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, p. 905-921

Let Ω be a smooth bounded domain in N ,N>1 and let n * . We prove here the existence of nonnegative solutions u n in BV(Ω), to the problem

(Pn)-divσ+2nΩu-1sign+(u)=0inΩ,σ·u=|u|inΩ,uisnotidenticallyzero,-σ·nu=uonΩ,

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 +.

Soit Ω un domaine borné et régulier dans N ,N>1 et soit n * . On montre dans cet article l’existence de solutions positives u n dans BV(Ω), au problème

(Pn)-divσ+2nΩu-1sign+(u)=0dansΩ,σ·u=|u|dansΩ,un'estpasidentiquementnulle,-σ·nu=usurΩ,

n est le vecteur normal sortant de Ω, et sign + (u) est une fonction dans L (Ω) définie par :

sign +(u).u=u+,0 sign +(u)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 +.

@article{AFST_2007_6_16_4_905_0,
     author = {Kra\"\i em, Mouna},
     title = {On some nonlinear partial differential equations involving the 1-Laplacian},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 16},
     number = {4},
     year = {2007},
     pages = {905-921},
     doi = {10.5802/afst.1170},
     zbl = {pre05363343},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2007_6_16_4_905_0}
}
Kraïem, Mouna. 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. http://www.numdam.org/item/AFST_2007_6_16_4_905_0/

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