Flat solutions of the 1-Laplacian equation

Orsina, Luigi [I-ROME] Ponce, Augusto [UCL]

For every $f\inL^N(\Omega)$ defined in an open bounded subset $\Omega$ of $\mathbb{R}^N$, we prove that a solution $u\inW^{1,1}_0(\Omega)$ of the 1-Laplacian equation -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.