# On some nonlinear partial diffrential equations involving the 1-Laplacian

1 AGM - Laboratoire d-Analyse, Géométrie et Modélisation

Abstract : Let $\Omega$ be a smooth bounded domain in $\R^N, N>1$ and let $n\in \N^*$. We are concerned here with the existence of nonnegative solutions $u n$ in $BV\Omega$, to the problem $$P n \begin{cases} -{ m div} \sigma +2n \left\int \Omega u -1 ight \ { m sign}^+ \ u=0 \ \quad \text{in} \ \Omega,\\ \sigma \cdot abla u= | abla u| \quad \text{in} \ \Omega, \\ u \ \text{ m is not identically zero} -\sigma \cdot \overrightarrow {n} u=u \quad \text{on}\ \partial\Omega, \end{cases}$$ where $\overrightarrow {n}$ denotes the unit outer normal to $\partial\Omega$, and ${ m sign}^+u$ denotes some $L^{\infty}\Omega$ function defined as: $${ m sign}^+ \ u. u =u^+, \ 0 \leq { m sign}^+u \leq 1.$$ Moreover, we prove the tight convergence of $u n$ towards one of the first eingenfunctions for the first $1-$Laplacian Operator $-\Delta 1$ on $\Omega$ when $n$ goes to $+\infty$.

Keywords : BV functions 1-Laplacian Operator

Author: Mouna Kraiem -

Source: https://hal.archives-ouvertes.fr/