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

• Published in: Calculus of variations and partial differential equations Vol. 63; no. 7
• Main Authors: Sabina de Lis, José C., Segura de León, Sergio
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.09.2024; Springer Nature B.V
• Subjects: Analysis; Calculus of Variations and Optimal Control; Optimization; Control; Dirichlet problem; Eigenvalues; Eigenvectors; Laplace transforms; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Systems Theory; Theoretical; 35P30; 35J92; 35B40
• ISSN: 0944-2669; 1432-0835
• DOI: 10.1007/s00526-024-02769-7

This work addresses several aspects of the dependence on p of the higher eigenvalues λ n to the Robin problem, - 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 ⊂ R N is a C 1 bounded domain, ν is the outer unit normal, Δ p u = div ( | ∇ u | p - 2 ∇ u ) stands for the p -Laplacian operator and b ∈ L ∞ ( ∂ Ω ) . Main results concern: (a) the existence of the limits of λ n as p → 1 , (b) the ‘limit problems’ satisfied by the ‘limit eigenpairs’, (c) the continuous dependence of λ n on p when 1 < p < ∞ and (d) the limit profile of the eigenfunctions as 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.