On the behaviour of the first eigenvalue of the p-Laplacian with Robin\n boundary conditions as p goes to 1

2021/10/28 by Francesco Della Pietra, Della Pietra, Francesco, Carlo Nitsch +5
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2110.15226

openalex publication_date 2021/10/28 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

In this paper we study the \Γ-limit, as p\→ 1, of the functional\n \n J_p(u)=
frac
displaystyle
int_
Omega |
nabla u|^p +
beta
int_{\n
partial
Omega} |u|^p
displaystyle
int_
Omega |u|^p,\n where \Ω is a smooth bounded open set in mathbb R^N, p>1 and\n\β is a real number. Among our results, for \β >-1, we derive an\nisoperimetric inequality for\n n
Lambda(
Omega,
beta)=
inf_{u
in BV(
Omega), u
not
equiv 0}\n
frac
displaystyle |Du|(
Omega) +
min(
beta,1)
int_{\n
partial
Omega} |u|
displaystyle
int_
Omega |u|\n which is the limit as p\→ 1⁺ of \λ(\Ω,p,\β)=\n \min_{u\∈ W^1,p(\Ω)} J_p(u). We show that among all\nbounded and smooth open sets with given volume, the ball maximizes\n\Λ(\Ω, \β) when \β \∈ (-1,0) and minimizes\n\Λ(\Ω, \β) when \β \∈[0, \∞).\n

Related

• Higher Robin eigenvalues for the p-Laplacian operator as p approaches 1
• Behaviour of solutions to p-Laplacian with Robin boundary conditions as p goes to 1
• A sharp estimate for the first Robin-Laplacian eigenvalue with negative boundary parameter
• Asymptotic behavior of the first Robin eigenvalue of nonlinear operators
• Behaviour of solutions to p -Laplacian with Robin boundary conditions as p goes to 1