Flat solutions of the 1-Laplacian equation

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 $f\notin L^N(\Omega )$ 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.

Dedicated to Jean Mawhin. Revised and extended version of a note written by the authors in 2012