Article

Applied Mathematics & Optimization

, Volume 65, Issue 1, pp 111-146

Principal Eigenvalue Minimization for an Elliptic Problem with Indefinite Weight and Robin Boundary Conditions

  • M. HintermüllerAffiliated withHumboldt-University of BerlinInstitute of Mathematics and Scientific Computing, University of Graz
  • , C.-Y. KaoAffiliated withDepartment of Mathematics and Computer Science, Claremont McKenna CollegeDepartment of Mathematics, The Ohio State University
  • , A. LaurainAffiliated withDepartment of Mathematics, Johann-von-Neumann-Haus Email author 

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Abstract

This paper focuses on the study of a linear eigenvalue problem with indefinite weight and Robin type boundary conditions. We investigate the minimization of the positive principal eigenvalue under the constraint that the absolute value of the weight is bounded and the total weight is a fixed negative constant. Biologically, this minimization problem is motivated by the question of determining the optimal spatial arrangement of favorable and unfavorable regions for a species to survive. For rectangular domains with Neumann boundary condition, it is known that there exists a threshold value such that if the total weight is below this threshold value then the optimal favorable region is like a section of a disk at one of the four corners; otherwise, the optimal favorable region is a strip attached to the shorter side of the rectangle. Here, we investigate the same problem with mixed Robin-Neumann type boundary conditions and study how this boundary condition affects the optimal spatial arrangement.

Keywords

Asymptotic analysis Principal eigenvalue Elliptic boundary value problem with indefinite weight Robin conditions Shape optimization