Higher Robin eigenvalues for the p-Laplacian operator as p approaches 1

2024/07/09 by José C. Sabina de Lis, Sergio Segura de León

paper · doi:10.1007/s00526-024-02769-7

Abstract

Abstract This work addresses several aspects of the dependence on p of the higher eigenvalues λ _n λ n to the Robin problem, \beginaligned \ -Δ _p u = λ |u|^p-2u · x∈ Ω ,
|∇ u|^p-2\dfrac∂ u∂ ν + b |u|^p-2u= 0 · x∈ ∂ Ω .. \endaligned - Δ p u = λ | u | p - 2 u x ∈ Ω , | ∇ u | p - 2 ∂ u ∂ ν + b | u | p - 2 u = 0 x ∈ ∂ Ω . Here, Ω ⊂ \mathbb R^N Ω ⊂ R N is a C¹ C 1 bounded domain, ν ν is the outer unit normal, Δ _p u = \text div (|∇ u|^p-2∇ u) Δ p u = div ( | ∇ u | p - 2 ∇ u ) stands for the p -Laplacian operator and b∈ L^∞ (∂ Ω ) b ∈ L ∞ ( ∂ Ω ) . Main results concern: (a) the existence of the limits of λ _n λ n as p→ 1 p → 1 , (b) the ‘limit problems’ satisfied by the ‘limit eigenpairs’, (c) the continuous dependence of λ _n λ n on p when 1 1 p ∞ and (d) the limit profile of the eigenfunctions asp→ 1 p → 1 . The latter study is performed in the one dimensional and radially symmetric cases. Corresponding properties on the Dirichlet and Neumann eigenvalues are also studied in these two special scenarios.

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