On some nonlinear partial diffrential equations involving the 1-Laplacian

2005/11/26 by Mouna Kraïem, Kraiem, Mouna
Computer Science · Mathematics · #35XX #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.48550/arxiv.math/0511640

openalex publication_date 2005/11/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Ω be a smooth bounded domain in \R^N, N>1 and let n∈ \N^*. We are concerned here with the existence of nonnegative solutions u_n in BV(Ω), to the problem (P_n) \begincases -\rm div σ+2n (∫_ Ωu -1) \rm sign⁺ (u)=0 in Ω, σ⋅ ∇ u= |∇ u| in Ω, u \rm is not identically zero, -σ⋅ \overrightarrow n u=u on ∂Ω, \endcases where \overrightarrow n denotes the unit outer normal to ∂Ω, and \rm sign⁺(u) denotes some L^∞(Ω) function defined as: \rm sign⁺ (u). u =u⁺, 0 ≤ \rm 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 +∞.

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