On some nonlinear partial differential equations involving the 1-Laplacian

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 (2007)

  • Volume: 16, Issue: 4, page 905-921
  • ISSN: 0240-2963

Abstract

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Let Ω be a smooth bounded domain in ℝ N , N > 1 and let n ∈ ℕ * . We prove here the existence of nonnegative solutions u n in B V ( Ω ) , to the problem ( P n ) - div σ + 2 n ∫ Ω u - 1 sign + ( u ) = 0 in Ω , σ · ∇ u = | ∇ u | in Ω , u is not identically zero , - σ · n → u = u on ∂ Ω , where n → denotes the unit outer normal to ∂ Ω , and sign + ( u ) denotes some L ∞ ( Ω ) function defined as: sign + ( u ) . u = u + , 0 ≤ sign + ( u ) ≤ 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 + ∞ .

How to cite

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Kraïem, Mouna. "On some nonlinear partial differential equations involving the 1-Laplacian." Annales de la faculté des sciences de Toulouse Mathématiques 16.4 (2007): 905-921. <http://eudml.org/doc/10073>.

@article{Kraïem2007,
abstract = {Let $\Omega $ be a smooth bounded domain in $\mathbb\{R\}^N, N&gt;1$ and let $n\in \mathbb\{N\}^*$. We prove here the existence of nonnegative solutions $u_n$ in $BV(\Omega )$, to the problem\begin\{equation*\} (P\_n) \{\left\lbrace \begin\{array\}\{ll\} -\operatorname\{div\} \sigma +2n \left(\int \_ \Omega u -1 \right) \ \operatorname\{sign\}^+ \ (u)=0 & \text\{in \} \Omega ,\\[2mm] \sigma \cdot \nabla u= |\nabla u| & \text\{in \} \Omega ,\\[2mm] u \text\{ is not identically zero\}, -\sigma \cdot \overrightarrow\{n\} u=u & \text\{on \} \partial \Omega , \end\{array\}\right.\} \end\{equation*\}where $\overrightarrow\{n\}$ denotes the unit outer normal to $\partial \Omega $, and $\{\rm sign\}^+(u)$ denotes some $L^\{\infty \}(\Omega )$ function defined as:\[\{\rm sign\}^+ \ (u). u =u^+, \ 0 \le \{\rm sign\}^+(u) \le 1.\]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 $.},
affiliation = {Université de Cergy Pontoise , Département de Mathématiques, 2, avenue Adolphe Chauvin, 95302 Cergy Pontoise Cedex, France},
author = {Kraïem, Mouna},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {1-Laplacian; first eigenvalue; first eigenfunction; penalization method},
language = {eng},
number = {4},
pages = {905-921},
publisher = {Université Paul Sabatier, Toulouse},
title = {On some nonlinear partial differential equations involving the 1-Laplacian},
url = {http://eudml.org/doc/10073},
volume = {16},
year = {2007},
}

TY - JOUR
AU - Kraïem, Mouna
TI - On some nonlinear partial differential equations involving the 1-Laplacian
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2007
PB - Université Paul Sabatier, Toulouse
VL - 16
IS - 4
SP - 905
EP - 921
AB - Let $\Omega $ be a smooth bounded domain in $\mathbb{R}^N, N&gt;1$ and let $n\in \mathbb{N}^*$. We prove here the existence of nonnegative solutions $u_n$ in $BV(\Omega )$, to the problem\begin{equation*} (P_n) {\left\lbrace \begin{array}{ll} -\operatorname{div} \sigma +2n \left(\int _ \Omega u -1 \right) \ \operatorname{sign}^+ \ (u)=0 & \text{in } \Omega ,\\[2mm] \sigma \cdot \nabla u= |\nabla u| & \text{in } \Omega ,\\[2mm] u \text{ is not identically zero}, -\sigma \cdot \overrightarrow{n} u=u & \text{on } \partial \Omega , \end{array}\right.} \end{equation*}where $\overrightarrow{n}$ denotes the unit outer normal to $\partial \Omega $, and ${\rm sign}^+(u)$ denotes some $L^{\infty }(\Omega )$ function defined as:\[{\rm sign}^+ \ (u). u =u^+, \ 0 \le {\rm sign}^+(u) \le 1.\]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 $.
LA - eng
KW - 1-Laplacian; first eigenvalue; first eigenfunction; penalization method
UR - http://eudml.org/doc/10073
ER -

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