On some nonlinear partial differential equations involving the 1-Laplacian

Mouna Kraïem

doi:10.5802/afst.1170

Mouna Kraïem ¹

¹ Université de Cergy Pontoise , Département de Mathématiques, 2, avenue Adolphe Chauvin, 95302 Cergy Pontoise Cedex, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 4, pp. 905-921.

Abstract
Abstract

Let $Ω$ be a smooth bounded domain in $ℝ^{N}, N > 1$ and let $n \in ℕ^{*}$ . We prove here the existence of nonnegative solutions $u_{n}$ in $B V (Ω)$ , to the problem

(P_{n}) \{\begin{matrix} - div σ + 2 n (\int_{Ω} u - 1) {sign}^{+} (u) = 0 & in Ω, \\ σ \cdot \nabla u = | \nabla u | & in Ω, \\ u is not identically zero, - σ \cdot \vec{n} u = u & on \partial Ω, \end{matrix}

where $\vec{n}$ denotes the unit outer normal to $\partial Ω$ , and ${sign}^{+} (u)$ denotes some $L^{\infty} (Ω)$ function defined as:

{sign}^{+} (u) . u = u^{+}, 0 \leq {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 $- Δ_{1}$ on $Ω$ when $n$ goes to $+ \infty$ .

Soit $Ω$ un domaine borné et régulier dans $ℝ^{N}, N > 1$ et soit $n \in ℕ^{*}$ . On montre dans cet article l’existence de solutions positives $u_{n}$ dans $B V (Ω)$ , au problème

(P_{n}) \{\begin{matrix} - div σ + 2 n (\int_{Ω} u - 1) {sign}^{+} (u) = 0 & dans Ω, \\ σ \cdot \nabla u = | \nabla u | & dans Ω, \\ u n'est pas identiquement nulle, - σ \cdot \vec{n} u = u & sur \partial Ω, \end{matrix}

où $\vec{n}$ est le vecteur normal sortant de $\partial Ω$ , et ${sign}^{+} (u)$ est une fonction dans $L^{\infty} (Ω)$ définie par :

{sign}^{+} (u) . u = u^{+}, 0 \leq {sign}^{+} (u) \leq 1 .

De plus, on montre la convergence de $u_{n}$ vers une des premières fonctions propres de l’opérateur $1 -$ Laplacian $- Δ_{1}$ sur $Ω$ quand $n$ tend vers $+ \infty$ .

DOI: 10.5802/afst.1170

Author's affiliations:

Mouna Kraïem ¹

¹ Université de Cergy Pontoise , Département de Mathématiques, 2, avenue Adolphe Chauvin, 95302 Cergy Pontoise Cedex, France

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     author = {Mouna Kra{\"\i}em},
     title = {On some nonlinear partial differential equations involving the {1-Laplacian}},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {905--921},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
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Mouna Kraïem. On some nonlinear partial differential equations involving the 1-Laplacian. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 16 (2007) no. 4, pp. 905-921. doi : 10.5802/afst.1170. https://afst.centre-mersenne.org/articles/10.5802/afst.1170/

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