Singularly perturbed spectral problems with Neumann boundary conditions. Issue 2 (1st February 2016)

• Record Type: Journal Article
• Authors: Piatnitski, A.; Rybalko, A.; Rybalko, V.
• Is Part Of: Complex variables and elliptic equations. Volume 61:Issue 2(2016)
• Page Start: 252
• Page End: 274
• Publication Date: 2016-02-01
• Subjects: singularly perturbed operator -- Neumann spectral problem -- Hamilton-Jacobi equation; 35B27 -- 35B40; Functions of complex variables -- Periodicals; Differential equations, Elliptic -- Periodicals; 515.905
• DOI: 10.1080/17476933.2015.1076396
• Languages: English
• ISSNs: 1747-6933

Abstract

Abstract : The paper deals with the Neumann spectral problem for a singularly perturbed second-order elliptic operator with bounded lower order terms. The main goal is to provide a refined description of the limit behaviour of the principal eigenvalue and eigenfunction. Using the logarithmic transformation, we reduce the studied problem to an additive eigenvalue problem for a singularly perturbed Hamilton–Jacobi equation. Then assuming that the Aubry set of the Hamiltonian consists of a finite number of points or limit cycles situated in the domain or on its boundary, we find the limit of the eigenvalue and formulate the selection criterion that allows us to choose a solution of the limit Hamilton–Jacobi equation which gives the logarithmic asymptotics of the principal eigenfunction.