Behaviour of solutions to p-Laplacian with Robin boundary conditions as p goes to 1

2022/06/07 by Francesco Della Pietra, Della Pietra, Francesco, Francescantonio Oliva +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2206.03337

openalex publication_date 2022/06/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the asymptotic behaviour, as p→ 1⁺, of the solutions of the following inhomogeneous Robin boundary value problem: \ -Δ_p u_p = f · amp; in Ω, |∇ u_p|^p-2∇ u_p⋅ ν+λ|u_p|^p-2u_p = g · amp; on ∂Ω,. where Ω is a bounded domain in \mathbb R^N with sufficiently smooth boundary, ν is its unit outward normal vector and Δ_p v is the p-Laplacian operator with p>1. The data f∈ L^N,∞(Ω) (which denotes the Marcinkiewicz space) and λ,g are bounded functions defined on ∂Ω with λ≥0. We find the threshold below which the family of p--solutions goes to 0 and above which this family blows up. As a second interest we deal with the 1-Laplacian problem formally arising by taking p→ 1⁺ in \eqrefpbabstract.

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