We study the asymptotic behaviour, as p→1+, of the solutions of the following inhomogeneous Robin boundary value problem: −Δpup=f in Ω, |∇up|p−2∇up⋅ν+λ|up|p−2up=g on ∂Ω, where Ω is a bounded domain in RN with sufficiently smooth boundary, ν is its unit outward normal vector and Δpv is the p-Laplacian operator with p>1. The data f∈LN,∞(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 (P).

Behaviour of solutions to p-Laplacian with Robin boundary conditions as p goes to 1 / Della Pietra, Francesco; Oliva, Francescantonio; Segura de León, Sergio. - In: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH. SECTION A. MATHEMATICS. - ISSN 0308-2105. - 154:1(2024), pp. 105-130. [10.1017/prm.2022.92]

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

Oliva, Francescantonio;
2024

Abstract

We study the asymptotic behaviour, as p→1+, of the solutions of the following inhomogeneous Robin boundary value problem: −Δpup=f in Ω, |∇up|p−2∇up⋅ν+λ|up|p−2up=g on ∂Ω, where Ω is a bounded domain in RN with sufficiently smooth boundary, ν is its unit outward normal vector and Δpv is the p-Laplacian operator with p>1. The data f∈LN,∞(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 (P).
2024
Robin boundary conditions; asymptotic behaviour; 1-Laplacian
01 Pubblicazione su rivista::01a Articolo in rivista
Behaviour of solutions to p-Laplacian with Robin boundary conditions as p goes to 1 / Della Pietra, Francesco; Oliva, Francescantonio; Segura de León, Sergio. - In: PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH. SECTION A. MATHEMATICS. - ISSN 0308-2105. - 154:1(2024), pp. 105-130. [10.1017/prm.2022.92]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1674538
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