On the first eigenpair of singularly perturbed operators with oscillating coefficients. Issue 1 (2nd January 2016)

• Record Type: Journal Article
• Authors: Piatnitski, A.; Rybalko, V.
• Is Part Of: Communications in partial differential equations. Volume 41:Issue 1(2016)
• Page Start: 1
• Page End: 31
• Publication Date: 2016-01-02
• Subjects: Dirichlet spectral problem -- homogenization -- singularly perturbed operators; 35P99 -- 35J75 -- 35D40 -- 35F21 -- 35F30; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2015.1091838
• Languages: English
• ISSNs: 0360-5302

Abstract

ABSTRACT: The paper deals with a Dirichlet spectral problem for a singularly perturbed second order elliptic operator with rapidly oscillating locally periodic coefficients. We study the limit behavior of the first eigenpair (ground state) of this problem. The main tool in deriving the limit \mbox(effective) problem is the viscosity solutions technique for Hamilton-Jacobi equations. The effective problem need not have a unique solution. We study the non-uniqueness issue in a particular case of zero potential and construct the higher order term of the ground state asymptotics.