Anisotropic p-Laplacian-Type Problems with Mixed Boundary Conditions and their Limit as p Anisotropic p-Laplacian-type Problems

• Published in: The Journal of geometric analysis Vol. 35; no. 8
• Main Authors: Apaza, Juan Pablo A., da Silva, João Vitor
• Format: Journal Article
• Language: English
• Published: New York Springer US 16.06.2025
• Subjects: Abstract Harmonic Analysis; Convex and Discrete Geometry; Differential Geometry; Dynamical Systems and Ergodic Theory; Fourier Analysis; Global Analysis and Analysis on Manifolds; Mathematics; Mathematics and Statistics; Anisotropic Laplacian; 35J94; 35P15; 35J92; Eigenvalue; Infinity Laplacian; Robin problems; Laplacian
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-025-02066-5

This work investigates a limit problem for an anisotropic p -Laplacian operator as p → ∞ within the framework of viscosity solutions. Specifically, we analyze the asymptotic behavior of an eigenvalue problem subject to Robin and mixed boundary conditions: - div H p ( ∇ u ) = Λ p | u | p - 2 u in Ω , H p ( ∇ u ) · ν + β p | u | p - 2 u = Λ p | u | p - 2 u on ∂ Ω . We demonstrate that the limit of the eigenfunction is a viscosity solution to an eigenvalue problem governed by an anisotropic ∞ -Laplacian, and we establish several geometric properties of the corresponding eigenvalues. Subsequently, utilizing the eigenvalues derived in the first part, we address the problem with a forcing term f ∈ L ∞ and a boundary condition g ∈ L ∞ . Furthermore, we investigate the asymptotic behavior of the associated solutions as p → ∞ : - div H p ( ∇ u ) = f in Ω , H p ( ∇ u ) · ν + β p | v | p - 2 v = g on ∂ Ω .