Flat solutions of the 1-Laplacian equation

TL;DR

This paper proves that solutions to the 1-Laplacian equation exhibit flat regions where the gradient vanishes, extending to cases with small data in weak spaces and BV minimizers, using Stampacchia's truncation method.

Contribution

It establishes the existence of flat solutions for the 1-Laplacian equation under broad conditions, including weak data and BV minimizers, which was previously not well understood.

Findings

Solutions have regions with zero gradient.

Flat regions occur even with small data in weak spaces.

Results apply to BV minimizers of the energy functional.

Abstract

For every $f \in L^N(\Omega)$ defined in an open bounded subset $\Omega$ of $\mathbb{R}^N$, we prove that a solution $u \in W_0^{1, 1}(\Omega)$ of the $1$-Laplacian equation ${-}\mathrm{div}{(\frac{\nabla u}{|\nabla u|})} = f$ in $\Omega$ satisfies $\nabla u = 0$ on a set of positive Lebesgue measure. The same property holds if $f \not\in L^N(\Omega)$ has small norm in the Marcinkiewicz space of weak-$L^{N}$ functions or if $u$ is a BV minimizer of the associated energy functional. The proofs rely on Stampacchia's truncation method.