# American Institute of Mathematical Sciences

2008, 5(2): 315-335. doi: 10.3934/mbe.2008.5.315

## Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains

 1 Department of Mathematics, The Ohio State University, Columbus, OH 43210, United States 2 Department of Mathematics, The Ohio State State University, Columbus, Ohio 43210 3 Mathematical Institute, Tohoku University, Sendai 980-8578, Japan

Received  November 2007 Revised  February 2008 Published  March 2008

This paper is concerned with an indefinite weight linear eigenvalue problem in cylindrical domains. We investigate the minimization of the positive principal eigenvalue under the constraint that the weight is bounded by a positive and a negative constant 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. Both our analysis and numerical simulations for rectangular domains indicate that there exists a threshold value such that if the total weight is below this threshold value, then the optimal favorable region is a circular-type domain at one of the four corners, and a strip at the one end with shorter edge otherwise.
Citation: Chiu-Yen Kao, Yuan Lou, Eiji Yanagida. Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains. Mathematical Biosciences & Engineering, 2008, 5 (2) : 315-335. doi: 10.3934/mbe.2008.5.315
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