Flat solutions of the 1-Laplacian equation
(2017) Universita degli Studi di Trieste. Istituto di Matematica. Rendiconti — Vol. 49, p. 41-51 (2017)
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- Abstract
- 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.
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Orsina, L., & Ponce, A. (2017). Flat solutions of the 1-Laplacian equation. Universita degli Studi di Trieste. Istituto di Matematica. Rendiconti, 49, 41-51. https://hdl.handle.net/2078.5/126313 (Original work published 2017)