicit boundary blow-up rates.
- Optimality of the regularity exponent is demonstrated.

## Abstract

It is shown that solutions to the anisotropic least gradient problem for boundary data $f \in L^p(\partial\Omega)$ lie in $L^{\frac{Np}{N-1}}(\Omega)$; the exponent is shown to be optimal. Moreover, the solutions are shown to be locally bounded with explicit bounds on the rate of blow-up of the solution near the boundary in two settings: in the anisotropic case on the plane and in the isotropic case in any dimension.